ALBERT

Beschreibung: We study the asymptotic consistency properties of $\alpha$-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the $\alpha$-Rényi divergence from the true posterior. Unique to our work is that we consider settings with $\alpha 〉 1$, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a `good' sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also extend our analysis to the setting where $\alpha$ equals one, corresponding to the minimizer of the reverse KL divergence, and to models with local latent variables. We also illustrate the existence of a good sequence with a number of examples. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods.

Beschreibung: Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2nd-order optimization method, with the Fisher information matrix acting as a substitute for the Hessian. In many important cases, the Fisher information matrix is shown to be equivalent to the Generalized Gauss-Newton matrix, which both approximates the Hessian, but also has certain properties that favor its use over the Hessian. This perspective turns out to have significant implications for the design of a practical and robust natural gradient optimizer, as it motivates the use of techniques like trust regions and Tikhonov regularization. Additionally, we make a series of contributions to the understanding of natural gradient and 2nd-order methods, including: a thorough analysis of the convergence speed of stochastic natural gradient descent (and more general stochastic 2nd-order methods) as applied to convex quadratics, a critical examination of the oft-used 'empirical' approximation of the Fisher matrix, and an analysis of the (approximate) parameterization invariance property possessed by natural gradient methods (which we show also holds for certain other curvature matrices, but notably not the Hessian).

Beschreibung: There has recently been much work on the "wide limit" of neural networks, where Bayesian neural networks (BNNs) are shown to converge to a Gaussian process (GP) as all hidden layers are sent to infinite width. However, these results do not apply to architectures that require one or more of the hidden layers to remain narrow. In this paper, we consider the wide limit of BNNs where some hidden layers, called "bottlenecks", are held at finite width. The result is a composition of GPs that we term a "bottleneck neural network Gaussian process" (bottleneck NNGP). Although intuitive, the subtlety of the proof is in showing that the wide limit of a composition of networks is in fact the composition of the limiting GPs. We also analyze theoretically a single-bottleneck NNGP, finding that the bottleneck induces dependence between the outputs of a multi-output network that persists through extreme post-bottleneck depths, and prevents the kernel of the network from losing discriminative power at extreme post-bottleneck depths.

Beschreibung: The computation of the distance to the true distribution is a key component of most state-of-the-art generative models. Inspired by prior works on the Sliced-Wasserstein Auto-Encoders (SWAE) and the Wasserstein Auto-Encoders with MMD-based penalty (WAE-MMD), we propose a new generative model - a Cramer-Wold Auto-Encoder (CWAE). A fundamental component of CWAE is the characteristic kernel, the construction of which is one of the goals of this paper, from here on referred to as the Cramer-Wold kernel. Its main distinguishing feature is that it has a closed-form of the kernel product of radial Gaussians. Consequently, CWAE model has a~closed-form for the distance between the posterior and the normal prior, which simplifies the optimization procedure by removing the need to sample in order to compute the loss function. At the same time, CWAE performance often improves upon WAE-MMD and SWAE on standard benchmarks.

Beschreibung: We consider a novel application of inverse reinforcement learning with behavioral economics constraints to model, learn and predict the commenting behavior of YouTube viewers. Each group of users is modeled as a rationally inattentive Bayesian agent which solves a contextual bandit problem. Our methodology integrates three key components. First, to identify distinct commenting patterns, we use deep embedded clustering to estimate framing information (essential extrinsic features) that clusters users into distinct groups. Second, we present an inverse reinforcement learning algorithm that uses Bayesian revealed preferences to test for rationality: does there exist a utility function that rationalizes the given data, and if yes, can it be used to predict commenting behavior? Finally, we impose behavioral economics constraints stemming from rational inattention to characterize the attention span of groups of users. The test imposes a Renyi mutual information cost constraint which impacts how the agent can select attention strategies to maximize their expected utility. After a careful analysis of a massive YouTube dataset, our surprising result is that in most YouTube user groups, the commenting behavior is consistent with optimizing a Bayesian utility with rationally inattentive constraints. The paper also highlights how the rational inattention model can accurately predict commenting behavior. The massive YouTube dataset and analysis used in this paper are available on GitHub and completely reproducible.

Beschreibung: Robust covariance matrix estimation is a fundamental task in statistics. The recent discovery on the connection between robust estimation and generative adversarial nets (GANs) suggests that it is possible to compute depth-like robust estimators using similar techniques that optimize GANs. In this paper, we introduce a general learning via classification framework based on the notion of proper scoring rules. This framework allows us to understand both matrix depth function, a technique of rate-optimal robust estimation, and various GANs through the lens of variational approximations of $f$-divergences induced by proper scoring rules. We then propose a new class of robust covariance matrix estimators in this framework by carefully constructing discriminators with appropriate neural network structures. These estimators are proved to achieve the minimax rate of covariance matrix estimation under Huber's contamination model. The results are also extended to robust scatter estimation for elliptical distributions. Our numerical results demonstrate the good performance of the proposed procedures under various settings against competitors in the literature.

Beschreibung: Vector autoregression (VAR) is a fundamental tool for modeling multivariate time series. However, as the number of component series is increased, the VAR model becomes overparameterized. Several authors have addressed this issue by incorporating regularized approaches, such as the lasso in VAR estimation. Traditional approaches address overparameterization by selecting a low lag order, based on the assumption of short range dependence, assuming that a universal lag order applies to all components. Such an approach constrains the relationship between the components and impedes forecast performance. The lasso-based approaches perform much better in high-dimensional situations but do not incorporate the notion of lag order selection. We propose a new class of hierarchical lag structures (HLag) that embed the notion of lag selection into a convex regularizer. The key modeling tool is a group lasso with nested groups which guarantees that the sparsity pattern of lag coefficients honors the VAR's ordered structure. The proposed HLag framework offers three basic structures, which allow for varying levels of flexibility, with many possible generalizations. A simulation study demonstrates improved performance in forecasting and lag order selection over previous approaches, and macroeconomic, financial, and energy applications further highlight forecasting improvements as well as HLag's convenient, interpretable output.

Beschreibung: The method of covariate adjustment is often used for estimation of total treatment effects from observational studies. Restricting attention to causal linear models, a recent article (Henckel et al., 2019) derived two novel graphical criteria: one to compare the asymptotic variance of linear regression treatment effect estimators that control for certain distinct adjustment sets and another to identify the optimal adjustment set that yields the least squares estimator with the smallest asymptotic variance. In this paper we show that the same graphical criteria can be used in non-parametric causal graphical models when treatment effects are estimated using non-parametrically adjusted estimators of the interventional means. We also provide a new graphical criterion for determining the optimal adjustment set among the minimal adjustment sets and another novel graphical criterion for comparing time dependent adjustment sets. We show that uniformly optimal time dependent adjustment sets do not always exist. For point interventions, we provide a sound and complete graphical criterion for determining when a non-parametric optimally adjusted estimator of an interventional mean, or of a contrast of interventional means, is semiparametric efficient under the non-parametric causal graphical model. In addition, when the criterion is not met, we provide a sound algorithm that checks for possible simplifications of the efficient influence function of the parameter. Finally, we find an interesting connection between identification and efficient covariate adjustment estimation. Specifically, we show that if there exists an identifying formula for an interventional mean that depends only on treatment, outcome and mediators, then the non-parametric optimally adjusted estimator can never be globally efficient under the causal graphical model.

Beschreibung: A conventional wisdom in statistical learning is that large models require strong regularization to prevent overfitting. Here we show that this rule can be violated by linear regression in the underdetermined $n\ll p$ situation under realistic conditions. Using simulations and real-life high-dimensional datasets, we demonstrate that an explicit positive ridge penalty can fail to provide any improvement over the minimum-norm least squares estimator. Moreover, the optimal value of ridge penalty in this situation can be negative. This happens when the high-variance directions in the predictor space can predict the response variable, which is often the case in the real-world high-dimensional data. In this regime, low-variance directions provide an implicit ridge regularization and can make any further positive ridge penalty detrimental. We prove that augmenting any linear model with random covariates and using minimum-norm estimator is asymptotically equivalent to adding the ridge penalty. We use a spiked covariance model as an analytically tractable example and prove that the optimal ridge penalty in this case is negative when $n\ll p$.

Beschreibung: We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., one with perfect accuracy on training set and near-zero generalization error ($\approx 0.01\%$). The goal is to shed light on two mysteries in deep neural networks: (i) a nonsmooth activation function like ReLU outperforms a smooth one like hyperbolic tangent; (ii) successful neural network architectures rely on having many layers, even though a shallow network can approximate any function arbitrarily well. We performed extensive experiments on the persistent homology of a wide range of point cloud data sets, both real and simulated. The results consistently demonstrate the following: (1) Neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simple one as it passes through the layers. No matter how complicated the topology of $M$ we begin with, when passed through a well-trained neural network $f : \mathbb{R}^d \to \mathbb{R}^p$, there is a vast reduction in the Betti numbers of both components $M_a$ and $M_b$; in fact they nearly always reduce to their lowest possible values: $\beta_k\bigl(f(M_i)\bigr) = 0$ for $k \ge 1$ and $\beta_0\bigl(f(M_i)\bigr) = 1$, $i =a, b$. (2) The reduction in Betti numbers is significantly faster for ReLU activation than for hyperbolic tangent activation as the former defines nonhomeomorphic maps that change topology, whereas the latter defines homeomorphic maps that preserve topology. (3) Shallow and deep networks transform data sets differently --- a shallow network operates mainly through changing geometry and changes topology only in its final layers, a deep one spreads topological changes more evenly across all layers.

Beschreibung: There is growing interest in large-scale machine learning and optimization over decentralized networks, e.g. in the context of multi-agent learning and federated learning. Due to the imminent need to alleviate the communication burden, the investigation of communication-efficient distributed optimization algorithms --- particularly for empirical risk minimization --- has flourished in recent years. A large fraction of these algorithms have been developed for the master/slave setting, relying on the presence of a central parameter server that can communicate with all agents. This paper focuses on distributed optimization over networks, or decentralized optimization, where each agent is only allowed to aggregate information from its neighbors over a network (namely, no centralized coordination is present). By properly adjusting the global gradient estimate via local averaging in conjunction with proper correction, we develop a communication-efficient approximate Newton-type method, called Network-DANE, which generalizes DANE to accommodate decentralized scenarios. Our key ideas can be applied, in a systematic manner, to obtain decentralized versions of other master/slave distributed algorithms. A notable development is Network-SVRG/SARAH, which employs variance reduction at each agent to further accelerate local computation. We establish linear convergence of Network-DANE and Network-SVRG for strongly convex losses, and Network-SARAH for quadratic losses, which shed light on the impacts of data homogeneity, network connectivity, and local averaging upon the rate of convergence. We further extend Network-DANE to composite optimization by allowing a nonsmooth penalty term. Numerical evidence is provided to demonstrate the appealing performance of our algorithms over competitive baselines, in terms of both communication and computation efficiency. Our work suggests that by performing a judiciously chosen amount of local communication and computation per iteration, the overall efficiency can be substantially improved.

Beschreibung: This paper presents a unified framework for supervised learning and inference procedures using the divide-and-conquer approach for high-dimensional correlated outcomes. We propose a general class of estimators that can be implemented in a fully distributed and parallelized computational scheme. Modeling, computational and theoretical challenges related to high-dimensional correlated outcomes are overcome by dividing data at both outcome and subject levels, estimating the parameter of interest from blocks of data using a broad class of supervised learning procedures, and combining block estimators in a closed-form meta-estimator asymptotically equivalent to estimates obtained by Hansen (1982)'s generalized method of moments (GMM) that does not require the entire data to be reloaded on a common server. We provide rigorous theoretical justifications for the use of distributed estimators with correlated outcomes by studying the asymptotic behaviour of the combined estimator with fixed and diverging number of data divisions. Simulations illustrate the finite sample performance of the proposed method, and we provide an R package for ease of implementation.

Beschreibung: An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the R package GPGame.

Beschreibung: Iterative Hard Thresholding (IHT) is a popular class of first-order greedy selection methods for loss minimization under cardinality constraint. The existing IHT-style algorithms, however, are proposed for minimizing the primal formulation. It is still an open issue to explore duality theory and algorithms for such a non-convex and NP-hard combinatorial optimization problem. To address this issue, we develop in this article a novel duality theory for $\ell_2$-regularized empirical risk minimization under cardinality constraint, along with an IHT-style algorithm for dual optimization. Our sparse duality theory establishes a set of sufficient and/or necessary conditions under which the original non-convex problem can be equivalently or approximately solved in a concave dual formulation. In view of this theory, we propose the Dual IHT (DIHT) algorithm as a super-gradient ascent method to solve the non-smooth dual problem with provable guarantees on primal-dual gap convergence and sparsity recovery. Numerical results confirm our theoretical predictions and demonstrate the superiority of DIHT to the state-of-the-art primal IHT-style algorithms in model estimation accuracy and computational efficiency.

Beschreibung: We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We first investigate distributed stochastic gradient methods (SGM), with mini-batches and multi-passes over the data. We show that optimal generalization error bounds (up to logarithmic factor) can be retained for distributed SGM provided that the partition level is not too large. We then extend our results to spectral algorithms (SA), including kernel ridge regression (KRR), kernel principal component regression, and gradient methods. Our results show that distributed SGM has a smaller theoretical computational complexity, compared with distributed KRR and classic SGM. Moreover, even for a general non-distributed SA, they provide optimal, capacity-dependent convergence rates, for the case that the regression function may not be in the RKHS in the well-conditioned regimes.

Beschreibung: In supervised learning, we typically leverage a fully labeled dataset to design methods for function estimation or prediction. In many practical situations, we are able to obtain alternative feedback, possibly at a low cost. A broad goal is to understand the usefulness of, and to design algorithms to exploit, this alternative feedback. In this paper, we consider a semi-supervised regression setting, where we obtain additional ordinal (or comparison) information for the unlabeled samples. We consider ordinal feedback of varying qualities where we have either a perfect ordering of the samples, a noisy ordering of the samples or noisy pairwise comparisons between the samples. We provide a precise quantification of the usefulness of these types of ordinal feedback in both nonparametric and linear regression, showing that in many cases it is possible to accurately estimate an underlying function with a very small labeled set, effectively escaping the curse of dimensionality. We also present lower bounds, that establish fundamental limits for the task and show that our algorithms are optimal in a variety of settings. Finally, we present extensive experiments on new datasets that demonstrate the efficacy and practicality of our algorithms and investigate their robustness to various sources of noise and model misspecification.

Beschreibung: We study the least-squares regression problem over a Hilbert space, covering nonparametric regression over a reproducing kernel Hilbert space as a special case. We first investigate regularized algorithms adapted to a projection operator on a closed subspace of the Hilbert space. We prove convergence results with respect to variants of norms, under a capacity assumption on the hypothesis space and a regularity condition on the target function. As a result, we obtain optimal rates for regularized algorithms with randomized sketches, provided that the sketch dimension is proportional to the effective dimension up to a logarithmic factor. As a byproduct, we obtain similar results for Nystr\"{o}m regularized algorithms. Our results provide optimal, distribution-dependent rates that do not have any saturation effect for sketched/Nystr\"{o}m regularized algorithms, considering both the attainable and non-attainable cases, in the well-conditioned regimes. We then study stochastic gradient methods with projection over the subspace, allowing multi-pass over the data and minibatches, and we derive similar optimal statistical convergence results.

Beschreibung: Recurrent Neural Networks have been widely used to process sequence data, but have long been criticized for their biological implausibility and training difficulties related to vanishing and exploding gradients. This paper presents a novel algorithm for training recurrent networks, target propagation through time (TPTT), that outperforms standard backpropagation through time (BPTT) on four out of the five problems used for testing. The proposed algorithm is initially tested and compared to BPTT on four synthetic time lag tasks, and its performance is also measured using the sequential MNIST data set. In addition, as TPTT uses target propagation, it allows for discrete nonlinearities and could potentially mitigate the credit assignment problem in more complex recurrent architectures.

Beschreibung: We present GluonCV and GluonNLP, the deep learning toolkits for computer vision and natural language processing based on Apache MXNet (incubating). These toolkits provide state-of-the-art pre-trained models, training scripts, and training logs, to facilitate rapid prototyping and promote reproducible research. We also provide modular APIs with flexible building blocks to enable efficient customization. Leveraging the MXNet ecosystem, the deep learning models in GluonCV and GluonNLP can be deployed onto a variety of platforms with different programming languages. The Apache 2.0 license has been adopted by GluonCV and GluonNLP to allow for software distribution, modification, and usage.

Beschreibung: We study derivative-free methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linear-quadratic systems, and study various settings of driving noise and reward feedback. Our main theoretical result provides an explicit bound on the sample or evaluation complexity: we show that these methods are guaranteed to converge to within any pre-specified tolerance of the optimal policy with a number of zero-order evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of one-point and two-point reward feedback. Our theory is corroborated by simulations of derivative-free methods in application to these systems. Along the way, we derive convergence rates for stochastic zero-order optimization algorithms when applied to a certain class of non-convex problems.

Beschreibung: This paper is concerned with a multivariate extension of Gaussian message passing applied to pairwise Markov graphs (MGs). Gaussian message passing applied to pairwise MGs is often labeled Gaussian belief propagation (GaBP) and can be used to approximate the marginal of each variable contained in the pairwise MG. We propose a multivariate extensionof GaBP (we label this GaBP-m) that can be used to estimate higher-dimensional marginals. Beyond the ability to estimatehigher-dimensional marginals, GaBP-m exhibits better convergence behavior than GaBP, and can also provide more accurate univariatemarginals. The theoretical results of this paper are based on an extension of the computation tree analysis conductedon univariate nodes to the multivariate case.The main contribution of this paper is the development of a convergence condition for GaBP-m that moves beyond the walk-summability of the precision matrix. Based on this convergence condition, we derived an upper bound for the number of iterations required for convergence of the GaBP-m algorithm.An upper bound on the dissimilarity between the approximate and exact marginal covariance matrices was established.We argue that GaBP-m is robust towards a certain change in variables, a property not shared by iterative solvers of linear systems, such as the conjugate gradient (CG) and preconditioned conjugate gradient (PCG) methods. The advantages of using GaBP-mover GaBP are also illustrated empirically.

Beschreibung: A common divide-and-conquer approach for Bayesian computation with big data is to partition the data, perform local inference for each piece separately, and combine the results to obtain a global posterior approximation. While being conceptually and computationally appealing, this method involves the problematic need to also split the prior for the local inferences; these weakened priors may not provide enough regularization for each separate computation, thus eliminating one of the key advantages of Bayesian methods. To resolve this dilemma while still retaining the generalizability of the underlying local inference method, we apply the idea of expectation propagation (EP) as a framework for distributed Bayesian inference. The central idea is to iteratively update approximations to the local likelihoods given the state of the other approximations and the prior. The present paper has two roles: we review the steps that are needed to keep EP algorithms numerically stable, and we suggest a general approach, inspired by EP, for approaching data partitioning problems in a way that achieves the computational benefits of parallelism while allowing each local update to make use of relevant information from the other sites. In addition, we demonstrate how the method can be applied in a hierarchical context to make use of partitioning of both data and parameters. The paper describes a general algorithmic framework, rather than a specific algorithm, and presents an example implementation for it.

Beschreibung: Principal component analysis (PCA) is a well-established tool in machine learning and data processing. The principal axes in PCA were shown to be equivalent to the maximum marginal likelihood estimator of the factor loading matrix in a latent factor model for the observed data, assuming that the latent factors are independently distributed as standard normal distributions. However, the independence assumption may be unrealistic for many scenarios such as modeling multiple time series, spatial processes, and functional data, where the outcomes are correlated. In this paper, we introduce the generalized probabilistic principal component analysis (GPPCA) to study the latent factor model for multiple correlated outcomes, where each factor is modeled by a Gaussian process. Our method generalizes the previous probabilistic formulation of PCA (PPCA) by providing the closed-form maximum marginal likelihood estimator of the factor loadings and other parameters. Based on the explicit expression of the precision matrix in the marginal likelihood that we derived, the number of the computational operations is linear to the number of output variables. Furthermore, we also provide the closed-form expression of the marginal likelihood when other covariates are included in the mean structure. We highlight the advantage of GPPCA in terms of the practical relevance, estimation accuracy and computational convenience. Numerical studies of simulated and real data confirm the excellent finite-sample performance of the proposed approach.

Beschreibung: We study the connections between spectral clustering and the problems of maximum margin clustering, and estimation of the components of level sets of a density function. Specifically, we obtain bounds on the eigenvectors of graph Laplacian matrices in terms of the between cluster separation, and within cluster connectivity. These bounds ensure that the spectral clustering solution converges to the maximum margin clustering solution as the scaling parameter is reduced towards zero. The sensitivity of maximum margin clustering solutions to outlying points is well known, but can be mitigated by first removing such outliers, and applying maximum margin clustering to the remaining points. If outliers are identified using an estimate of the underlying probability density, then the remaining points may be seen as an estimate of a level set of this density function. We show that such an approach can be used to consistently estimate the components of the level sets of a density function under very mild assumptions.

Beschreibung: Over the past decades, numerous loss functions have been been proposed for a variety of supervised learning tasks, including regression, classification, ranking, and more generally structured prediction. Understanding the core principles and theoretical properties underpinning these losses is key to choose the right loss for the right problem, as well as to create new losses which combine their strengths. In this paper, we introduce Fenchel-Young losses, a generic way to construct a convex loss function for a regularized prediction function. We provide an in-depth study of their properties in a very broad setting, covering all the aforementioned supervised learning tasks, and revealing new connections between sparsity, generalized entropies, and separation margins. We show that Fenchel-Young losses unify many well-known loss functions and allow to create useful new ones easily. Finally, we derive efficient predictive and training algorithms, making Fenchel-Young losses appealing both in theory and practice.

Beschreibung: This paper develops deterministic upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model. In particular, our bounds exploit r-nonbacktracking walks and Fortuin-Kasteleyn-Ginibre (FKG) type inequalities, and are computed by message passing algorithms. Further, we provide parameterized versions of the bounds that control the trade-off between efficiency and accuracy. Finally, the tightness of the bounds is illustrated on various network models.

Beschreibung: The accuracy and complexity of kernel learning algorithms is determined by the set of kernels over which it is able to optimize. An ideal set of kernels should: admit a linear parameterization (tractability); be dense in the set of all kernels (accuracy); and every member should be universal so that the hypothesis space is infinite-dimensional (scalability). Currently, there is no class of kernel that meets all three criteria - e.g. Gaussians are not tractable or accurate; polynomials are not scalable. We propose a new class that meet all three criteria - the Tessellated Kernel (TK) class. Specifically, the TK class: admits a linear parameterization using positive matrices; is dense in all kernels; and every element in the class is universal. This implies that the use of TK kernels for learning the kernel can obviate the need for selecting candidate kernels in algorithms such as SimpleMKL and parameters such as the bandwidth. Numerical testing on soft margin Support Vector Machine (SVM) problems show that algorithms using TK kernels outperform other kernel learning algorithms and neural networks. Furthermore, our results show that when the ratio of the number of training data to features is high, the improvement of TK over MKL increases significantly.

Beschreibung: We present a probabilistic framework for studying adversarial attacks on discrete data. Based on this framework, we derive a perturbation-based method, Greedy Attack, and a scalable learning-based method, Gumbel Attack, that illustrate various tradeoffs in the design of attacks. We demonstrate the effectiveness of these methods using both quantitative metrics and human evaluation on various state-of-the-art models for text classification, including a word-based CNN, a character-based CNN and an LSTM. As an example of our results, we show that the accuracy of character-based convolutional networks drops to the level of random selection by modifying only five characters through Greedy Attack.

Beschreibung: This paper presents a new open source Python framework for causal discovery from observational data and domain background knowledge, aimed at causal graph and causal mechanism modeling. The cdt package implements an end-to-end approach, recovering the direct dependencies (the skeleton of the causal graph) and the causal relationships between variables. It includes algorithms from the `Bnlearn' and `Pcalg' packages, together with algorithms for pairwise causal discovery such as ANM.

Beschreibung: We develop ancestral Gumbel-Top-$k$ sampling: a generic and efficient method for sampling without replacement from discrete-valued Bayesian networks, which includes multivariate discrete distributions, Markov chains and sequence models. The method uses an extension of the Gumbel-Max trick to sample without replacement by finding the top $k$ of perturbed log-probabilities among all possible configurations of a Bayesian network. Despite the exponentially large domain, the algorithm has a complexity linear in the number of variables and sample size $k$. Our algorithm allows to set the number of parallel processors $m$, to trade off the number of iterations versus the total cost (iterations times $m$) of running the algorithm. For $m = 1$ the algorithm has minimum total cost, whereas for $m = k$ the number of iterations is minimized, and the resulting algorithm is known as Stochastic Beam Search. We provide extensions of the algorithm and discuss a number of related algorithms. We analyze the properties of Gumbel-Top-$k$ sampling and compare against alternatives on randomly generated Bayesian networks with different levels of connectivity. In the context of (deep) sequence models, we show its use as a method to generate diverse but high-quality translations and statistical estimates of translation quality and entropy.

Beschreibung: We consider the problem of jointly estimating multiple inverse covariance matrices from high-dimensional data consisting of distinct classes. An $\ell_2$-penalized maximum likelihood approach is employed. The suggested approach is flexible and generic, incorporating several other $\ell_2$-penalized estimators as special cases. In addition, the approach allows specification of target matrices through which prior knowledge may be incorporated and which can stabilize the estimation procedure in high-dimensional settings. The result is a targeted fused ridge estimator that is of use when the precision matrices of the constituent classes are believed to chiefly share the same structure while potentially differing in a number of locations of interest. It has many applications in (multi)factorial study designs. We focus on the graphical interpretation of precision matrices with the proposed estimator then serving as a basis for integrative or meta-analytic Gaussian graphical modeling. Situations are considered in which the classes are defined by data sets and subtypes of diseases. The performance of the proposed estimator in the graphical modeling setting is assessed through extensive simulation experiments. Its practical usability is illustrated by the differential network modeling of 12 large-scale gene expression data sets of diffuse large B-cell lymphoma subtypes. The estimator and its related procedures are incorporated into the R-package rags2ridges.

Beschreibung: We propose graph-dependent implicit regularisation strategies for synchronised distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, single-machine serial statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding constraints to the empirical risk minimisation rule. Particularly for distributed methods, the use of implicit regularisation allows the algorithm to remain simple, without projections or dual methods. To prove our results, we establish graph-independent generalisation bounds for Distributed SGD that match the single-machine serial SGD setting (using algorithmic stability), and we establish graph-dependent optimisation bounds that are of independent interest. We present numerical experiments to show that the qualitative nature of the upper bounds we derive can be representative of real behaviours.

Beschreibung: In many applications, observed data are influenced by some combination of latent causes. For example, suppose sensors are placed inside a building to record responses such as temperature, humidity, power consumption and noise levels. These random, observed responses are typically affected by many unobserved, latent factors (or features) within the building such as the number of individuals, the turning on and off of electrical devices, power surges, etc. These latent factors are usually present for a contiguous period of time before disappearing; further, multiple factors could be present at a time. This paper develops new probabilistic methodology and inference methods for random object generation influenced by latent features exhibiting temporal persistence. Every datum is associated with subsets of a potentially infinite number of hidden, persistent features that account for temporal dynamics in an observation. The ensuing class of dynamic models constructed by adapting the Indian Buffet Process — a probability measure on the space of random, unbounded binary matrices — finds use in a variety of applications arising in operations, signal processing, biomedicine, marketing, image analysis, etc. Illustrations using synthetic and real data are provided.

Beschreibung: We study bipartite community detection in networks, or more generally the network biclustering problem. We present a fast two-stage procedure based on spectral initialization followed by the application of a pseudo-likelihood classifier twice. Under mild regularity conditions, we establish the weak consistency of the procedure (i.e., the convergence of the misclassification rate to zero) under a general bipartite stochastic block model. We show that the procedure is optimal in the sense that it achieves the optimal convergence rate that is achievable by a biclustering oracle, adaptively over the whole class, up to constants. This is further formalized by deriving a minimax lower bound over a class of biclustering problems. The optimal rate we obtain sharpens some of the existing results and generalizes others to a wide regime of average degree growth, from sparse networks with average degrees growing arbitrarily slowly to fairly dense networks with average degrees of order $\sqrt{n}$. As a special case, we recover the known exact recovery threshold in the $\log n$ regime of sparsity. To obtain the consistency result, as part of the provable version of the algorithm, we introduce a sub-block partitioning scheme that is also computationally attractive, allowing for distributed implementation of the algorithm without sacrificing optimality. The provable algorithm is derived from a general class of pseudo-likelihood biclustering algorithms that employ simple EM type updates. We show the effectiveness of this general class by numerical simulations.

Beschreibung: Parametrised state space models in the form of recurrent networks are often used in machine learning to learn from data streams exhibiting temporal dependencies. To break the black box nature of such models it is important to understand the dynamical features of the input-driving time series that are formed in the state space. We propose a framework for rigorous analysis of such state representations in vanishing memory state space models such as echo state networks (ESN). In particular, we consider the state space a temporal feature space and the readout mapping from the state space a kernel machine operating in that feature space. We show that: (1) The usual ESN strategy of randomly generating input-to-state, as well as state coupling leads to shallow memory time series representations, corresponding to cross-correlation operator with fast exponentially decaying coefficients; (2) Imposing symmetry on dynamic coupling yields a constrained dynamic kernel matching the input time series with straightforward exponentially decaying motifs or exponentially decaying motifs of the highest frequency; (3) Simple ring (cycle) high-dimensional reservoir topology specified only through two free parameters can implement deep memory dynamic kernels with a rich variety of matching motifs. We quantify richness of feature representations imposed by dynamic kernels and demonstrate that for dynamic kernel associated with cycle reservoir topology, the kernel richness undergoes a phase transition close to the edge of stability.

Beschreibung: We consider the problem of learning causal models from observational data generated by linear non-Gaussian acyclic causal models with latent variables. Without considering the effect of latent variables, the inferred causal relationships among the observed variables are often wrong. Under faithfulness assumption, we propose a method to check whether there exists a causal path between any two observed variables. From this information, we can obtain the causal order among the observed variables. The next question is whether the causal effects can be uniquely identified as well. We show that causal effects among observed variables cannot be identified uniquely under mere assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we are able to propose an efficient method that identifies the set of all possible causal effects that are compatible with the observational data. We present additional structural conditions on the causal graph under which causal effects among observed variables can be determined uniquely. Furthermore, we provide necessary and sufficient graphical conditions for unique identification of the number of variables in the system. Experiments on synthetic data and real-world data show the effectiveness of our proposed algorithm for learning causal models.

Beschreibung: Given a response $Y$ and a vector $X = (X^1, \dots, X^d)$ of $d$ predictors, we investigate the problem of inferring direct causes of $Y$ among the vector $X$. Models for $Y$ that use all of its causal covariates as predictors enjoy the property of being invariant across different environments or interventional settings. Given data from such environments, this property has been exploited for causal discovery. Here, we extend this inference principle to situations in which some (discrete-valued) direct causes of $ Y $ are unobserved. Such cases naturally give rise to switching regression models. We provide sufficient conditions for the existence, consistency and asymptotic normality of the MLE in linear switching regression models with Gaussian noise, and construct a test for the equality of such models. These results allow us to prove that the proposed causal discovery method obtains asymptotic false discovery control under mild conditions. We provide an algorithm, make available code, and test our method on simulated data. It is robust against model violations and outperforms state-of-the-art approaches. We further apply our method to a real data set, where we show that it does not only output causal predictors, but also a process-based clustering of data points, which could be of additional interest to practitioners.

Beschreibung: Sufficient dimension reduction (SDR) is a very useful concept for exploratory analysis and data visualization in regression, especially when the number of covariates is large. Many SDR methods have been proposed for regression with a continuous response, where the central subspace (CS) is the target of estimation. Various conditions, such as the linearity condition and the constant covariance condition, are imposed so that these methods can estimate at least a portion of the CS. In this paper we study SDR for regression and discriminant analysis with categorical response. Motivated by the exploratory analysis and data visualization aspects of SDR, we propose a new geometric framework to reformulate the SDR problem in terms of manifold optimization and introduce a new concept called Maximum Separation Subspace (MASES). The MASES naturally preserves the “sufficiency” in SDR without imposing additional conditions on the predictor distribution, and directly inspires a semi-parametric estimator. Numerical studies show MASES exhibits superior performance as compared with competing SDR methods in specific settings.

Beschreibung: This paper considers a Bayesian approach to graph-based semi-supervised learning. We show that if the graph parameters are suitably scaled, the graph-posteriors converge to a continuum limit as the size of the unlabeled data set grows. This consistency result has profound algorithmic implications: we prove that when consistency holds, carefully designed Markov chain Monte Carlo algorithms have a uniform spectral gap, independent of the number of unlabeled inputs. Numerical experiments illustrate and complement the theory.

Beschreibung: Consider an unknown smooth function $f: [0,1]^d \rightarrow \mathbb{R}$, and assume we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + \eta_i) \bmod 1$, for $x_i \in [0,1]^d$, where $\eta_i$ denotes the noise. Given the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust estimates of the clean samples $f(x_i) \bmod 1$. We formulate a natural approach for solving this problem, which works with angular embeddings of the noisy mod 1 samples over the unit circle, inspired by the angular synchronization framework. This amounts to solving a smoothness regularized least-squares problem -- a quadratically constrained quadratic program (QCQP) -- where the variables are constrained to lie on the unit circle. Our proposed approach is based on solving its relaxation, which is a trust-region sub-problem and hence solvable efficiently. We provide theoretical guarantees demonstrating its robustness to noise for adversarial, as well as random Gaussian and Bernoulli noise models. To the best of our knowledge, these are the first such theoretical results for this problem. We demonstrate the robustness and efficiency of our proposed approach via extensive numerical simulations on synthetic data, along with a simple least-squares based solution for the unwrapping stage, that recovers the original samples of $f$ (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo $1$ samples. Finally, we also consider two other approaches for denoising the modulo 1 samples that leverage tools from Riemannian optimization on manifolds, including a Burer-Monteiro approach for a semidefinite programming relaxation of our formulation. For the two-dimensional version of the problem, which has applications in synthetic aperture radar interferometry (InSAR), we are able to solve instances of real-world data with a million sample points in under 10 seconds, on a personal laptop.

Beschreibung: Great attention has been paid to Big Data in recent years. Such data hold promise for scientific discoveries but also pose challenges to analyses. One potential challenge is noise accumulation. In this paper, we explore noise accumulation in high dimensional two-group classification. First, we revisit a previous assessment of noise accumulation with principal component analyses, which yields a different threshold for discriminative ability than originally identified. Then we extend our scope to its impact on classifiers developed with three common machine learning approaches---random forest, support vector machine, and boosted classification trees. We simulate four scenarios with differing amounts of signal strength to evaluate each method. After determining noise accumulation may affect the performance of these classifiers, we assess factors that impact it. We conduct simulations by varying sample size, signal strength, signal strength proportional to the number predictors, and signal magnitude with random forest classifiers. These simulations suggest that noise accumulation affects the discriminative ability of high-dimensional classifiers developed using common machine learning methods, which can be modified by sample size, signal strength, and signal magnitude. We developed the measure total signal index (TSI) to track the trends of total signal and noise accumulation.

Beschreibung: pyts is an open-source Python package for time series classification. This versatile toolbox provides implementations of many algorithms published in the literature, preprocessing functionalities, and data set loading utilities. pyts relies on the standard scientific Python packages numpy, scipy, scikit-learn, joblib, and numba, and is distributed under the BSD-3-Clause license. Documentation contains installation instructions, a detailed user guide, a full API description, and concrete self-contained examples.

Beschreibung: High dimensional data often contain multiple facets, and several clustering patterns can co-exist under different variable subspaces, also known as the views. While multi-view clustering algorithms were proposed, the uncertainty quantification remains difficult --- a particular challenge is in the high complexity of estimating the cluster assignment probability under each view, and sharing information among views. In this article, we propose an approximate Bayes approach --- treating the similarity matrices generated over the views as rough first-stage estimates for the co-assignment probabilities; in its Kullback-Leibler neighborhood, we obtain a refined low-rank matrix, formed by the pairwise product of simplex coordinates. Interestingly, each simplex coordinate directly encodes the cluster assignment uncertainty. For multi-view clustering, we let each view draw a parameterization from a few candidates, leading to dimension reduction. With high model flexibility, the estimation can be efficiently carried out as a continuous optimization problem, hence enjoys gradient-based computation. The theory establishes the connection of this model to a random partition distribution under multiple views. Compared to single-view clustering approaches, substantially more interpretable results are obtained when clustering brains from a human traumatic brain injury study, using high-dimensional gene expression data.

Beschreibung: This paper is a broad and accessible survey of the methods we have at our disposal for Monte Carlo gradient estimation in machine learning and across the statistical sciences: the problem of computing the gradient of an expectation of a function with respect to parameters defining the distribution that is integrated; the problem of sensitivity analysis. In machine learning research, this gradient problem lies at the core of many learning problems, in supervised, unsupervised and reinforcement learning. We will generally seek to rewrite such gradients in a form that allows for Monte Carlo estimation, allowing them to be easily and efficiently used and analysed. We explore three strategies---the pathwise, score function, and measure-valued gradient estimators---exploring their historical development, derivation, and underlying assumptions. We describe their use in other fields, show how they are related and can be combined, and expand on their possible generalisations. Wherever Monte Carlo gradient estimators have been derived and deployed in the past, important advances have followed. A deeper and more widely-held understanding of this problem will lead to further advances, and it is these advances that we wish to support.

Beschreibung: Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, $N$, due to the cubic (in $N$) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on $M \ll N$ inducing variables to form an approximation at a cost of $\mathcal{O}\left(NM^2\right)$. While the computational cost appears linear in $N$, the true complexity depends on how $M$ must scale with $N$ to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how $M$ needs to grow with $N$ to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with $M \ll N$. Specifically, for the popular squared exponential kernel and $D$-dimensional Gaussian distributed covariates, $M = \mathcal{O}((\log N)^D)$ suffice and a method with an overall computational cost of $\mathcal{O}\left(N(\log N)^{2D}(\log \log N)^2\right)$ can be used to perform inference.

Beschreibung: Convolutional neural networks typically consist of many convolutional layers followed by one or more fully connected layers. While convolutional layers map between high-order activation tensors, the fully connected layers operate on flattened activation vectors. Despite empirical success, this approach has notable drawbacks. Flattening followed by fully connected layers discards multilinear structure in the activations and requires many parameters. We address these problems by incorporating tensor algebraic operations that preserve multilinear structure at every layer. First, we introduce Tensor Contraction Layers (TCLs) that reduce the dimensionality of their input while preserving their multilinear structure using tensor contraction. Next, we introduce Tensor Regression Layers (TRLs), which express outputs through a low-rank multilinear mapping from a high-order activation tensor to an output tensor of arbitrary order. We learn the contraction and regression factors end-to-end, and produce accurate nets with fewer parameters. Additionally, our layers regularize networks by imposing low-rank constraints on the activations (TCL) and regression weights (TRL). Experiments on ImageNet show that, applied to VGG and ResNet architectures, TCLs and TRLs reduce the number of parameters compared to fully connected layers by more than 65% while maintaining or increasing accuracy. In addition to the space savings, our approach's ability to leverage topological structure can be crucial for structured data such as MRI. In particular, we demonstrate significant performance improvements over comparable architectures on three tasks associated with the UK Biobank dataset.

Beschreibung: Structure learning of Gaussian graphical models typically involves careful tuning of penalty parameters, which balance the tradeoff between data fidelity and graph sparsity. Unfortunately, this tuning is often a “black art” requiring expert experience or brute-force search. It is therefore tempting to develop tuning-free algorithms that can determine the sparsity of the graph adaptively from the observed data in an automatic fashion. In this paper, we propose a novel approach, named BISN (Bayesian inference of Sparse Networks), for automatic Gaussian graphical model selection. Specifically, we regard the off-diagonal entries in the precision matrix as random variables and impose sparse-promoting horseshoe priors on them, resulting in automatic sparsity determination. With the help of stochastic gradients, an efficient variational Bayes algorithm is derived to learn the model. We further propose a decaying recursive stochastic gradient (DRSG) method to reduce the variance of the stochastic gradients and to accelerate the convergence. Our theoretical analysis shows that the time complexity of BISN scales only quadratically with the dimension, whereas the theoretical time complexity of the state-of-the-art methods for automatic graphical model selection is typically a third-order function of the dimension. Furthermore, numerical results show that BISN can achieve comparable or better performance than the state-of-the-art methods in terms of structure recovery, and yet its computational time is several orders of magnitude shorter, especially for large dimensions.

Beschreibung: Transfer learning, where a model is first pre-trained on a data-rich task before being fine-tuned on a downstream task, has emerged as a powerful technique in natural language processing (NLP). The effectiveness of transfer learning has given rise to a diversity of approaches, methodology, and practice. In this paper, we explore the landscape of transfer learning techniques for NLP by introducing a unified framework that converts all text-based language problems into a text-to-text format. Our systematic study compares pre-training objectives, architectures, unlabeled data sets, transfer approaches, and other factors on dozens of language understanding tasks. By combining the insights from our exploration with scale and our new “Colossal Clean Crawled Corpus”, we achieve state-of-the-art results on many benchmarks covering summarization, question answering, text classification, and more. To facilitate future work on transfer learning for NLP, we release our data set, pre-trained models, and code.

Beschreibung: Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted considerable attention because of their low computational cost in each iteration. However, these methods might suffer from poor performance when the Hessian is hard to be approximate well in a computation-efficient way. To overcome this dilemma, we resort to Nesterov's acceleration to improve the convergence performance of these second-order methods and propose accelerated approximate Newton. We give the theoretical convergence analysis of accelerated approximate Newton and show that Nesterov's acceleration can improve the convergence rate. Accordingly, we propose an accelerated regularized sub-sampled Newton (ARSSN) which performs much better than the conventional regularized sub-sampled Newton empirically and theoretically. Moreover, we show that ARSSN has better performance than classical first-order methods empirically.

Beschreibung: We consider learning problems over training sets in which both, the number of training examples and the dimension of the feature vectors, are large. To solve these problems we propose the random parallel stochastic algorithm (RAPSA). We call the algorithm random parallel because it utilizes multiple parallel processors to operate on a randomly chosen subset of blocks of the feature vector. RAPSA is doubly stochastic since each processor utilizes a random set of functions to compute the stochastic gradient associated with a randomly chosen sets of variable coordinates. Algorithms that are parallel in either of these dimensions exist, but RAPSA is the first attempt at a methodology that is parallel in both the selection of blocks and the selection of elements of the training set. In RAPSA, processors utilize the randomly chosen functions to compute the stochastic gradient component associated with a randomly chosen block. The technical contribution of this paper is to show that this minimally coordinated algorithm converges to the optimal classifier when the training objective is strongly convex. Moreover, we present an accelerated version of RAPSA (ARAPSA) that incorporates the objective function curvature information by premultiplying the descent direction by a Hessian approximation matrix. We further extend the results for asynchronous settings and show that if the processors perform their updates without any coordination the algorithms are still convergent to the optimal argument. RAPSA and its extensions are then numerically evaluated on a linear estimation problem and a binary image classification task using the MNIST handwritten digit dataset.

Beschreibung: As artificial intelligence algorithms make further inroads in high-stakes societal applications, there are increasing calls from multiple stakeholders for these algorithms to explain their outputs. To make matters more challenging, different personas of consumers of explanations have different requirements for explanations. Toward addressing these needs, we introduce AI Explainability 360, an open-source Python toolkit featuring ten diverse and state-of-the-art explainability methods and two evaluation metrics. Equally important, we provide a taxonomy to help entities requiring explanations to navigate the space of interpretation and explanation methods, not only those in the toolkit but also in the broader literature on explainability. For data scientists and other users of the toolkit, we have implemented an extensible software architecture that organizes methods according to their place in the AI modeling pipeline. The toolkit is not only the software, but also guidance material, tutorials, and an interactive web demo to introduce AI explainability to different audiences. Together, our toolkit and taxonomy can help identify gaps where more explainability methods are needed and provide a platform to incorporate them as they are developed.

Beschreibung: Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. The SOEC objective and constraints contain expectations defined with respect to complex distributions or large data sets, leading to high computational complexity when solved by the algorithms that use exact functions and their gradients. Recent stochastic first order methods exhibit low computational complexity when handling SOECs but guarantee near-feasibility and near-optimality only at convergence. These methods may thus return highly infeasible solutions when heuristically terminated, as is often the case, due to theoretical convergence criteria being highly conservative. This issue limits the use of first order methods in several applications where the SOEC constraints encode implementation requirements. We design a stochastic feasible level set method (SFLS) for SOECs that has low complexity and emphasizes feasibility before convergence. Specifically, our level-set method solves a root-finding problem by calling a novel first order oracle that computes a stochastic upper bound on the level-set function by extending mirror descent and online validation techniques. We establish that SFLS maintains a high-probability feasible solution at each root-finding iteration and exhibits favorable complexity compared to state-of-the-art deterministic feasible level set and stochastic subgradient methods. Numerical experiments on three diverse applications highlight how SFLS finds feasible solutions with small optimality gaps with lower complexity than the former approaches.

Beschreibung: The goal of noisy high-dimensional phase retrieval is to estimate an $s$-sparse parameter $\boldsymbol{\beta}^*\in \mathbb{R}^d$ from $n$ realizations of the model $Y = (\mathbf{X}^T \boldsymbol{\beta}^*)^2 + \varepsilon$. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which $Y = f(\mathbf{X}^T \boldsymbol{\beta}^*, \varepsilon)$ with unknown $f$ and $\operatorname{Cov}(Y, (\mathbf{X}^T \boldsymbol{\beta}^*)^2) 〉 0$. For example, MPR encompasses $Y = h(|\mathbf{X}^T \boldsymbol{\beta}^*|) + \varepsilon$ with increasing $h$ as a special case. Despite the generality of the MPR model, it eludes the reach of most existing semi-parametric estimators. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex programs and provably recovers the direction of $\boldsymbol{\beta}^*$. Furthermore, we prove that our procedure is minimax optimal over the class of MPR models. Interestingly, our minimax analysis characterizes the statistical price of misspecifying the link function in phase retrieval models. Our theory is backed up by thorough numerical results.

Beschreibung: We present a framework for compactly summarizing many recent results in efficient and/or biologically plausible online training of recurrent neural networks (RNN). The framework organizes algorithms according to several criteria: (a) past vs. future facing, (b) tensor structure, (c) stochastic vs. deterministic, and (d) closed form vs. numerical. These axes reveal latent conceptual connections among several recent advances in online learning. Furthermore, we provide novel mathematical intuitions for their degree of success. Testing these algorithms on two parametric task families shows that performances cluster according to our criteria. Although a similar clustering is also observed for pairwise gradient alignment, alignment with exact methods does not explain ultimate performance. This suggests the need for better comparison metrics.

Beschreibung: First-order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, was recently introduced to adjust iteratively the learning rate for each coordinate. Despite great practical success in deep learning, their behavior and performance on more general loss functions are not well understood. In this paper, we derive a non-autonomous system of differential equations, which is the continuous-time limit of adaptive optimization methods. We study the convergence of its trajectories and give conditions under which the differential system, underlying all adaptive algorithms, is suitable for optimization. We discuss convergence to a critical point in the non-convex case and give conditions for the dynamics to avoid saddle points and local maxima. For convex loss function, we introduce a suitable Lyapunov functional which allows us to study its rate of convergence. Several other properties of both the continuous and discrete systems are briefly discussed. The differential system studied in the paper is general enough to encompass many other classical algorithms (such as Heavy Ball and Nesterov's accelerated method) and allow us to recover several known results for these algorithms.

Beschreibung: Commonly-used clustering algorithms usually find ellipsoidal, spherical or other regular-structured clusters, but are more challenged when the underlying groups lack formal structure or definition. Syncytial clustering is the name that we introduce for methods that merge groups obtained from standard clustering algorithms in order to reveal complex group structure in the data. Here, we develop a distribution-free fully-automated syncytial clustering algorithm that can be used with $k$-means and other algorithms. Our approach estimates the cumulative distribution function of the normed residuals from an appropriately fit $k$-groups model and calculates the estimated nonparametric overlap between each pair of clusters. Groups with high pairwise overlap are merged as long as the estimated generalized overlap decreases. Our methodology is always a top performer in identifying groups with regular and irregular structures in several datasets and can be applied to datasets with scatter or incomplete records. The approach is also used to identify the distinct kinds of gamma ray bursts in the Burst and Transient Source Experiment 4Br catalog and the distinct kinds of activation in a functional Magnetic Resonance Imaging study.

Beschreibung: We prove the convergence to minima and estimates on the rate of convergence for the stochastic gradient descent method in the case of not necessarily locally convex nor contracting objective functions. In particular, the analysis relies on a quantitative use of mini-batches to control the loss of iterates to non-attracted regions. The applicability of the results to simple objective functions arising in machine learning is shown.

Beschreibung: In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular function is also coordinate-wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasi-linear time. Both these results improve upon prior work (Bian et al. 2017; Soma and Yoshida, 2017). Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

Beschreibung: We propose a novel methodology for representation learning on graph-structured data, in which a stack of Bayesian Networks learns different distributions of a vertex's neighbourhood. Through an incremental construction policy and layer-wise training, we can build deeper architectures with respect to typical graph convolutional neural networks, with benefits in terms of context spreading between vertices. First, the model learns from graphs via maximum likelihood estimation without using target labels. Then, a supervised readout is applied to the learned graph embeddings to deal with graph classification and vertex classification tasks, showing competitive results against neural models for graphs. The computational complexity is linear in the number of edges, facilitating learning on large scale data sets. By studying how depth affects the performances of our model, we discover that a broader context generally improves performances. In turn, this leads to a critical analysis of some benchmarks used in literature.

Beschreibung: Estimation of Distribution Algorithms (EDAs) are metaheuristics where learning a model and sampling new solutions replaces the variation operators recombination and mutation used in standard Genetic Algorithms. The choice of these models as well as the corresponding training processes are subject to the bias/variance tradeoff, also known as under- and overfitting: simple models cannot capture complex interactions between problem variables, whereas complex models are susceptible to modeling random noise. This paper suggests using Denoising Autoencoders (DAEs) as generative models within EDAs (DAE-EDA). The resulting DAE-EDA is able to model complex probability distributions. Furthermore, overfitting is less harmful, since DAEs overfit by learning the identity function. This overfitting behavior introduces unbiased random noise into the samples, which is no major problem for the EDA but just leads to higher population diversity. As a result, DAE-EDA runs for more generations before convergence and searches promising parts of the solution space more thoroughly. We study the performance of DAE-EDA on several combinatorial single-objective optimization problems. In comparison to the Bayesian Optimization Algorithm, DAE-EDA requires a similar number of evaluations of the objective function but is much faster and can be parallelized efficiently, making it the preferred choice especially for large and difficult optimization problems.

Beschreibung: The accuracy of probability distributions inferred using machine-learning algorithms heavily depends on data availability and quality. In practical applications it is therefore fundamental to investigate the robustness of a statistical model to misspecification of some of its underlying probabilities. In the context of graphical models, investigations of robustness fall under the notion of sensitivity analyses. These analyses consist in varying some of the model's probabilities or parameters and then assessing how far apart the original and the varied distributions are. However, for Gaussian graphical models, such variations usually make the original graph an incoherent representation of the model's conditional independence structure. Here we develop an approach to sensitivity analysis which guarantees the original graph remains valid after any probability variation and we quantify the effect of such variations using different measures. To achieve this we take advantage of algebraic techniques to both concisely represent conditional independence and to provide a straightforward way of checking the validity of such relationships. Our methods are demonstrated to be robust and comparable to standard ones, which can break the conditional independence structure of the model, using an artificial example and a medical real-world application.

Beschreibung: Peak detection in genomic data involves segmenting counts of DNA sequence reads aligned to different locations of a chromosome. The goal is to detect peaks with higher counts, and filter out background noise with lower counts. Most existing algorithms for this problem are unsupervised heuristics tailored to patterns in specific data types. We propose a supervised framework for this problem, using optimal changepoint detection models with learned penalty functions. We propose the first dynamic programming algorithm that is guaranteed to compute the optimal solution to changepoint detection problems with constraints between adjacent segment mean parameters. Implementing this algorithm requires the choice of penalty parameter that determines the number of segments that are estimated. We show how the supervised learning ideas of Rigaill et al. (2013) can be used to choose this penalty. We compare the resulting implementation of our algorithm to several baselines in a benchmark of labeled ChIP-seq data sets with two different patterns (broad H3K36me3 data and sharp H3K4me3 data). Whereas baseline unsupervised methods only provide accurate peak detection for a single pattern, our supervised method achieves state-of-the-art accuracy in all data sets. The log-linear timings of our proposed dynamic programming algorithm make it scalable to the large genomic data sets that are now common. Our implementation is available in the PeakSegOptimal R package on CRAN.

Beschreibung: Graphical models are ubiquitous tools to describe the interdependence between variables measured simultaneously such as large-scale gene or protein expression data. Gaussian graphical models (GGMs) are well-established tools for probabilistic exploration of dependence structures using precision matrices and they are generated under a multivariate normal joint distribution. However, they suffer from several shortcomings since they are based on Gaussian distribution assumptions. In this article, we propose a Bayesian quantile based approach for sparse estimation of graphs. We demonstrate that the resulting graph estimation is robust to outliers and applicable under general distributional assumptions. Furthermore, we develop efficient variational Bayes approximations to scale the methods for large data sets. Our methods are applied to a novel cancer proteomics data dataset where-in multiple proteomic antibodies are simultaneously assessed on tumor samples using reverse-phase protein arrays (RPPA) technology.

Beschreibung: An accurate model of a patient’s individual survival distribution can help determine the appropriate treatment for terminal patients. Unfortunately, risk scores (for example from Cox Proportional Hazard models) do not provide survival probabilities, single-time probability models (for instance the Gail model, predicting 5 year probability) only provide for a single time point, and standard Kaplan-Meier survival curves provide only population averages for a large class of patients, meaning they are not specific to individual patients. This motivates an alternative class of tools that can learn a model that provides an individual survival distribution for each subject, which gives survival probabilities across all times, such as extensions to the Cox model, Accelerated Failure Time, an extension to Random Survival Forests, and Multi-Task Logistic Regression. This paper first motivates such 'individual survival distribution' (ISD) models, and explains how they differ from standard models. It then discusses ways to evaluate such models — namely Concordance, 1-Calibration, Integrated Brier score, and versions of L1-loss — then motivates and defines a novel approach, 'D-Calibration', which determines whether a model's probability estimates are meaningful. We also discuss how these measures differ, and use them to evaluate several ISD prediction tools over a range of survival data sets. We also provide a code base for all of these survival models and evaluation measures, at https://github.com/haiderstats/ISDEvaluation.

Beschreibung: Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Stormer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of step-size and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algorithms such as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we provide a general framework for sharpening mixing time bounds of Markov chains initialized at a substantial distance from the target distribution over continuous spaces. We apply this sharpening device to the Metropolized random walk and Langevin algorithms, thereby obtaining improved mixing time bounds from a non-warm initial distribution.

Beschreibung: pyDML is an open-source python library that provides a wide range of distance metric learning algorithms. Distance metric learning can be useful to improve similarity learning algorithms, such as the nearest neighbors classifier, and also has other applications, like dimensionality reduction. The pyDML package currently provides more than 20 algorithms, which can be categorized, according to their purpose, in: dimensionality reduction algorithms, algorithms to improve nearest neighbors or nearest centroids classifiers, information theory based algorithms or kernel based algorithms, among others. In addition, the library also provides some utilities for the visualization of classifier regions, parameter tuning and a stats website with the performance of the implemented algorithms. The package relies on the scipy ecosystem, it is fully compatible with scikit-learn, and is distributed under GPLv3 license. Source code and documentation can be found at https://github.com/jlsuarezdiaz/pyDML.

Beschreibung: Predicting a new response from a covariate is a challenging task in regression, which raises new question since the era of high-dimensional data. In this paper, we are interested in the inverse regression method from a theoretical viewpoint. Theoretical results for the well-known Gaussian linear model are well-known, but the curse of dimensionality has increased the interest of practitioners and theoreticians into generalization of those results for various estimators, calibrated for the high-dimension context. We propose to focus on inverse regression. It is known to be a reliable and efficient approach when the number of features exceeds the number of observations. Indeed, under some conditions, dealing with the inverse regression problem associated to a forward regression problem drastically reduces the number of parameters to estimate, makes the problem tractable and allows to consider more general distributions, as elliptical distributions. When both the responses and the covariates are multivariate, estimators constructed by the inverse regression are studied in this paper, the main result being explicit asymptotic prediction regions for the response. The performances of the proposed estimators and prediction regions are also analyzed through a simulation study and compared with usual estimators.

Beschreibung: We propose a new stochastic first-order algorithmic framework to solve stochastic composite nonconvex optimization problems that covers both finite-sum and expectation settings. Our algorithms rely on the SARAH estimator and consist of two steps: a proximal gradient and an averaging step making them different from existing nonconvex proximal-type algorithms. The algorithms only require an average smoothness assumption of the nonconvex objective term and additional bounded variance assumption if applied to expectation problems. They work with both constant and dynamic step-sizes, while allowing single sample and mini-batches. In all these cases, we prove that our algorithms can achieve the best-known complexity bounds in terms of stochastic first-order oracle. One key step of our methods is the new constant and dynamic step-sizes resulting in the desired complexity bounds while improving practical performance. Our constant step-size is much larger than existing methods including proximal SVRG scheme in the single sample case. We also specify our framework to the non-composite case that covers existing state-of-the-arts in terms of oracle complexity bounds. Our update also allows one to trade-off between step-sizes and mini-batch sizes to improve performance. We test the proposed algorithms on two composite nonconvex problems and neural networks using several well-known data sets.

Beschreibung: Bayesian Optimisation (BO) refers to a suite of techniques for global optimisation of expensive black box functions, which use introspective Bayesian models of the function to efficiently search for the optimum. While BO has been applied successfully in many applications, modern optimisation tasks usher in new challenges where conventional methods fail spectacularly. In this work, we present Dragonfly, an open source Python library for scalable and robust BO. Dragonfly incorporates multiple recently developed methods that allow BO to be applied in challenging real world settings; these include better methods for handling higher dimensional domains, methods for handling multi-fidelity evaluations when cheap approximations of an expensive function are available, methods for optimising over structured combinatorial spaces, such as the space of neural network architectures, and methods for handling parallel evaluations. Additionally, we develop new methodological improvements in BO for selecting the Bayesian model, selecting the acquisition function, and optimising over complex domains with different variable types and additional constraints. We compare Dragonfly to a suite of other packages and algorithms for global optimisation and demonstrate that when the above methods are integrated, they enable significant improvements in the performance of BO. The Dragonfly library is available at dragonfly.github.io.

Beschreibung: Testing the hypothesis of parallelism is a fundamental statistical problem arising from many applied sciences. In this paper, we develop a nonparametric parallelism test for inferring whether the trends are parallel in treatment and control groups. In particular, the proposed nonparametric parallelism test is a Wald type test based on a smoothing spline ANOVA (SSANOVA) model which can characterize the complex patterns of the data. We derive that the asymptotic null distribution of the test statistic is a Chi-square distribution, unveiling a new version of Wilks phenomenon. Notably, we establish the minimax sharp lower bound of the distinguishable rate for the nonparametric parallelism test by using the information theory, and further prove that the proposed test is minimax optimal. Simulation studies are conducted to investigate the empirical performance of the proposed test. DNA methylation and neuroimaging studies are presented to illustrate potential applications of the test. The software is available at https://github.com/BioAlgs/Parallelism.

Beschreibung: Gaussian belief propagation (GaBP) is a message-passing algorithm that can be used to perform approximate inference on a pairwise Markov graph (MG) constructed from a multivariate Gaussian distribution in canonical parameterization. The output of GaBP is a set of approximate univariate marginals for each variable in the pairwise MG. An extension of GaBP (labeled GaBP-m), allowing for the approximation of higher-dimensional marginal distributions, was explored by Kamper et al. (2019). The idea is to create an MG in which each node is allowed to receive more than one variable. As in the univariate case, the multivariate extension does not necessarily converge in loopy graphs and, even if convergence occurs, is not guaranteed to provide exact inference. To address the problem of convergence, we consider a multivariate extension of the principle of node regularization proposed by Kamper et al. (2018). We label this algorithm slow GaBP-m (sGaBP-m), where the term 'slow' relates to the damping effect of the regularization on the message passing. We prove that, given sufficient regularization, this algorithm will converge and provide the exact marginal means at convergence, regardless of the way variables are assigned to nodes. The selection of the degree of regularization is addressed through the use of a heuristic, which is based on a tree representation of sGaBP-m. As a further contribution, we extend other GaBP variants in the literature to allow for higher-dimensional marginalization. We show that our algorithm compares favorably with these variants, both in terms of convergence speed and inference quality.

Beschreibung: We propose and analyze a novel theoretical and algorithmic framework for structured prediction. While so far the term has referred to discrete output spaces, here we consider more general settings, such as manifolds or spaces of probability measures. We define structured prediction as a problem where the output space lacks a vectorial structure. We identify and study a large class of loss functions that implicitly defines a suitable geometry on the problem. The latter is the key to develop an algorithmic framework amenable to a sharp statistical analysis and yielding efficient computations. When dealing with output spaces with infinite cardinality, a suitable implicit formulation of the estimator is shown to be crucial.

Beschreibung: tslearn is a general-purpose Python machine learning library for time series that offers tools for pre-processing and feature extraction as well as dedicated models for clustering, classification and regression. It follows scikit-learn's Application Programming Interface for transformers and estimators, allowing the use of standard pipelines and model selection tools on top of tslearn objects. It is distributed under the BSD-2-Clause license, and its source code is available at https://github.com/tslearn-team/tslearn.

Beschreibung: Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures, in an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings. By considering group invariance from the perspective of probabilistic symmetry, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of probability distributions that are invariant or equivariant under the action of a compact group. Our representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We demonstrate that examples from the recent literature are special cases, and develop the details of the general program for exchangeable sequences and arrays.

Beschreibung: Decision forests, including Random Forests and Gradient Boosting Trees, have recently demonstrated state-of-the-art performance in a variety of machine learning settings. Decision forests are typically ensembles of axis-aligned decision trees; that is, trees that split only along feature dimensions. In contrast, many recent extensions to decision forests are based on axis-oblique splits. Unfortunately, these extensions forfeit one or more of the favorable properties of decision forests based on axis-aligned splits, such as robustness to many noise dimensions, interpretability, or computational efficiency. We introduce yet another decision forest, called “Sparse Projection Oblique Randomer Forests” (SPORF). SPORF trees recursively split along very sparse random projections. Our method significantly improves accuracy over existing state-of-the-art algorithms on a standard benchmark suite for classification with $〉100$ problems of varying dimension, sample size, and number of classes. To illustrate how SPORF addresses the limitations of both axis-aligned and existing oblique decision forest methods, we conduct extensive simulated experiments. SPORF typically yields improved performance over existing decision forest methods, while mitigating computational efficiency and scalability and maintaining interpretability. Very sparse random projections can be incorporated into gradient boosted trees to obtain potentially similar gains.

Beschreibung: Linear discriminant analysis (LDA) is a popular classifier that is built on the assumption of common population covariance matrix across classes. The performance of LDA depends heavily on the quality of estimating the mean vectors and the population covariance matrix. This issue becomes more challenging in high-dimensional settings where the number of features is of the same order as the number of training samples. Several techniques for estimating the covariance matrix can be found in the literature. One of the most popular approaches are estimators based on using a regularized sample covariance matrix, giving the name regularized LDA (R-LDA) to the corresponding classifier. These estimators are known to be more resilient to the sample noise than the traditional sample covariance matrix estimator. However, the main challenge of the regularization approach is the choice of the optimal regularization parameter, as an arbitrary choice could lead to severe degradation of the classifier performance. In this work, we propose an improved LDA classifier based on the assumption that the covariance matrix follows a spiked covariance model. The main principle of our proposed technique is the design of a parametrized inverse covariance matrix estimator, the parameters of which are shown to be easily optimized. Numerical simulations, using both real and synthetic data, show that the proposed classifier yields better classification performance than the classical R-LDA while requiring lower computational complexity.

Beschreibung: We study the convergence rate of the optimal quantization for a probability measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer $x^{(n)}\in(\mathbb{R}^{d})^{K}$ of $\mu_{n}$ at level $K$; the other one is the convergence rate of the distortion function valued at $x^{(n)}$, called the “performance” of $x^{(n)}$. Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution $\mu$ with finite second moment but possibly unbounded support. As an application, we show an upper bound with a convergence rate $\mathcal{O}(\frac{\log n}{\sqrt{n}})$ of the mean performance for the empirical measure of the multidimensional normal distribution $\mathcal{N}(m, \Sigma)$ and of distributions with hyper-exponential tails. This extends the results from Biau et al. (2008) obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level $K$ but suboptimal in $n$ by applying results in Fournier and Guillin (2015).

Beschreibung: We propose a general algorithmic framework for Bayesian model selection. A spike-and-slab Laplacian prior is introduced to model the underlying structural assumption. Using the notion of effective resistance, we derive an EM-type algorithm with closed-form iterations to efficiently explore possible candidates for Bayesian model selection. The deterministic nature of the proposed algorithm makes it more scalable to large-scale and high-dimensional data sets compared with existing stochastic search algorithms. When applied to sparse linear regression, our framework recovers the EMVS algorithm by Ročková and George (2014) as a special case. We also discuss extensions of our framework using tools from graph algebra to incorporate complex Bayesian models such as biclustering and submatrix localization. Extensive simulation studies and real data applications are conducted to demonstrate the superior performance of our methods over its frequentist competitors such as $\ell_0$ or $\ell_1$ penalization.

Beschreibung: In short-term portfolio optimization (SPO), some financial characteristics like the expected return and the true covariance might be dynamic. Then there are only a small window size $w$ of observations that are sufficiently close to the current moment and reliable to make estimations. $w$ is usually much smaller than the number of assets $d$, which leads to a typical undersampled problem. Worse still, the asset price relatives are not likely subject to any proper distributions. These facts violate the statistical assumptions of the traditional covariance estimates and invalidate their statistical efficiency and consistency in risk measurement. In this paper, we propose to reconsider the function of covariance estimates in the perspective of operators, and establish a rank-one covariance estimate in the principal rank-one tangent space at the observation matrix. Moreover, we propose a loss control scheme with this estimate, which effectively catches the instantaneous risk structure and avoids extreme losses. We conduct extensive experiments on $7$ real-world benchmark daily or monthly data sets with stocks, funds and portfolios from diverse regional markets to show that the proposed method achieves state-of-the-art performance in comprehensive downside risk metrics and gains good investing incomes as well. It offers a novel perspective of rank-related approaches for undersampled estimations in SPO.

Beschreibung: One way to define the randomness of a fixed individual sequence is to ask how hard it is to predict relative to a given loss function. A sequence is memoryless if, with respect to average loss, no continuous function can predict the next entry of the sequence from a finite window of previous entries better than a constant prediction. For squared loss, memoryless sequences are known to have stochastic attributes analogous to those of truly random sequences. In this paper, we address the question of how changing the loss function changes the set of memoryless sequences, and in particular, the stochastic attributes they possess. For convex differentiable losses we establish that the statistic or property elicited by the loss determines the identity and stochastic attributes of the corresponding memoryless sequences. We generalize these results to convex non-differentiable losses, under additional assumptions, and to non-convex Bregman divergences. In particular, our results show that any Bregman divergence has the same set of memoryless sequences as squared loss. We apply our results to price calibration in prediction markets.

Beschreibung: Automated recommendation of machine learning algorithms is receiving a large deal of attention, not only because they can recommend the most suitable algorithms for a new task, but also because they can support efficient hyper-parameter tuning, leading to better machine learning solutions. The automated recommendation can be implemented using meta-learning, learning from previous learning experiences, to create a meta-model able to associate a data set to the predictive performance of machine learning algorithms. Although a large number of publications report the use of meta-learning, reproduction and comparison of meta-learning experiments is a difficult task. The literature lacks extensive and comprehensive public tools that enable the reproducible investigation of the different meta-learning approaches. An alternative to deal with this difficulty is to develop a meta-feature extractor package with the main characterization measures, following uniform guidelines that facilitate the use and inclusion of new meta-features. In this paper, we propose two Meta-Feature Extractor (MFE) packages, written in both Python and R, to fill this lack. The packages follow recent frameworks for meta-feature extraction, aiming to facilitate the reproducibility of meta-learning experiments.

Beschreibung: Cornac is an open-source Python framework for multimodal recommender systems. In addition to core utilities for accessing, building, evaluating, and comparing recommender models, Cornac is distinctive in putting emphasis on recommendation models that leverage auxiliary information in the form of a social network, item textual descriptions, product images, etc. Such multimodal auxiliary data supplement user-item interactions (e.g., ratings, clicks), which tend to be sparse in practice. To facilitate broad adoption and community contribution, Cornac is publicly available at https://github.com/PreferredAI/cornac, and it can be installed via Anaconda or the Python Package Index (pip). Not only is it well-covered by unit tests to ensure code quality, but it is also accompanied with a detailed documentation, tutorials, examples, and several built-in benchmarking data sets.

Beschreibung: This paper focuses on generalization performance analysis for distributed algorithms in the framework of learning theory. Taking distributed kernel ridge regression (DKRR) for example, we succeed in deriving its optimal learning rates in expectation and providing theoretically optimal ranges of the number of local processors. Due to the gap between theory and experiments, we also deduce optimal learning rates for DKRR in probability to essentially reflect the generalization performance and limitations of DKRR. Furthermore, we propose a communication strategy to improve the learning performance of DKRR and demonstrate the power of communications in DKRR via both theoretical assessments and numerical experiments.

Beschreibung: The gold standard for discovering causal relations is by means of experimentation. Over the last decades, alternative methods have been proposed that can infer causal relations between variables from certain statistical patterns in purely observational data. We introduce Joint Causal Inference (JCI), a novel approach to causal discovery from multiple data sets from different contexts that elegantly unifies both approaches. JCI is a causal modeling framework rather than a specific algorithm, and it can be implemented using any causal discovery algorithm that can take into account certain background knowledge. JCI can deal with different types of interventions (e.g., perfect, imperfect, stochastic, etc.) in a unified fashion, and does not require knowledge of intervention targets or types in case of interventional data. We explain how several well-known causal discovery algorithms can be seen as addressing special cases of the JCI framework, and we also propose novel implementations that extend existing causal discovery methods for purely observational data to the JCI setting. We evaluate different JCI implementations on synthetic data and on flow cytometry protein expression data and conclude that JCI implementations can considerably outperform state-of-the-art causal discovery algorithms.

Beschreibung: Probabilistic graphical models provide a flexible yet parsimonious framework for modeling dependencies among nodes in networks. There is a vast literature on parameter estimation and consistent model selection for graphical models. However, in many of the applications, scientists are also interested in quantifying the uncertainty associated with the estimated parameters and selected models, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for statistical inference on edge parameters in pairwise graphical models based on generalized Hyvarinen scoring rule. Hyvarinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form, which is a common problem for graphical models, including Ising models and truncated Gaussian graphical models. Our estimator allows us to perform statistical inference for general graphical models whereas the existing works mostly focus on statistical inference for Gaussian graphical models where finding normalizing constant is computationally tractable. Under mild conditions that are typically assumed in the literature for consistent estimation, we prove that our proposed estimator is $\sqrt{n}$-consistent and asymptotically normal, which allows us to construct confidence intervals and build hypothesis tests for edge parameters. Moreover, we show how our proposed method can be applied to test hypotheses that involve a large number of model parameters simultaneously. We illustrate validity of our estimator through extensive simulation studies on a diverse collection of data-generating processes.

Beschreibung: Despite the rich literature, the linear convergence of alternating direction method of multipliers (ADMM) has not been fully understood even for the convex case. For example, the linear convergence of ADMM can be empirically observed in a wide range of applications arising in statistics, machine learning, and related areas, while existing theoretical results seem to be too stringent to be satisfied or too ambiguous to be checked and thus why the ADMM performs linear convergence for these applications still seems to be unclear. In this paper, we systematically study the local linear convergence of ADMM in the context of convex optimization through the lens of variational analysis. We show that the local linear convergence of ADMM can be guaranteed without the strong convexity of objective functions together with the full rank assumption of the coefficient matrices, or the full polyhedricity assumption of their subdifferential; and it is possible to discern the local linear convergence for various concrete applications, especially for some representative models arising in statistical learning. We use some variational analysis techniques sophisticatedly; and our analysis is conducted in the most general proximal version of ADMM with Fortin and Glowinski's larger step size so that all major variants of the ADMM known in the literature are covered.

Beschreibung: Latent variable models allow capturing the hidden structure underlying the data. In particular, feature allocation models represent each observation by a linear combination of latent variables. These models are often used to make predictions either for new observations or for missing information in the original data, as well as to perform exploratory data analysis. Although there is an extensive literature on latent feature allocation models for homogeneous datasets, where all the attributes that describe each object are of the same (continuous or discrete) type, there is no general framework for practical latent feature modeling for heterogeneous datasets. In this paper, we introduce a general Bayesian nonparametric latent feature allocation model suitable for heterogeneous datasets, where the attributes describing each object can be arbitrary combinations of real-valued, positive real-valued, categorical, ordinal and count variables. The proposed model presents several important properties. First, it is suitable for heterogeneous data while keeping the properties of conjugate models, which enables us to develop an inference algorithm that presents linear complexity with respect to the number of objects and attributes per MCMC iteration. Second, the Bayesian nonparametric component allows us to place a prior distribution on the number of features required to capture the latent structure in the data. Third, the latent features in the model are binary-valued, which facilitates the interpretability of the obtained latent features in exploratory data analysis. Finally, a software package, called GLFM toolbox, is made publicly available for other researchers to use and extend. It is available at https://ivaleram.github.io/GLFM/. We show the flexibility of the proposed model by solving both prediction and data analysis tasks on several real-world datasets.

Beschreibung: We study nonconvex optimization problems, where the objective function is either an average of $n$ nonconvex functions or the expectation of some stochastic function. We propose a new stochastic gradient descent algorithm based on nested variance reduction, namely, Stochastic Nested Variance-Reduced Gradient descent ($\text{SNVRG}$). Compared with conventional stochastic variance reduced gradient ($\text{SVRG}$) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, $\text{SNVRG}$ converges to an $\epsilon$-approximate first-order stationary point within $\tilde O(n\land\epsilon^{-2}+\epsilon^{-3}\land n^{1/2}\epsilon^{-2})$ number of stochastic gradient evaluations. This improves the best known gradient complexity of $\text{SVRG}$ $O(n+n^{2/3}\epsilon^{-2})$ and that of $\text{SCSG}$ $O(n\land \epsilon^{-2}+\epsilon^{-10/3}\land n^{2/3}\epsilon^{-2})$. For gradient dominated functions, $\text{SNVRG}$ also achieves better gradient complexity than the state-of-the-art algorithms. Based on $\text{SNVRG}$, we further propose two algorithms that can find local minima faster than state-of-the-art algorithms in both finite-sum and general stochastic (online) nonconvex optimization. In particular, for finite-sum optimization problems, the proposed $\text{SNVRG}+\text{Neon2}^{\text{finite}}$ algorithm achieves $\tilde{O}(n^{1/2}\epsilon^{-2}+n\epsilon_H^{-3}+n^{3/4}\epsilon_H^{-7/2})$ gradient complexity to converge to an $(\epsilon, \epsilon_H)$-second-order stationary point, which outperforms $\text{SVRG}+\text{Neon2}^{\text{finite}}$ (Allen-Zhu and Li, 2018), the best existing algorithm, in a wide regime. For general stochastic optimization problems, the proposed $\text{SNVRG}+\text{Neon2}^{\text{online}}$ achieves $\tilde{O}(\epsilon^{-3}+\epsilon_H^{-5}+\epsilon^{-2}\epsilon_H^{-3})$ gradient complexity, which is better than both $\text{SVRG}+\text{Neon2}^{\text{online}}$ (Allen-Zhu and Li, 2018) and $\text{Natasha2}$ (Allen-Zhu, 2018a) in certain regimes. Thorough experimental results on different nonconvex optimization problems back up our theory.

Beschreibung: This paper describes AI-Toolbox, a C++ software library that contains reinforcement learning and planning algorithms, and supports both single and multi agent problems, as well as partial observability. It is designed for simplicity and clarity, and contains extensive documentation of its API and code. It supports Python to enable users not comfortable with C++ to take advantage of the library's speed and functionality. AI-Toolbox is free software, and is hosted online at https://github.com/Svalorzen/AI-Toolbox.

Beschreibung: A tacit assumption in linear regression is that (response, predictor)-pairs correspond to identical observational units. A series of recent works have studied scenarios in which this assumption is violated under terms such as “Unlabeled Sensing and “Regression with Unknown Permutation”. In this paper, we study the setup of multiple response variables and a notion of mismatches that generalizes permutations in order to allow for missing matches as well as for one-to-many matches. A two-stage method is proposed under the assumption that most pairs are correctly matched. In the first stage, the regression parameter is estimated by handling mismatches as contaminations, and subsequently the generalized permutation is estimated by a basic variant of matching. The approach is both computationally convenient and equipped with favorable statistical guarantees. Specifically, it is shown that the conditions for permutation recovery become considerably less stringent as the number of responses $m$ per observation increase. Particularly, for $m = \Omega(\log n)$, the required signal-to-noise ratio no longer depends on the sample size $n$. Numerical results on synthetic and real data are presented to support the main findings of our analysis.

Beschreibung: We consider the problem of inferring simplified topological substructures—which we term backbones—in metric and non-metric graphs. Intuitively, these are subgraphs with ‘few’ nodes, multifurcations, and cycles, that model the topology of the original graph well. We present a multistep procedure for inferring these backbones. First, we encode local (geometric) information of each vertex in the original graph by means of the boundary coefficient (BC) to identify ‘core’ nodes in the graph. Next, we construct a forest representation of the graph, termed an f-pine, that connects every node of the graph to a local ‘core’ node. The final backbone is then inferred from the f-pine through CLOF (Constrained Leaves Optimal subForest), a novel graph optimization problem we introduce in this paper. On a theoretical level, we show that CLOF is NP-hard for general graphs. However, we prove that CLOF can be efficiently solved for forest graphs, a surprising fact given that CLOF induces a nontrivial monotone submodular set function maximization problem on tree graphs. This result is the basis of our method for mining backbones in graphs through forest representation. We qualitatively and quantitatively confirm the applicability, effectiveness, and scalability of our method for discovering backbones in a variety of graph-structured data, such as social networks, earthquake locations scattered across the Earth, and high-dimensional cell trajectory data

Beschreibung: In this paper, we propose a data-adaptive non-parametric kernel learning framework in margin based kernel methods. In model formulation, given an initial kernel matrix, a data-adaptive matrix with two constraints is imposed in an entry-wise scheme. Learning this data-adaptive matrix in a formulation-free strategy enlarges the margin between classes and thus improves the model flexibility. The introduced two constraints are imposed either exactly (on small data sets) or approximately (on large data sets) in our model, which provides a controllable trade-off between model flexibility and complexity with theoretical demonstration. In algorithm optimization, the objective function of our learning framework is proven to be gradient-Lipschitz continuous. Thereby, kernel and classifier/regressor learning can be efficiently optimized in a unified framework via Nesterov's acceleration. For the scalability issue, we study a decomposition-based approach to our model in the large sample case. The effectiveness of this approximation is illustrated by both empirical studies and theoretical guarantees. Experimental results on various classification and regression benchmark data sets demonstrate that our non-parametric kernel learning framework achieves good performance when compared with other representative kernel learning based algorithms.

Beschreibung: Two popular approaches to dimensionality reduction are principal component analysis, which projects onto a small number of well-chosen but non-interpretable directions, and feature selection, which selects a small number of the original features. Feature selection can be abstracted as selecting the subset of columns of a matrix $X \in \mathbb{R}^{N \times d}$ which minimize the approximation error, i.e., the norm of the residual after projecting $X$ onto the space spanned by the selected columns. Such a combinatorial optimization is usually impractical, and there has been interest in polynomial-cost, random subset selection algorithms that favour small values of this approximation error. We propose sampling from a projection determinantal point process, a repulsive distribution over column indices that favours diversity among the selected columns. We bound the ratio of the expected approximation error over the optimal error of PCA. These bounds improve over the state-of-the-art bounds of volume sampling when some realistic structural assumptions are satisfied for $X$. Numerical experiments suggest that our bounds are tight, and that our algorithms have comparable performance with the double phase algorithm, often considered the practical state-of-the-art.

Beschreibung: Cluster analysis is a fundamental tool for pattern discovery of complex heterogeneous data. Prevalent clustering methods mainly focus on vector or matrix-variate data and are not applicable to general-order tensors, which arise frequently in modern scientific and business applications. Moreover, there is a gap between statistical guarantees and computational efficiency for existing tensor clustering solutions due to the nature of their non-convex formulations. In this work, we bridge this gap by developing a provable convex formulation of tensor co-clustering. Our convex co-clustering (CoCo) estimator enjoys stability guarantees and its computational and storage costs are polynomial in the size of the data. We further establish a non-asymptotic error bound for the CoCo estimator, which reveals a surprising “blessing of dimensionality” phenomenon that does not exist in vector or matrix-variate cluster analysis. Our theoretical findings are supported by extensive simulated studies. Finally, we apply the CoCo estimator to the cluster analysis of advertisement click tensor data from a major online company. Our clustering results provide meaningful business insights to improve advertising effectiveness.

Beschreibung: scikit-survival is an open-source Python package for time-to-event analysis fully compatible with scikit-learn. It provides implementations of many popular machine learning techniques for time-to-event analysis, including penalized Cox model, Random Survival Forest, and Survival Support Vector Machine. In addition, the library includes tools to evaluate model performance on censored time-to-event data. The documentation contains installation instructions, interactive notebooks, and a full description of the API. scikit-survival is distributed under the GPL-3 license with the source code and detailed instructions available at https://github.com/sebp/scikit-survival

Beschreibung: The challenge in controlling stochastic systems in which low-probability events can set the system on catastrophic trajectories is to develop a robust ability to respond to such events without significantly compromising the optimality of the baseline control policy. This paper presents CelluDose, a stochastic simulation-trained deep reinforcement learning adaptive feedback control prototype for automated precision drug dosing targeting stochastic and heterogeneous cell proliferation. Drug resistance can emerge from random and variable mutations in targeted cell populations; in the absence of an appropriate dosing policy, emergent resistant subpopulations can proliferate and lead to treatment failure. Dynamic feedback dosage control holds promise in combatting this phenomenon, but the application of traditional control approaches to such systems is fraught with challenges due to the complexity of cell dynamics, uncertainty in model parameters, and the need in medical applications for a robust controller that can be trusted to properly handle unexpected outcomes. Here, training on a sample biological scenario identified single-drug and combination therapy policies that exhibit a $100\%$ success rate at suppressing cell proliferation and responding to diverse system perturbations while establishing low-dose no-event baselines. These policies were found to be highly robust to variations in a key model parameter subject to significant uncertainty and unpredictable dynamical changes.

Beschreibung: This paper is concerned with the quadratic regression problem, where the goal is to find the unknown state (numerical parameters) of a system modeled by a set of equations that are quadratic in the state. We focus on the setting when a subset of equations of fixed cardinality is subject to errors of arbitrary magnitudes (potentially adversarial). We develop two methods to address this problem, which are both based on conic optimization and are able to accept any available prior knowledge on the solution as an input. We derive sufficient conditions for guaranteeing the correct recovery of the unknown state for each method and show that one method provides a better accuracy while the other one scales better to large-scale systems. The obtained conditions consist in bounds on the number of bad measurements each method can tolerate without producing a nonzero estimation error. In the case when no prior knowledge is available, we develop an iterative-based conic optimization technique. It is proved that the proposed methods allow up to half of the total number of measurements to be grossly erroneous.The efficacy of the developed methods is demonstrated in different case studies, including data analytics for a European power grid.

Beschreibung: In this paper, we study the dynamic assortment optimization problem over a finite selling season of length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products under a cardinality constraint, and the customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contextual information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and make the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\tilde{O}(d\sqrt{T})$, where $d$ is the dimension of the feature and $\tilde{O}$ suppresses logarithmic dependence. We further establish a lower bound $\Omega(d\sqrt{T}/{K})$, where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization problem for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.

Beschreibung: We propose a general framework of using a multi-level log-Gaussian Cox process to model repeatedly observed point processes with complex structures; such type of data has become increasingly available in various areas including medical research, social sciences, economics, and finance due to technological advances. A novel nonparametric approach is developed to efficiently and consistently estimate the covariance functions of the latent Gaussian processes at all levels. To predict the functional principal component scores, we propose a consistent estimation procedure by maximizing the conditional likelihood of super-positions of point processes. We further extend our procedure to the bivariate point process case in which potential correlations between the processes can be assessed. Asymptotic properties of the proposed estimators are investigated, and the effectiveness of our procedures is illustrated through a simulation study and an application to a stock trading dataset.

Beschreibung: We propose a novel inherently interpretable machine learning method that bases decisions on few relevant examples that we call prototypes. Our method, ProtoAttend, can be integrated into a wide range of neural network architectures including pre-trained models. It utilizes an attention mechanism that relates the encoded representations to samples in order to determine prototypes. Protoattend yields superior results in three high impact problems without sacrificing accuracy of the original model: (1)it enables high-quality interpretability that outputs samples most relevant to the decision-making (i.e. a sample-based interpretability method); (2) it achieves state of the art confidence estimation by quantifying the mismatch across prototype labels; and (3) it obtains state of the art in distribution mismatch detection. All these can be achieved with minimal additional test time and a practically viable training time computational cost.