Revisiting Fixed Support Wasserstein Barycenter: Computational Hardness and Efficient Algorithms

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$. We show first that the constraint matrix arising from the linear programming (LP) representation of the FS-WBP is totally unimodular when $m \geq 3$ and $n = 2$, but not totally unimodular when $m \geq 3$ and $n \geq 3$. This result answers an open problem, since it shows that the FS-WBP is not a minimum-cost flow problem and therefore cannot be solved efficiently using linear programming. Building on this negative result, we propose and analyze a simple and efficient variant of the iterative Bregman projection (IBP) algorithm, currently the most widely adopted algorithm to solve the FS-WBP. The algorithm is an accelerated IBP algorithm which achieves the complexity bound of $\widetilde{\mathcal{O}}(mn^{7/3}/\varepsilon)$. This bound is better than that obtained for the standard IBP algorithm---$\widetilde{\mathcal{O}}(mn^{2}/\varepsilon^2)$---in terms of $\varepsilon$, and that of accelerated primal-dual gradient algorithm---$\widetilde{\mathcal{O}}(mn^{5/2}/\varepsilon)$---in terms of $n$. Empirical study demonstrates that the acceleration promised by the theory is real in practice.

Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis

We address the problem of policy evaluation in discounted Markov decision processes, and provide instance-dependent guarantees on the $\ell_\infty$-error under a generative model. We establish both asymptotic and non-asymptotic versions of local minimax lower bounds for policy evaluation, thereby providing an instance-dependent baseline by which to compare algorithms. Theory-inspired simulations show that the widely-used temporal difference (TD) algorithm is strictly suboptimal when evaluated in a non-asymptotic setting, even when combined with Polyak-Ruppert iterate averaging. We remedy this issue by introducing and analyzing variance-reduced forms of stochastic approximation, showing that they achieve non-asymptotic, instance-dependent optimality up to logarithmic factors.

Post-Estimation Smoothing: A Simple Baseline for Learning with Side Information

Observational data are often accompanied by natural structural indices, such as time stamps or geographic locations, which are meaningful to prediction tasks but are often discarded. We leverage semantically meaningful indexing data while ensuring robustness to potentially uninformative or misleading indices. We propose a post-estimation smoothing operator as a fast and effective method for incorporating structural index data into prediction. Because the smoothing step is separate from the original predictor, it applies to a broad class of machine learning tasks, with no need to retrain models. Our theoretical analysis details simple conditions under which post-estimation smoothing will improve accuracy over that of the original predictor. Our experiments on large scale spatial and temporal datasets highlight the speed and accuracy of post-estimation smoothing in practice. Together, these results illuminate a novel way to consider and incorporate the natural structure of index variables in machine learning.

Robustness Guarantees for Mode Estimation with an Application to Bandits

Mode estimation is a classical problem in statistics with a wide range of applications in machine learning. Despite this, there is little understanding in its robustness properties under possibly adversarial data contamination. In this paper, we give precise robustness guarantees as well as privacy guarantees under simple randomization. We then introduce a theory for multi-armed bandits where the values are the modes of the reward distributions instead of the mean. We prove regret guarantees for the problems of top arm identification, top m-arms identification, contextual modal bandits, and infinite continuous arms top arm recovery. We show in simulations that our algorithms are robust to perturbation of the arms by adversarial noise sequences, thus rendering modal bandits an attractive choice in situations where the rewards may have outliers or adversarial corruptions.

Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives

We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their initial conditions, to provide a simple characterization of convergence rates. In many cases, closed-form expressions are obtained that relate algorithm parameters to the convergence rate. The analysis encompasses discrete time and continuous time, as well as time-invariant and time-variant formulations, and is not limited to a convex or Euclidean setting. In addition, the article rigorously establishes why symplectic discretization schemes are important for momentum-based optimization algorithms, and provides a characterization of algorithms that exhibit accelerated convergence.

Provable Meta-Learning of Linear Representations

Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning---a key tool for performing meta-learning---learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression---in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features, showing that our algorithms are optimal up to logarithmic factors.

On Thompson Sampling with Langevin Algorithms

Thompson sampling is a methodology for multi-armed bandit problems that is known to enjoy favorable performance in both theory and practice. It does, however, have a significant limitation computationally, arising from the need for samples from posterior distributions at every iteration. We propose two Markov Chain Monte Carlo (MCMC) methods tailored to Thompson sampling to address this issue. We construct quickly converging Langevin algorithms to generate approximate samples that have accuracy guarantees, and we leverage novel posterior concentration rates to analyze the regret of the resulting approximate Thompson sampling algorithm. Further, we specify the necessary hyper-parameters for the MCMC procedure to guarantee optimal instance-dependent frequentist regret while having low computational complexity. In particular, our algorithms take advantage of both posterior concentration and a sample reuse mechanism to ensure that only a constant number of iterations and a constant amount of data is needed in each round. The resulting approximate Thompson sampling algorithm has logarithmic regret and its computational complexity does not scale with the time horizon of the algorithm.

Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games

We consider multi-agent learning via online gradient descent (OGD) in a class of games called $\lambda$-cocoercive games, a broad class of games that admits many Nash equilibria and that properly includes strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on $\lambda$-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of the problem parameter (e.g., the cocoercive constant $\lambda$) and show, via a novel double-stopping-time technique, that this adaptive algorithm achieves the same finite-time last-iterate convergence rate as its non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results---first qualitative almost sure convergence, then quantitative finite-time convergence rates---all under non-decreasing step-sizes. These results fill in several gaps in the existing multi-agent online learning literature, where three aspects---finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms---have not been previously explored.

Robust Optimization for Fairness with Noisy Protected Groups

Many existing fairness criteria for machine learning involve equalizing or achieving some metric across \textit{protected groups} such as race or gender groups. However, practitioners trying to audit or enforce such group-based criteria can easily face the problem of noisy or biased protected group information. We study this important practical problem in two ways. First, we study the consequences of na{\"i}vely only relying on noisy protected groups: we provide an upper bound on the fairness violations on the true groups $G$ when the fairness criteria are satisfied on noisy groups $\hat{G}$. Second, we introduce two new approaches using robust optimization that, unlike the na{\"i}ve approach of only relying on $\hat{G}$, are guaranteed to satisfy fairness criteria on the true protected groups $G$ while minimizing a training objective. We provide theoretical guarantees that one such approach converges to an optimal feasible solution. Using two case studies, we empirically show that the robust approaches achieve better true group fairness guarantees than the na{\"i}ve approach.

Decision-Making with Auto-Encoding Variational Bayes

To make decisions based on a model fit by Auto-Encoding Variational Bayes (AEVB), practitioners typically use importance sampling to estimate a functional of the posterior distribution. The variational distribution found by AEVB serves as the proposal distribution for importance sampling. However, this proposal distribution may give unreliable (high variance) importance sampling estimates, thus leading to poor decisions. We explore how changing the objective function for learning the variational distribution, while continuing to learn the generative model based on the ELBO, affects the quality of downstream decisions. For a particular model, we characterize the error of importance sampling as a function of posterior variance and show that proposal distributions learned with evidence upper bounds are better. Motivated by these theoretical results, we propose a novel variant of the VAE. In addition to experimenting with MNIST, we present a full-fledged application of the proposed method to single-cell RNA sequencing. In this challenging instance of multiple hypothesis testing, the proposed method surpasses the current state of the art.