We provide novel theoretical insights on structured prediction in the context of efficient convex surrogate loss minimization with consistency guarantees. For any task loss, we construct a convex surrogate that can be optimized via stochastic gradient descent and we prove tight bounds on the so-called "calibration function" relating the excess surrogate risk to the actual risk. In contrast to prior related work, we carefully monitor the effect of the exponential number of classes in the learning guarantees as well as on the optimization complexity. As an interesting consequence, we formalize the intuition that some task losses make learning harder than others, and that the classical 0-1 loss is ill-suited for structured prediction.

Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work \citep{jang2016categorical, maddison2016concrete} has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates. In this work, we combine the two approaches through a novel control variate that produces low-variance, \emph{unbiased} gradient estimates. Then, we introduce a novel continuous relaxation and show that the tightness of the relaxation can be adapted online, removing it as a hyperparameter. We show state-of-the-art variance reduction on several benchmark generative modeling tasks, generally leading to faster convergence to a better final log likelihood.

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.

In the online multiple testing problem, p-values corresponding to different null hypotheses are presented one by one, and the decision of whether to reject a hypothesis must be made immediately, after which the next p-value is presented. Alpha-investing algorithms to control the false discovery rate were first formulated by Foster and Stine and have since been generalized and applied to various settings, varying from quality-preserving databases for science to multiple A/B tests for internet commerce. This paper improves the class of generalized alpha-investing algorithms (GAI) in four ways : (a) we show how to uniformly improve the power of the entire class of GAI procedures under independence by awarding more alpha-wealth for each rejection, giving a near win-win resolution to a dilemma raised by Javanmard and Montanari, (b) we demonstrate how to incorporate prior weights to indicate domain knowledge of which hypotheses are likely to be null or non-null, (c) we allow for differing penalties for false discoveries to indicate that some hypotheses may be more meaningful/important than others, (d) we define a new quantity called the \emph{decaying memory false discovery rate, or $\memfdr$} that may be more meaningful for applications with an explicit time component, using a discount factor to incrementally forget past decisions and alleviate some potential problems that we describe and name ``piggybacking'' and ``alpha-death''. Our GAI++ algorithms incorporate all four generalizations (a, b, c, d) simulatenously, and reduce to more powerful variants of earlier algorithms when the weights and decay are all set to unity.

In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. We derive sample-complexity bounds on the uniform convergence rate of the empirical revenues to the true revenues, assuming a bound on the k'th moment of the valuations, for any (possibly fractional) k > 1. For uniform convergence in the limit, we give a complete characterization and a zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform convergence almost never occurs.

We introduce TrustVI, a fast second-order algorithm for black-box variational inference based on trust-region optimization and the reparameterization trick. At each iteration, TrustVI proposes and assesses a step based on minibatches of draws from the variational distribution. The algorithm provably converges to a stationary point. We implement TrustVI in the Stan framework and compare it to ADVI. TrustVI typically converges in tens of iterations to a solution at least as good as the one that ADVI reaches in thousands of iterations. TrustVI iterations can be more computationally expensive, but total computation is typically an order of magnitude less in our experiments.

A problem that has been of recent interest in statistical inference, machine learning and signal processing is that of understanding the asymptotic behavior of regularized least squares solutions under random measurement matrices (or dictionaries). The Least Absolute Shrinkage and Selection Operator (LASSO or least-squares with $\ell_1$ regularization) is perhaps one of the most interesting examples. Precise expressions for the asymptotic performance of LASSO have been obtained for a number of different cases, in particular when the elements of the dictionary matrix are sampled independently from a Gaussian distribution. It has also been empirically observed that the resulting expressions remain valid when the entries of the dictionary matrix are independently sampled from certain non-Gaussian distributions. In this paper, we confirm these observations theoretically when the distribution is sub-Gaussian. We further generalize the previous expressions for a broader family of regularization functions and under milder conditions on the underlying random, possibly non-Gaussian, dictionary matrix. In particular, we establish the universality of the asymptotic statistics (e.g., the average quadratic risk) of LASSO with non-Gaussian dictionaries.

This paper takes a step towards temporal reasoning in a dynamically changing video, not in the pixel space that constitutes its frames, but in a latent space that describes the non-linear dynamics of the objects in its world. We introduce the Kalman variational auto-encoder, a framework for unsupervised learning of sequential data that disentangles two latent representations: an object's representation, coming from a recognition model, and a latent state describing its dynamics. As a result, the evolution of the world can be imagined and missing data imputed, both without the need to generate high dimensional frames at each time step. The model is trained end-to-end on videos of a variety of simulated physical systems, and outperforms competing methods in generative and missing data imputation tasks.

In this paper we study the well-known greedy coordinate descent (GCD) algorithm to solve $\ell_1$-regularized problems and improve GCD by the two popular strategies: Nesterov's acceleration and stochastic optimization. Firstly, we propose a new rule for greedy selection based on an $\ell_1$-norm square approximation which is nontrivial to solve but convex; then an efficient algorithm called ``SOft ThreshOlding PrOjection (SOTOPO)'' is proposed to exactly solve the $\ell_1$-regularized $\ell_1$-norm square approximation problem, which is induced by the new rule. Based on the new rule and the SOTOPO algorithm, the Nesterov's acceleration and stochastic optimization strategies are then successfully applied to the GCD algorithm. The resulted algorithm called accelerated stochastic greedy coordinate descent (ASGCD) has the optimal convergence rate $O(\sqrt{1/\epsilon})$; meanwhile, it reduces the iteration complexity of greedy selection up to a factor of sample size. Both theoretically and empirically, we show that ASGCD has better performance for high-dimensional and dense problems with sparse solution.

Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L^2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

We show that the margin distribution --- normalized by a spectral complexity parameter --- is strongly predictive of neural network generalization performance. Namely, we 1) Use the margin distribution to correctly predict whether deep neural networks generalize under changes to label distribution such as randomization. That is, the margin distribution accurately predicts the difficulty of deep learning tasks. We further show that normalizing the margin by the network's spectral complexity is critical to obtaining this predictive power, and finally use the margin distribution to compare the generalization performance of multiple networks across different datasets on even terms. Our corresponding generalization bound places these results on rigorous theoretical footing.

We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically ''decoupled'', with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research.