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Register a new userAccelerated Linear Convergence of Stochastic Momentum Methods in Wasserstein Distances
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Momentum methods such as Polyaks heavy ball (HB) method, Nesterovs accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stochastic oracle model where noisy estimates of the gradients are available. For strongly convex problems, we show that the distribution of the iterates of AG converges with the accelerated $O(sqrt{kappa}log(1/varepsilon))$ linear rate to a ball of radius $varepsilon$ centered at a unique invariant distribution in the 1-Wasserstein metric where $kappa$ is the condition number as long as the noise variance is smaller than an explicit upper bound we can provide. Our analysis also certifies linear convergence rates as a function of the stepsize, momentum parameter and the noise variance; recovering the accelerated rates in the noiseless case and quantifying the level of noise that can be tolerated to achieve a given performance. In the special case of strongly convex quadratic objectives, we can show accelerated linear rates in the $p$-Wasserstein metric for any $pgeq 1$ with improved sensitivity to noise for both AG and HB through a non-asymptotic analysis under some additional assumptions on the noise structure. Our analysis for HB and AG also leads to improved non-asymptotic convergence bounds in suboptimality for both deterministic and stochastic settings which is of independent interest. To the best of our knowledge, these are the first linear convergence results for stochastic momentum methods under the stochastic oracle model. We also extend our results to the APG method and weakly convex functions showing accelerated rates when the noise magnitude is sufficiently small.
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Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Ascent (GDA) algorithm. In the special case of mixtures of two linear regression models, we show that WMLR enjoys global convergence and generalization guarantees. We prove that WMLRs sample complexity grows linearly with the dimension of data. Finally, we discuss the application of WMLR to the federated learning task where the training samples are collected by multiple agents in a network. Unlike the Expectation Maximization algorithm, WMLR directly extends to the distributed, federated learning setting. We support our theoretical results through several numerical experiments, which highlight our frameworks ability to handle the federated learning setting with mixture models.
We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete probability measures, which are endowed with Wasserstein distance metrics. We propose several variants of this problem, which admit fast optimization algorithms, by exploiting the connection to the problem of finding Wasserstein barycenters. Consistency properties are established for the estimates of both local and global clusters. Finally, experimental results with both synthetic and real data are presented to demonstrate the flexibility and scalability of the proposed approach.
We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.
We undertake a precise study of the asymptotic and non-asymptotic properties of stochastic approximation procedures with Polyak-Ruppert averaging for solving a linear system $bar{A} theta = bar{b}$. When the matrix $bar{A}$ is Hurwitz, we prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity. The CLT characterizes the exact asymptotic covariance matrix, which is the sum of the classical Polyak-Ruppert covariance and a correction term that scales with the step size. Under assumptions on the tail of the noise distribution, we prove a non-asymptotic concentration inequality whose main term matches the covariance in CLT in any direction, up to universal constants. When the matrix $bar{A}$ is not Hurwitz but only has non-negative real parts in its eigenvalues, we prove that the averaged LSA procedure actually achieves an $O(1/T)$ rate in mean-squared error. Our results provide a more refined understanding of linear stochastic approximation in both the asymptotic and non-asymptotic settings. We also show various applications of the main results, including the study of momentum-based stochastic gradient methods as well as temporal difference algorithms in reinforcement learning.
Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely many training samples that will perform well on unseen test samples. This learning task is difficult even if all training and test samples are drawn from the same distribution---especially if the dimension of the uncertainty is large relative to the training sample size. Wasserstein distributionally robust optimization seeks data-driven decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In this tutorial we will argue that this approach has many conceptual and computational benefits. Most prominently, the optimal decisions can often be computed by solving tractable convex optimization problems, and they enjoy rigorous out-of-sample and asymptotic consistency guarantees. We will also show that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.
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