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On The Convergence Of Stochastic Primal-Dual Hybrid Gradient

Ahmet Alacaoglu, Volkan Cevher

In this paper, we analyze the recently proposed stochastic primal-dual hybrid gradient (SPDHG) algorithm and provide new theoretical results. In particular, we prove almost sure convergence of the iterates to a solution with convexity and linear convergence with further structure, using standard step sizes independent of strong convexity or other regularity constants. In the general convex case, we also prove the \scrO (1/k) convergence rate for the ergodic sequence, on expected primal-dual gap function. Our assumption for linear convergence is metric subregularity, which is satisfied for strongly convex-strongly concave problems in addition to many nonsmooth and/or nonstrongly convex problems, such as linear programs, Lasso, and support vector machines. We also provide numerical evidence showing that SPDHG with standard step sizes shows a competitive practical performance against its specialized strongly convex variant SPDHG-\mu and other state-of-the-art algorithms, including variance reduction methods.

SIAM PUBLICATIONS2022

Adaptation in Stochastic Algorithms: From Nonsmooth Optimization to Min-Max Problems and Beyond

Ahmet Alacaoglu

Stochastic gradient descent (SGD) and randomized coordinate descent (RCD) are two of the workhorses for training modern automated decision systems. Intriguingly, convergence properties of these methods are not well-established as we move away from the specific case of smooth minimization. In this dissertation, we focus on related problems of nonsmooth optimization and min-max optimization to improve the theoretical understanding of stochastic algorithms.First, we study SGD-based adaptive algorithms and propose a regret analysis framework overcoming the limitations of the existing ones in the convex case. In the nonconvex case, we prove convergence of an adaptive gradient algoritm for solving constrained weakly convex optimization, generalizing the previously known results on unconstrained smooth optimization. We also propose an algorithm combining Nesterov's smoothing with SGD to solve convex problems with infinitely many linear constraints, with optimal rates.Then, we move on to convex-concave min-max problems with bilinear coupling and analyze primal-dual coordinate descent (PDCD) algorithms. We obtain the first PDCD methods with the optimal

\mathcal{O}(1/k)

rate on the the standard optimality measure expected primal-dual gap, which was an open question since 2014. Our analysis also aims to explain the practical behavior of these algorithms by showing that the last iterate enjoys adaptive linear convergence without altering the parameters, depending on a certain error bound condition. Furthermore, we propose an algorithm combining the favorable properties of two branches of PDCD methods: the new method uses large step sizes with dense data and its per-iteration cost depends on the number of nonzeros of the data matrix. Thanks to these unique properties, this method enjoys compelling practical performance complementing its rigorous theoretical guarantees.Next, we consider monotone variational inequalities that generalize convex-concave min-max problems with nonbilinear coupling. We introduce variance reduced algorithms that converge under the same set of assumptions as their deterministic counterparts and improve the best-known complexities for solving convex-concave min-max problems with finite-sum structure. Optimality of our algorithms for this problem class is established in a recent work via matching lower bounds. Finally, we show our preliminary results on policy optimization methods for solving two player zero-sum Markov games for competitive reinforcement learning (RL). Even though this is a nonconvex-nonconcave min-max problem in general, thanks to the special structure, it is tractable to find an approximate Nash equilibrium. We introduce an algorithm that improves the best-known sample complexity of policy gradient methods. This development combines tools from RL and stochastic primal-dual optimization, showing the importance of techniques from convex-concave optimization.

EPFL2021

Convergence of adaptive algorithms for constrained weakly convex optimization

Ahmet Alacaoglu, Volkan Cevher, Yurii Malitskyi

We analyze the adaptive first order algorithm AMSGrad, for solving a constrained stochastic optimization problem with a weakly convex objective. We prove the

\tilde O(t^{-1/2})

rate of convergence for the squared norm of the gradient of Moreau envelope, which is the standard stationarity measure for this class of problems. It matches the known rates that adaptive algorithms enjoy for the specific case of unconstrained smooth nonconvex stochastic optimization. Our analysis works with mini-batch size of 1, constant first and second order moment parameters, and possibly unbounded optimization domains. Finally, we illustrate the applications and extensions of our results to specific problems and algorithms.

2021