arXiv subjects

Weiqiang Zheng

Publications and source records attributed to Weiqiang Zheng.

28 records · Page 2Linked to original sources

arXiv2025

Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Self-play via online learning is one of the premier ways to solve large-scale two-player zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $δ>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/δ$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

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arXiv2024

Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusion

We study constrained comonotone min-max optimization, a structured class of nonconvex-nonconcave min-max optimization problems, and their generalization to comonotone inclusion. In our first contribution, we extend the Extra Anchored Gradient (EAG) algorithm, originally proposed by Yoon and Ryu (2021) for unconstrained min-max optimization, to constrained comonotone min-max optimization and comonotone inclusion, achieving an optimal convergence rate of $O\left(\frac{1}{T}\right)$ among all first-order methods. Additionally, we prove that the algorithm's iterations converge to a point in the solution set. In our second contribution, we extend the Fast Extra Gradient (FEG) algorithm, as developed by Lee and Kim (2021), to constrained comonotone min-max optimization and comonotone inclusion, achieving the same $O\left(\frac{1}{T}\right)$ convergence rate. This rate is applicable to the broadest set of comonotone inclusion problems yet studied in the literature. Our analyses are based on simple potential function arguments, which might be useful for analyzing other accelerated algorithms.

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arXiv2024

Near-Optimal Policy Optimization for Correlated Equilibrium in General-Sum Markov Games

We study policy optimization algorithms for computing correlated equilibria in multi-player general-sum Markov Games. Previous results achieve $O(T^{-1/2})$ convergence rate to a correlated equilibrium and an accelerated $O(T^{-3/4})$ convergence rate to the weaker notion of coarse correlated equilibrium. In this paper, we improve both results significantly by providing an uncoupled policy optimization algorithm that attains a near-optimal $\tilde{O}(T^{-1})$ convergence rate for computing a correlated equilibrium. Our algorithm is constructed by combining two main elements (i) smooth value updates and (ii) the optimistic-follow-the-regularized-leader algorithm with the log barrier regularizer.

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arXiv2023

Learning Thresholds with Latent Values and Censored Feedback

In this paper, we investigate a problem of actively learning threshold in latent space, where the unknown reward $g(γ, v)$ depends on the proposed threshold $γ$ and latent value $v$ and it can be $only$ achieved if the threshold is lower than or equal to the unknown latent value. This problem has broad applications in practical scenarios, e.g., reserve price optimization in online auctions, online task assignments in crowdsourcing, setting recruiting bars in hiring, etc. We first characterize the query complexity of learning a threshold with the expected reward at most $ε$ smaller than the optimum and prove that the number of queries needed can be infinitely large even when $g(γ, v)$ is monotone with respect to both $γ$ and $v$. On the positive side, we provide a tight query complexity $\tildeΘ(1/ε^3)$ when $g$ is monotone and the CDF of value distribution is Lipschitz. Moreover, we show a tight $\tildeΘ(1/ε^3)$ query complexity can be achieved as long as $g$ satisfies one-sided Lipschitzness, which provides a complete characterization for this problem. Finally, we extend this model to an online learning setting and demonstrate a tight $Θ(T^{2/3})$ regret bound using continuous-arm bandit techniques and the aforementioned query complexity results.

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arXiv2023

Uncoupled and Convergent Learning in Two-Player Zero-Sum Markov Games with Bandit Feedback

We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is uncoupled, convergent, and rational, with non-asymptotic convergence rates. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $O(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $O(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a path convergence rate, which is a new notion of convergence we defined, of $O(t^{-\frac{1}{10}})$. Our algorithm removes the coordination and prior knowledge requirement of [Wei et al., 2021], which pursued the same goals as us for irreducible Markov games. Our algorithm is related to [Chen et al., 2021, Cen et al., 2021] and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

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arXiv2023

Doubly Optimal No-Regret Learning in Monotone Games

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow $O(\frac{1}{\sqrt{T}})$ last-iterate convergence rate to a Nash equilibrium. While the $O(\frac{1}{\sqrt{T}})$ rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal $O(\sqrt{T})$ regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal $O(\frac{1}{T})$ last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an $O(\log T)$ individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art $O(\sqrt{T})$ bound.

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arXiv2023

Accelerated Single-Call Methods for Constrained Min-Max Optimization

We study first-order methods for constrained min-max optimization. Existing methods either require two gradient calls or two projections in each iteration, which may be costly in some applications. In this paper, we first show that a variant of the Optimistic Gradient (OG) method, a single-call single-projection algorithm, has $O(\frac{1}{\sqrt{T}})$ best-iterate convergence rate for inclusion problems with operators that satisfy the weak Minty variation inequality (MVI). Our second result is the first single-call single-projection algorithm -- the Accelerated Reflected Gradient (ARG) method that achieves the optimal $O(\frac{1}{T})$ last-iterate convergence rate for inclusion problems that satisfy negative comonotonicity. Both the weak MVI and negative comonotonicity are well-studied assumptions and capture a rich set of non-convex non-concave min-max optimization problems. Finally, we show that the Reflected Gradient (RG) method, another single-call single-projection algorithm, has $O(\frac{1}{\sqrt{T}})$ last-iterate convergence rate for constrained convex-concave min-max optimization, answering an open problem of [Heish et al, 2019]. Our convergence rates hold for standard measures such as the tangent residual and the natural residual.

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arXiv2022

Beyond the Worst Case: Semi-Random Complexity Analysis of Winner Determination

The computational complexity of winner determination is a classical and important problem in computational social choice. Previous work based on worst-case analysis has established NP-hardness of winner determination for some classic voting rules, such as Kemeny, Dodgson, and Young. In this paper, we revisit the classical problem of winner determination through the lens of semi-random analysis, which is a worst average-case analysis where the preferences are generated from a distribution chosen by the adversary. Under a natural class of semi-random models that are inspired by recommender systems, we prove that winner determination remains hard for Dodgson, Young, and some multi-winner rules such as the Chamberlin-Courant rule and the Monroe rule. Under another natural class of semi-random models that are extensions of the Impartial Culture, we show that winner determination is hard for Kemeny, but is easy for Dodgson. This illustrates an interesting separation between Kemeny and Dodgson.

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arXiv2022

Tight Last-Iterate Convergence of the Extragradient and the Optimistic Gradient Descent-Ascent Algorithm for Constrained Monotone Variational Inequalities

The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient algorithm by Korpelevich [1976] and the optimistic gradient descent-ascent algorithm by Popov [1980] are arguably the two most classical and popular methods for solving monotone variational inequalities. Despite their long histories, the following major problem remains open. What is the last-iterate convergence rate of the extragradient algorithm or the optimistic gradient descent-ascent algorithm for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing that both the extragradient algorithm and the optimistic gradient descent-ascent algorithm have a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020a,b]. Our rate is measured in terms of the standard gap function. At the core of our results lies a non-standard performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. We use the tangent residual (or a slight variation of the tangent residual) as the the potential function in our analysis of the extragradient algorithm (or the optimistic gradient descent-ascent algorithm) and prove that it is non-increasing between two consecutive iterates.

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arXiv2020

The Smoothed Complexity of Computing Kemeny and Slater Rankings

The computational complexity of winner determination under common voting rules is a classical and fundamental topic in the field of computational social choice. Previous work has established the NP-hardness of winner determination under some commonly-studied voting rules, especially the Kemeny rule and the Slater rule. In a recent blue-sky paper, Baumeister, Hogrebe, and Rothe (2020) questioned the relevance of the worst-case nature of NP-hardness in social choice and proposed to conduct smoothed complexity analysis (Spielman and Teng 2009) under Blaser and Manthey (2015)'s framework. In this paper, we develop the first smoothed complexity results for winner determination in voting. We illustrate the inappropriateness of Blaser and Manthey (2015)'s smoothed complexity framework in social choice contexts by proving a paradoxical result, which states that the exponential-time brute force search algorithm is smoothed poly-time according to their definition. We then prove the smoothed hardness of Kemeny and Slater using the classical smoothed complexity analysis, and prove a parameterized typical-case smoothed easiness result for Kemeny. Overall, our results show that smoothed complexity analysis in computational social choice is a challenging and fruitful topic.

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