AISTATS 2020 Papers

In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal estimator is the square-root Lasso, formulated along with its derivatives as a “non-smooth + non-smooth” optimization problem. Modern techniques to solve these include smoothing the datafitting term, to benefit from fast efficient proximal algorithms. In this work we show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators. Thanks to our theoretical analysis, we provide some guidelines on how to set the smoothing hyperparameter, and illustrate on synthetic data the interest of such guidelines.

We study the problem of estimating the expected reward of the optimal policy in the stochastic disjoint linear bandit setting. We prove that for certain settings it is possible to obtain an accurate estimate of the optimal policy value even with a sublinear number of samples, where a linear set would be needed to reliably estimate the reward that can be obtained by any policy. We establish near matching information theoretic lower bounds, showing that our algorithm achieves near optimal estimation error. Finally, we demonstrate the effectiveness of our algorithm on joke recommendation and cancer inhibition dosage selection problems using real datasets.

We exploit minimally faithful inversion of graphical model structures to specify sparse continuous normalizing flows (CNFs) for amortized inference. We find that the sparsity of this factorization can be exploited to reduce the numbers of parameters in the neural network, adaptive integration steps of the flow, and consequently FLOPs at both training and inference time without decreasing performance in comparison to unconstrained flows. By expressing the structure inversion as a compilation pass in a probabilistic programming language, we are able to apply it in a novel way to models as complex as convolutional neural networks. Furthermore, we extend the training objective for CNFs in the context of inference amortization to the symmetric Kullback-Leibler divergence, and demonstrate its theoretical and practical advantages.

A number of applications (e.g., AI bot tournaments, sports, peer grading, crowdsourcing) use pairwise comparison data and the Bradley-Terry-Luce (BTL) model to evaluate a given collection of items (e.g., bots, teams, students, search results). Past work has shown that under the BTL model, the widely-used maximum-likelihood estimator (MLE) is minimax-optimal in estimating the item parameters, in terms of the mean squared error. However, another important desideratum for designing estimators is fairness. In this work, we consider one specific type of fairness, which is the notion of bias in statistics. We show that the MLE incurs a suboptimal rate in terms of bias. We then propose a simple modification to the MLE, which "stretches" the bounding box of the maximum-likelihood optimizer by a small constant factor from the underlying ground truth domain. We show that this simple modification leads to an improved rate in bias, while maintaining minimax-optimality in the mean squared error. In this manner, our proposed class of estimators provably improves fairness in the sense of bias without loss in accuracy.

Active learning is a framework in which the learning machine can select the samples to be used for training. This technique is promising, particularly when the cost of data acquisition and labeling is high. In active learning, determining the timing at which learning should be stopped is a critical issue. In this study, we propose a criterion for automatically stopping active learning. The proposed stopping criterion is based on the difference in the expected generalization errors and hypothesis testing. We derive a novel upper bound for the difference in expected generalization errors before and after obtaining a new training datum based on PAC-Bayesian theory. Unlike ordinary PAC-Bayesian bounds, though, the proposed bound is deterministic; hence, there is no uncontrollable trade-off between the confidence and tightness of the inequality. We combine the upper bound with a statistical test to derive a stopping criterion for active learning. We demonstrate the effectiveness of the proposed method via experiments with both artificial and real datasets.

We present two stochastic variance-reduced PCA algorithms and their convergence analyses. By deriving explicit forms of step size, epoch length and batch size to ensure the optimal runtime, we show that the proposed algorithms can attain the optimal runtime with any batch sizes. Also, we establish global convergence of the algorithms based on a novel approach, which studies the optimality gap as a ratio of two expectation terms. The framework in our analysis is general and can be used to analyze other stochastic variance-reduced PCA algorithms and improve their analyses. Moreover, we introduce practical implementations of the algorithms which do not require hyper-parameters. The experimental results show that the proposed methodsd outperform other stochastic variance-reduced PCA algorithms regardless of the batch size.

Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional problems. In this paper, we first present a Stochastic Recursive Variance-Reduced Cubic regularization method (SRVRC) using a recursively updated semi-stochastic gradient and Hessian estimators. It enjoys improved gradient and Hessian complexities to find an $(\epsilon, \sqrt{\epsilon})$-approximate local minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon SRVRC, we further propose a Hessian-free SRVRC algorithm, namely SRVRC$_{\text{free}}$, which only needs $\tilde O(n\epsilon^{-2} \land \epsilon^{-3})$ stochastic gradient and Hessian-vector product computations, where $n$ is the number of component functions in the finite-sum objective and $\epsilon$ is the optimization precision. This outperforms the best-known result $\tilde O(\epsilon^{-3.5})$ achieved by stochastic cubic regularization algorithm proposed in \cite{tripuraneni2018stochastic}.

Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles tend to collapse. As a remedy, we generalize POS to a stochastic setting by injecting random noise into particle updates, thus termed stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop non-asymptotic convergence theory for the SPOS framework (related to SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein distance w.r.t. the numbers of particles and iterations. Somewhat surprisingly, with the same number of updates (not too large) for each particle, our theory suggests adopting more particles does not necessarily lead to a better approximation of a target distribution, due to limited computational budget and numerical errors. This phenomenon is also observed in SVGD and verified via a synthetic experiment. Extensive experimental results verify our theory and demonstrate the effectiveness of our proposed framework.

Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks. This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters. In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks. We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits. In all setups, our methods prove to be competitive with existing methods and betterthan the baselines.

In this paper, we investigate the impact of context diversity on stochastic linear contextual bandits. As opposed to the previous view that contexts lead to more difficult bandit learning, we show that when the contexts are sufficiently diverse, the learner is able to utilize the information obtained during exploitation to shorten the exploration process, thus achieving reduced regret. We design the LinUCB-d algorithm, and propose a novel approach to analyze its regret performance. The main theoretical result is that under the diverse context assumption, the cumulative expected regret of LinUCB-d is bounded by a constant. As a by-product, our results improve the previous understanding of LinUCB and strengthen its performance guarantee.

Motivated by recommendation problems in music streaming platforms, we propose a nonstationary stochastic bandit model in which the expected reward of an arm depends on the number of rounds that have passed since the arm was last pulled. After proving that finding an optimal policy is NP-hard even when all model parameters are known, we introduce a class of ranking policies provably approximating, to within a constant factor, the expected reward of the optimal policy. We show an algorithm whose regret with respect to the best ranking policy is bounded by $\widetilde{\scO}\big(\!\sqrt{kT}\big)$, where $k$ is the number of arms and $T$ is time. Our algorithm uses only $\scO\big(k\ln\ln T)$ switches, which helps when switching between policies is costly. As constructing the class of learning policies requires ordering the arms according to their expectations, we also bound the number of pulls required to do so. Finally, we run experiments to compare our algorithm against UCB on different problem instances.

An essential problem in automated machine learning (AutoML) is that of model selection. A unique challenge in the sequential setting is the fact that the optimal model itself may vary over time, depending on the distribution of features and labels available up to each point in time. In this paper, we propose a novel Bayesian optimization (BO) algorithm to tackle the challenge of model selection in this setting. This is accomplished by treating the performance at each time step as its own black-box function. In order to solve the resulting multiple black-box function optimization problem jointly and efficiently, we exploit potential correlations among black-box functions using deep kernel learning (DKL). To the best of our knowledge, we are the first to formulate the problem of stepwise model selection (SMS) for sequence prediction, and to design and demonstrate an efficient joint-learning algorithm for this purpose. Using multiple real-world datasets, we verify that our proposed method outperforms both standard BO and multi-objective BO algorithms on a variety of sequence prediction tasks.

Gradient-based approximate inference methods, such as Stein variational gradient descent (SVGD) \cite{liu2016stein}, provide simple and general-purpose inference engines for differentiable continuous distributions. However, existing forms of SVGD can not be directly applied to discrete distributions. In this work, we fill this gap by proposing a simple general-purpose framework that transforms discrete distributions to equivalent piecewise continuous distribution, on which we apply gradient-free Stein variational gradient descent to perform efficient approximate inference. Our empirical results show that our method outperforms traditional algorithms such as Gibbs sampling and discontinuous Hamiltonian Monte Carlo on various challenging benchmarks of discrete graphical models. We demonstrate that our method provides a promising tool for learning ensembles of binarized neural network (BNN), outperforming other widely used ensemble methods on learning binarized AlexNet on CIFAR-10. In addition, such transform can be straightforwardly employed in gradient-free kernelized Stein discrepancy to perform goodness-of-fit (GOF) test on discrete distributions. Our proposed method outperforms existing GOF test methods for intractable discrete distributions.

Local graph clustering methods aim to find small clusters in very large graphs. These methods take as input a graph and a seed node, and they return as output a good cluster in a running time that depends on the size of the output cluster but that is independent of the size of the input graph. In this paper, we adopt a statistical perspective on local graph clustering, and we analyze the performance of the l1-regularized PageRank method for the recovery of a single target cluster, given a seed node inside the cluster. Assuming the target cluster has been generated by a random model, we present two results. In the first, we show that the optimal support of l1-regularized PageRank recovers the full target cluster, with bounded false positives. In the second, we show that if the seed node is connected solely to the target cluster then the optimal support of l1-regularized PageRank recovers exactly the target cluster. We also show empirically that l1-regularized PageRank has a state-of-the-art performance on many real graphs, demonstrating the superiority of the method.

We consider the problem of graph logistic regression, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analyzed in a high-dimensional regime under a structural assumption. The optimal statistical rates are derived, and an estimator based on penalized maximum likelihood is shown to attain it. The algorithmic aspects of this problem are also studied, and optimal rates under computational constraints are derived, and shown to differ from the information-theoretic rates - under a complexity assumption.

Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincaré constant. As a by-product of the estimation of the Poincaré constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.

We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under suitable scaling on the weights, is a stochastic process whose finite-dimensional distributions are multivariate stable distributions. The limiting process is referred to as the stable process, and it generalizes the class of Gaussian processes recently obtained as infinite wide limits of NNs (Matthews at al., 2018b). Parameters of the stable process can be computed via an explicit recursion over the layers of the network. Our result contributes to the theory of fully connected feed-forward deep NNs, and it paves the way to expand recent lines of research that rely on Gaussian infinite wide limits.

Comparing data defined over space and time is notoriously hard. It involves quantifying both spatial and temporal variability while taking into account the chronological structure of the data. Dynamic Time Warping (DTW) computes a minimal cost alignment between time series that preserves the chronological order but is inherently blind to spatio-temporal shifts. In this paper, we propose Spatio-Temporal Alignments (STA), a new differentiable formulation of DTW that captures spatial and temporal variability. Spatial differences between time samples are captured using regularized Optimal transport. While temporal alignment cost exploits a smooth variant of DTW called soft-DTW. We show how smoothing DTW leads to alignment costs that increase quadratically with time shifts. The costs are expressed using an unbalanced Wasserstein distance to cope with observations that are not probabilities. Experiments on handwritten letters and brain imaging data confirm our theoretical findings and illustrate the effectiveness of STA as a dissimilarity for spatio-temporal data.

In this paper, we propose a general framework for sparse and low-rank tensor estimation from cubic sketchings. A two-stage non-convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent, which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability. The non-asymptotic analysis sheds light on an interplay between optimization error and statistical error. The proposed procedure is shown to be rate-optimal under certain conditions. As a technical by-product, novel high-order concentration inequalities are derived for studying high-moment sub-Gaussian tensors. An interesting tensor formulation illustrates the potential application to high-order interaction pursuit in high-dimensional linear regression.