Weiqiang Zheng

Mete Şeref Ahunbay, Weiqiang Zheng, Tao Lin

We show that in discretised first-price auctions with complete information, if the buyers learn to bid with online gradient ascent, in time-average the outcome is (almost) the efficient outcome of the second-price auction. Our proof rests on two novel innovations in the analysis of online gradient ascent in normal-form games, which may be useful in a wider range of applications. First, we develop a potential-function-based argument for the analysis of gradient ascent in normal-form games, allowing us to deduce that certain strategies will not be played in time-average. We provide sufficient conditions which ensure this argument can be applied iteratively, resulting in a procedure reminiscent of iterative elimination of dominated strategies. Second, we develop a novel class of cubic "candidate potential functions", classifying a family of quadratic strategy modifications on the probability simplex against which online gradient ascent incurs no regret.

Yue Wu, Weiqiang Zheng, Yang Cai et al.

We revisit the convergence guarantees of the Extragradient (EG) method for unconstrained biaffine min-max optimization. It is known that EG with a fixed stepsize achieves a $Θ(T^{-1/2})$ last-iterate convergence rate, which is slower than the optimal $\mathcal{O}(T^{-1})$ rate attainable by incorporating additional mechanisms such as anchoring. Motivated by recent advances showing that dynamic stepsizes alone can significantly accelerate gradient descent, we ask whether dynamic stepsizes can similarly accelerate the last-iterate convergence of EG. We present the first positive result in this direction. Specifically, we provide a deterministic dynamic stepsize schedule that accelerates the convergence rate of EG to $\mathcal{O}(T^{-2/3+\varepsilon})$ for any $\varepsilon > 0$. We also show that this rate is tight when the extrapolation and update steps of EG use the same stepsize. We then show that allowing different stepsizes for the extrapolation and update steps further improves the convergence rate to the near-optimal $\mathcal{O}(T^{-1+\varepsilon})$. Our analysis reduces stepsize scheduling to an optimization problem, whose solution leads to a stepsize schedule that follows (a discretization of) a power-law distribution. Our proposed stepsize schedules and analysis extend to other methods, such as Optimistic Gradient (OG), and suggest broader applicability to general min-max optimization problems.

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, Gabriele Farina, Julien Grand-Clément et al.

We study last-iterate convergence properties of algorithms for solving two-player zero-sum games based on Regret Matching$^+$ (RM$^+$). Despite their widespread use for solving real games, virtually nothing is known about their last-iterate convergence. A major obstacle to analyzing RM-type dynamics is that their regret operators lack Lipschitzness and (pseudo)monotonicity. We start by showing numerically that several variants used in practice, such as RM$^+$, predictive RM$^+$ and alternating RM$^+$, all lack last-iterate convergence guarantees even on a simple $3\times 3$ matrix game. We then prove that recent variants of these algorithms based on a smoothing technique, extragradient RM$^{+}$ and smooth Predictive RM$^+$, enjoy asymptotic last-iterate convergence (without a rate), $1/\sqrt{t}$ best-iterate convergence, and when combined with restarting, linear-rate last-iterate convergence. Our analysis builds on a new characterization of the geometric structure of the limit points of our algorithms, marking a significant departure from most of the literature on last-iterate convergence. We believe that our analysis may be of independent interest and offers a fresh perspective for studying last-iterate convergence in algorithms based on non-monotone operators.

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

We consider bidding in repeated Bayesian first-price auctions. Bidding algorithms that achieve optimal regret have been extensively studied, but their strategic robustness to the seller's manipulation remains relatively underexplored. Bidding algorithms based on no-swap-regret algorithms achieve both desirable properties, but are suboptimal in terms of statistical and computational efficiency. In contrast, online gradient ascent is the only algorithm that achieves $O(\sqrt{TK})$ regret and strategic robustness [KSS24], where $T$ denotes the number of auctions and $K$ the number of bids. In this paper, we explore whether simple online linear optimization (OLO) algorithms suffice for bidding algorithms with both desirable properties. Our main result shows that sublinear linearized regret is sufficient for strategic robustness. Specifically, we construct simple black-box reductions that convert any OLO algorithm into a strategically robust no-regret bidding algorithm, in both known and unknown value distribution settings. For the known value distribution case, our reduction yields a bidding algorithm that achieves $O(\sqrt{T \log K})$ regret and strategic robustness (with exponential improvement on the $K$-dependence compared to [KSS24]). For the unknown value distribution case, our reduction gives a bidding algorithm with high-probability $O(\sqrt{T (\log K+\log(T/δ)})$ regret and strategic robustness, while removing the bounded density assumption made in [KSS24].

Yang Cai, Weiqiang Zheng

Aligning large language models (LLMs) to serve users with heterogeneous and potentially conflicting preferences is a central challenge for personalized and trustworthy AI. We formalize an ideal notion of universal alignment through test-time scaling: for each prompt, the model produces $k\ge 1$ candidate responses and a user selects their preferred one. We introduce $(k,f(k))$-robust alignment, which requires the $k$-output model to have win rate $f(k)$ against any other single-output model, and asymptotic universal alignment (U-alignment), which requires $f(k)\to 1$ as $k\to\infty$. Our main result characterizes the optimal convergence rate: there exists a family of single-output policies whose $k$-sample product policies achieve U-alignment at rate $f(k)=\frac{k}{k+1}$, and no method can achieve a faster rate in general. We show that popular post-training methods, including Nash learning from human feedback (NLHF), can fundamentally underutilize the benefits of test-time scaling. Even though NLHF is optimal for $k=1$, sampling from the resulting (often deterministic) policy cannot guarantee win rates above $\tfrac{1}{2}$ except for an arbitrarily small slack. This stems from a lack of output diversity: existing alignment methods can collapse to a single majority-preferred response, making additional samples redundant. In contrast, our approach preserves output diversity and achieves the optimal test-time scaling rate. In particular, we propose a family of symmetric multi-player alignment games and prove that any symmetric Nash equilibrium policy of the $(k+1)$-player alignment game achieves the optimal $(k,\frac{k}{k+1})$-robust alignment. Finally, we provide theoretical convergence guarantees for self-play learning dynamics in these games and extend the framework to opponents that also generate multiple responses.

Yixin Liu, Argyris Oikonomou, Weiqiang Zheng et al.

Many alignment methods, including reinforcement learning from human feedback (RLHF), rely on the Bradley-Terry reward assumption, which is not always sufficient to capture the full range and complexity of general human preferences. We explore RLHF under a general preference framework by modeling the alignment problem as a two-player zero-sum game in a game-theoretic framework, where the Nash equilibrium policy guarantees a 50% win rate against any competing policy. However, previous self-play algorithms for finding the Nash policy either diverge or only converge to a Nash policy in a modified game, even in a simple synthetic setting, thereby failing to maintain the 50% win rate guarantee against all other policies. We propose a meta-algorithm, Convergent Meta Alignment Algorithm (COMAL), for language model alignment with general preferences, inspired by convergent algorithms in game theory. We provide theoretical analysis that our meta-algorithm converges to an exact Nash policy in the last iterate and demonstrate its effectiveness on a range of synthetic and preference optimization datasets. COMAL is simple and can be integrated with many existing methods designed for preference optimization with minimal changes, and empirically it consistently maintains above 60.2% and 56.8% win rates, when applied to Llama-3-8B-Instruct and Qwen2.5-7B, against all compared algorithms under controlled evaluations.

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 $\widetilde{O}(\sqrt{d} T^{-\frac{1}{6}})$.

Yang Cai, Gabriele Farina, Julien Grand-Clément et al.

Non-ergodic convergence of learning dynamics in games is widely studied recently because of its importance in both theory and practice. Recent work (Cai et al., 2024) showed that a broad class of learning dynamics, including Optimistic Multiplicative Weights Update (OMWU), can exhibit arbitrarily slow last-iterate convergence even in simple $2 \times 2$ matrix games, despite many of these dynamics being known to converge asymptotically in the last iterate. It remains unclear, however, whether these algorithms achieve fast non-ergodic convergence under weaker criteria, such as best-iterate convergence. We show that for $2\times 2$ matrix games, OMWU achieves an $O(T^{-1/6})$ best-iterate convergence rate, in stark contrast to its slow last-iterate convergence in the same class of games. Furthermore, we establish a lower bound showing that OMWU does not achieve any polynomial random-iterate convergence rate, measured by the expected duality gaps across all iterates. This result challenges the conventional wisdom that random-iterate convergence is essentially equivalent to best-iterate convergence, with the former often used as a proxy for establishing the latter. Our analysis uncovers a new connection to dynamic regret and presents a novel two-phase approach to best-iterate convergence, which could be of independent interest.

Jiahao Zhang, Tao Lin, Weiqiang Zheng et al. · harvard, pku

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.

Yang Cai, Gabriele Farina, Julien Grand-Clément et al.

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.

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.

Xiaotie Deng, Xinyan Hu, Tao Lin et al.

The convergence properties of learning dynamics in repeated auctions is a timely and important question, with numerous applications in, e.g., online advertising markets. This work focuses on repeated first-price auctions where bidders with fixed values learn to bid using mean-based algorithms -- a large class of online learning algorithms that include popular no-regret algorithms such as Multiplicative Weights Update and Follow the Perturbed Leader. We completely characterize the learning dynamics of mean-based algorithms, under two notions of convergence: (1) time-average: the fraction of rounds where bidders play a Nash equilibrium converges to 1; (2) last-iterate: the mixed strategy profile of bidders converges to a Nash equilibrium. Specifically, the results depend on the number of bidders with the highest value: - If the number is at least three, the dynamics almost surely converges to a Nash equilibrium of the auction, in both time-average and last-iterate. - If the number is two, the dynamics almost surely converges to a Nash equilibrium in time-average but not necessarily last-iterate. - If the number is one, the dynamics may not converge to a Nash equilibrium in time-average or last-iterate. Our discovery opens up new possibilities in the study of the convergence of learning dynamics.