Yang Cai

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

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.

Yang Cai, Weiqiang Zheng

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.

Yang Cai, Argyris Oikonomou, Weiqiang Zheng

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.

Yang Cai, Argyris Oikonomou, Weiqiang Zheng

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.

Yang Cai, Constantinos Daskalakis, Haipeng Luo et al.

Learning and computation of equilibria are central problems in game theory, theory of computation, and artificial intelligence. In this work, we introduce proximal regret, a new notion of regret based on proximal operators that lies strictly between external and swap regret. When every player employs a no-proximal-regret algorithm in a general convex game, the empirical distribution of play converges to proximal correlated equilibria (PCE), a refinement of coarse correlated equilibria. Our framework unifies several emerging notions in online learning and game theory-such as gradient equilibrium and semicoarse correlated equilibrium-and introduces new ones. Our main result shows that the classic Online Gradient Descent (GD) algorithm achieves an optimal $O(\sqrt{T})$ bound on proximal regret, revealing that GD, without modification, minimizes a stronger regret notion than external regret. This provides a new explanation for the empirically superior performance of gradient descent in online learning and games. We further extend our analysis to Mirror Descent in the Bregman setting and to Optimistic Gradient Descent, which yields faster convergence in smooth convex games.

Yang Cai, Michael I. Jordan, Tianyi Lin et al.

Equilibrium computation on Riemannian manifolds provides a unifying framework for numerous problems in machine learning and data analytics. One of the simplest yet most fundamental methods is Riemannian gradient descent (RGD). While its Euclidean counterpart has been extensively studied, it remains unclear how the manifold curvature affects RGD in game-theoretic settings. This paper addresses this gap by establishing new convergence results for \textit{geodesic strongly monotone} games. Our key result shows that RGD attains last-iterate linear convergence in a \textit{geometry-agnostic} fashion, a key property for applications in machine learning. We extend this guarantee to stochastic and adaptive variants -- SRGD and FARGD -- and establish that: (i) the sample complexity of SRGD is geometry-agnostic and optimal with respect to noise; (ii) FARGD matches the convergence rate of its non-adaptive counterpart up to constant factors, while avoiding reliance on the condition number. Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings, underscoring the surprising power of RGD -- despite its simplicity -- in solving a wide spectrum of machine learning problems.

Yang Cai, Weiqiang Zheng

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.

Yang Cai, Zhe Feng, Christopher Liaw et al.

We propose a new Markov Decision Process (MDP) model for ad auctions to capture the user response to the quality of ads, with the objective of maximizing the long-term discounted revenue. By incorporating user response, our model takes into consideration all three parties involved in the auction (advertiser, auctioneer, and user). The state of the user is modeled as a user-specific click-through rate (CTR) with the CTR changing in the next round according to the set of ads shown to the user in the current round. We characterize the optimal mechanism for this MDP as a Myerson's auction with a notion of modified virtual value, which relies on the value distribution of the advertiser, the current user state, and the future impact of showing the ad to the user. Leveraging this characterization, we design a sample-efficient and computationally-efficient algorithm which outputs an approximately optimal policy that requires only sample access to the true MDP and the value distributions of the bidders. Finally, we propose a simple mechanism built upon second price auctions with personalized reserve prices and show it can achieve a constant-factor approximation to the optimal long term discounted revenue.

Yang Cai, Constantinos Daskalakis, Haipeng Luo et al.

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $Φ$-equilibria introduced by Greenwald and Jafari [2003], which is guaranteed to exist for an arbitrary set of strategy modifications $Φ$ even in non-concave games [Stolz and Lugosi, 2007]. However, the tractability of $Φ$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $Φ$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $Φ$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Φ$-equilibria. Additionally, we explore cases where $Φ$ is infinite but consists of local modifications. We show that approximating local $Φ$-equilibria beyond the first-order stationary regime is computationally intractable. In contrast, within this regime, we show Online Gradient Descent efficiently converges to $Φ$-equilibria for several natural infinite families of modifications, including a new structural family of modifications inspired by the well-studied proximal operator.

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

The convergence of online learning algorithms in games under self-play is a fundamental question in game theory and machine learning. Among various notions of convergence, last-iterate convergence is particularly desirable, as it reflects the actual decisions made by the learners and captures the day-to-day behavior of the learning dynamics. While many algorithms are known to converge in the average-iterate, achieving last-iterate convergence typically requires considerably more effort in both the design and the analysis of the algorithm. Somewhat surprisingly, we show in this paper that for a large family of games, there exists a simple black-box reduction that transforms the average iterates of an uncoupled learning dynamics into the last iterates of a new uncoupled learning dynamics, thus also providing a reduction from last-iterate convergence to average-iterate convergence. Our reduction applies to games where each player's utility is linear in both their own strategy and the joint strategy of all opponents. This family includes two-player bimatrix games and generalizations such as multi-player polymatrix games. By applying our reduction to the Optimistic Multiplicative Weights Update algorithm, we obtain new state-of-the-art last-iterate convergence rates for uncoupled learning dynamics in multi-player zero-sum polymatrix games: (1) an $O(\frac{\log d}{T})$ last-iterate convergence rate under gradient feedback, representing an exponential improvement in the dependence on the dimension $d$ (i.e., the maximum number of actions available to either player); and (2) an $\widetilde{O}(d^{\frac{1}{5}} T^{-\frac{1}{5}})$ last-iterate convergence rate under bandit feedback, improving upon the previous best rates of $\widetilde{O}(\sqrt{d} T^{-\frac{1}{8}})$ and $\widetilde{O}(\sqrt{d} T^{-\frac{1}{6}})$.

Xi Chen, Yang Cai, Yuan Wu et al.

Recently, MBConv blocks, initially designed for efficiency in resource-limited settings and later adapted for cutting-edge image classification performances, have demonstrated significant potential in image classification tasks. Despite their success, their application in semantic segmentation has remained relatively unexplored. This paper introduces a novel adaptation of MBConv blocks specifically tailored for semantic segmentation. Our modification stems from the insight that semantic segmentation requires the extraction of more detailed spatial information than image classification. We argue that to effectively perform multi-scale semantic segmentation, each branch of a U-Net architecture, regardless of its resolution, should possess equivalent segmentation capabilities. By implementing these changes, our approach achieves impressive mean Intersection over Union (IoU) scores of 84.5% and 84.0% on the Cityscapes test and validation datasets, respectively, demonstrating the efficacy of our proposed modifications in enhancing semantic segmentation performance.

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

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.

Yang Cai, Constantinos Daskalakis

Machine learning has developed a variety of tools for learning and representing high-dimensional distributions with structure. Recent years have also seen big advances in designing multi-item mechanisms. Akin to overfitting, however, these mechanisms can be extremely sensitive to the Bayesian prior that they target, which becomes problematic when that prior is only approximately known. At the same time, even if access to the exact Bayesian prior is given, it is known that optimal or even approximately optimal multi-item mechanisms run into sample, computational, representation and communication intractability barriers. We consider a natural class of multi-item mechanism design problems with very large numbers of items, but where the bidders' value distributions can be well-approximated by a topic model akin to those used in recommendation systems with very large numbers of possible recommendations. We propose a mechanism design framework for this setting, building on a recent robustification framework by Brustle et al., which disentangles the statistical challenge of estimating a multi-dimensional prior from the task of designing a good mechanism for it, and robustifies the performance of the latter against the estimation error of the former. We provide an extension of this framework appropriate for our setting, which allows us to exploit the expressive power of topic models to reduce the effective dimensionality of the mechanism design problem and remove the dependence of its computational, communication and representation complexity on the number of items.

Johaness Brustle, Yang Cai, Constantinos Daskalakis

We study the sample complexity of learning revenue-optimal multi-item auctions. We obtain the first set of positive results that go beyond the standard but unrealistic setting of item-independence. In particular, we consider settings where bidders' valuations are drawn from correlated distributions that can be captured by Markov Random Fields or Bayesian Networks -- two of the most prominent graphical models. We establish parametrized sample complexity bounds for learning an up-to-$\varepsilon$ optimal mechanism in both models, which scale polynomially in the size of the model, i.e.~the number of items and bidders, and only exponential in the natural complexity measure of the model, namely either the largest in-degree (for Bayesian Networks) or the size of the largest hyper-edge (for Markov Random Fields). We obtain our learnability results through a novel and modular framework that involves first proving a robustness theorem. We show that, given only ``approximate distributions'' for bidder valuations, we can learn a mechanism whose revenue is nearly optimal simultaneously for all ``true distributions'' that are close to the ones we were given in Prokhorov distance. Thus, to learn a good mechanism, it suffices to learn approximate distributions. When item values are independent, learning in Prokhorov distance is immediate, hence our framework directly implies the main result of Gonczarowski and Weinberg. When item values are sampled from more general graphical models, we combine our robustness theorem with novel sample complexity results for learning Markov Random Fields or Bayesian Networks in Prokhorov distance, which may be of independent interest. Finally, in the single-item case, our robustness result can be strengthened to hold under an even weaker distribution distance, the Lévy distance.

Jessie Huang, Fa Wu, Doina Precup et al.

We propose a framework for ensuring safe behavior of a reinforcement learning agent when the reward function may be difficult to specify. In order to do this, we rely on the existence of demonstrations from expert policies, and we provide a theoretical framework for the agent to optimize in the space of rewards consistent with its existing knowledge. We propose two methods to solve the resulting optimization: an exact ellipsoid-based method and a method in the spirit of the "follow-the-perturbed-leader" algorithm. Our experiments demonstrate the behavior of our algorithm in both discrete and continuous problems. The trained agent safely avoids states with potential negative effects while imitating the behavior of the expert in the other states.

Yang Cai, Constantinos Daskalakis

We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given "approximate distributions," and we seek to compute an auction whose revenue is approximately optimal simultaneously for all "true distributions" that are close to the given ones. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of previously run, potentially non-truthful auctions. Our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works, which have provided algorithms that learn approximately optimal auctions in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting. Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension, and are of independent interest.

Yang Cai, Constantinos Daskalakis, Christos H. Papadimitriou

We propose an optimum mechanism for providing monetary incentives to the data sources of a statistical estimator such as linear regression, so that high quality data is provided at low cost, in the sense that the sum of payments and estimation error is minimized. The mechanism applies to a broad range of estimators, including linear and polynomial regression, kernel regression, and, under some additional assumptions, ridge regression. It also generalizes to several objectives, including minimizing estimation error subject to budget constraints. Besides our concrete results for regression problems, we contribute a mechanism design framework through which to design and analyze statistical estimators whose examples are supplied by workers with cost for labeling said examples.