Since Polyak’s pioneering work, heavy ball (HB) momentum has been widely studied in minimization. However, its role in min-max games remains largely unexplored. As a key component of practical min-max algorithms like Adam, this gap limits their effectiveness. In this paper, we present a continuous-time analysis for HB with simultaneous and alternating update schemes in min-max games. Locally, we prove smaller momentum enhances algorithmic stability by enabling local convergence across a wider range of step sizes, with alternating updates generally converging faster. Globally, we study the implicit regularization of HB, and find smaller momentum guides algorithms trajectories towards shallower slope regions of the loss landscapes, with alternating updates amplifying this effect. Surprisingly, all these phenomena differ from those observed in minimization, where larger momentum yields similar effects. Our results reveal fundamental differences between HB in min-max games and minimization, and numerical experiments further validate our theoretical results.

We investigate the accuracy of prediction in deterministic learning dynamics of zero-sum games with random initializations, specifically focusing on observer uncertainty and its relationship to the evolution of covariances. Zero-sum games are a prominent field of interest in machine learning due to their various applications. Concurrently, the accuracy of prediction in dynamical systems from mechanics has long been a classic subject of investigation since the discovery of the Heisenberg Uncertainty Principle. This principle employs covariance and standard deviation of particle states to measure prediction accuracy. In this study, we bring these two approaches together to analyze the Follow-the-Regularized-Leader (FTRL) algorithm in two-player zero-sum games. We provide growth rates of covariance information for continuous-time FTRL, as well as its two canonical discretization methods (Euler and Symplectic). A Heisenberg-type inequality is established for FTRL. Our analysis and experiments also show that employing Symplectic discretization enhances the accuracy of prediction in learning dynamics.

Last-iterate behaviors of learning algorithms in repeated two-player zero-sum games have been extensively studied due to their wide applications in machine learning and related tasks. Typical algorithms that exhibit the last-iterate convergence property include optimistic and extra-gradient methods. However, most existing results establish these properties under the assumption that the game is time-independent. Recently, (Feng et al., 2023) studied the last-iterate behaviors of optimistic and extra-gradient methods in games with a time-varying payoff matrix, and proved that in an unconstrained periodic game, extra-gradient method converges to the equilibrium while optimistic method diverges. This finding challenges the conventional wisdom that these two methods are expected to behave similarly as they do in time-independent games. However, compared to unconstrained games, games with constrains are more common both in practical and theoretical studies. In this paper, we investigate the last-iterate behaviors of optimistic and extra-gradient methods in the constrained periodic games, demonstrating that similar separation results for last-iterate convergence also hold in this setting.

In this work, we propose introduce a variant of online stochastic gradient descent and prove it converges to Nash equilibria and simultaneously it has sublinear regret for the class of congestion games in the semi-bandit feedback setting. Our proposed method admits convergence rates depending only polynomially on the number of players and the number of facilities, but not on the size of the action set, which can be exponentially large in terms of the number of facilities. Moreover, the running time of our method has polynomial-time dependence on the implicit description of the game. Our analysis exploits techniques from convex geometry, in particular Caratheodory's theorem and recent advances in non-convex stochastic optimization. This work improves upon and answers an open question from (Cui et al 2022).

Last-iterate convergence has received extensive study in two player zero-sum games starting from bilinear, convex-concave up to settings that satisfy the MVI condition. Typical methods that exhibit last-iterate convergence for the aforementioned games include extra-gradient (EG) and optimistic gradient descent ascent (OGDA). However, all the established last-iterate convergence results hold for the restrictive setting where the underlying repeated game does not change over time. Recently, a line of research has focused on regret analysis of OGDA in time-varying games, i.e., games where payoffs evolve with time; the last-iterate behavior of OGDA and EG in time-varying environments remains unclear though. In this paper, we study the last-iterate behavior of various algorithms in two types of unconstrained, time-varying, bilinear zero-sum games: periodic and convergent perturbed games. These models expand upon the usual repeated game formulation and incorporate external environmental factors, such as the seasonal effects on species competition and vanishing external noise. In periodic games, we prove that EG will converge while OGDA and momentum method will diverge. This is quite surprising, as to the best of our knowledge, it is the first result that indicates EG and OGDA have qualitatively different last-iterate behaviors and do not exhibit similar behavior. In convergent perturbed games, we prove all these algorithms converge as long as the game itself stabilizes with a faster rate than $1/t$.

Adaptive first-order methods in optimization have widespread ML applications due to their ability to adapt to non-convex landscapes. However, their convergence guarantees are typically stated in terms of vanishing gradient norms, which leaves open the issue of converging to undesirable saddle points (or even local maxima). In this paper, we focus on the AdaGrad family of algorithms - from scalar to full-matrix preconditioning - and we examine the question of whether the method’s trajectories avoid saddle points. A major challenge that arises here is that AdaGrad’s step-size (or, more accurately, the method’s preconditioner) evolves over time in a filtration-dependent way, i.e., as a function of all gradients observed in earlier iterations; as a result, avoidance results for methods with a constant or vanishing step-size do not apply. We resolve this challenge by combining a series of step-size stabilization arguments with a recursive representation of the AdaGrad preconditioner that allows us to employ center-stable techniques and ultimately show that the induced trajectories avoid saddle points from almost any initial condition.

We consider nonconvex optimization problem with constraint that is a product of simplices. A commonly used algorithm in solving this type of problem is the Multiplicative Weights Update (MWU), an algorithm that is widely used in game theory, machine learning and multi agent systems. Despite it has been known that MWU avoids saddle points, there is a question that remains unaddressed: ``Is there an accelerated version of MWU that avoids saddle points provably?'' In this paper we provide a positive answer to above question. We provide an accelerated MWU based on Riemannian Accelerated Gradient Descent, and prove that the Riemannian Accelerated Gradient Descent, thus the accelerated MWU, avoid saddle points.

In a recent series of papers it has been established that variants of Gradient Descent/Ascent and Mirror Descent exhibit last iterate convergence in convex-concave zero-sum games. Specifically, Daskalakis et al 2018, Liang-Stokes 2019, show last iterate convergence of the so called “Optimistic Gradient Descent/Ascent" for the case of \textit{unconstrained} min-max optimization. Moreover, in Mertikopoulos et al 2019 the authors show that Mirror Descent with an extra gradient step displays last iterate convergence for convex-concave problems (both constrained and unconstrained), though their algorithm uses \textit{vanishing stepsizes}. In this work, we show that "Optimistic Multiplicative-Weights Update (OMWU)" with \textit{constant stepsize}, exhibits last iterate convergence locally for convex-concave games, generalizing the results of Daskalakis and Panageas 2019 where last iterate convergence of OMWU was shown only for the \textit{bilinear case}. To the best of our knowledge, this is the first result about last-iterate convergence for constrained zero sum games (beyond the bilinear case) in which the dynamics use constant step-sizes.

Sampling is a fundamental and arguably very important task with numerous applications in Machine Learning. One approach to sample from a high dimensional distribution $e^{-f}$ for some function $f$ is the Langevin Algorithm (LA). Recently, there has been a lot of progress in showing fast convergence of LA even in cases where $f$ is non-convex, notably \cite{VW19}, \cite{MoritaRisteski} in which the former paper focuses on functions $f$ defined in $\mathbb{R}^n$ and the latter paper focuses on functions with symmetries (like matrix completion type objectives) with manifold structure. Our work generalizes the results of \cite{VW19} where $f$ is defined on a manifold $M$ rather than $\mathbb{R}^n$. From technical point of view, we show that KL decreases in a geometric rate whenever the distribution $e^{-f}$ satisfies a log-Sobolev inequality on $M$.

Non-concave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers ([Ge et al. 2015, Lee et al 2017] and references therein) indicate that first order methods work well and avoid saddles points. Results as in [Lee \etal 2017], however, are limited to the unconstrained case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on constrained non-concave maximization. We analyze a variant of a well-established algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem $\max_{\mathbf{x} \in D} P(\mathbf{x})$, where $P$ is non-concave, twice continuously differentiable and $D$ is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order KKT conditions, by combining techniques from dynamical systems as well as taking advantage of a recent connection between Baum Eagon inequality and MWU [Palaiopanos et al 2017].

In a series of papers [Lee et al 2016], [Panageas and Piliouras 2017], [Lee et al 2019], it was established that some of the most commonly used first order methods almost surely (under random initializations) and with step-size being small enough, avoid strict saddle points, as long as the objective function $f$ is $C^2$ and has Lipschitz gradient. The key observation was that first order methods can be studied from a dynamical systems perspective, in which instantiations of Center-Stable manifold theorem allow for a global analysis. The results of the aforementioned papers were limited to the case where the step-size $\alpha$ is constant, i.e., does not depend on time (and typically bounded from the inverse of the Lipschitz constant of the gradient of $f$). It remains an open question whether or not the results still hold when the step-size is time dependent and vanishes with time. In this paper, we resolve this question on the affirmative for gradient descent, mirror descent, manifold descent and proximal point. The main technical challenge is that the induced (from each first order method) dynamical system is time non-homogeneous and the stable manifold theorem is not applicable in its classic form. By exploiting the dynamical systems structure of the aforementioned first order methods, we are able to prove a stable manifold theorem that is applicable to time non-homogeneous dynamical systems and generalize the results in [Lee et al 2019] for time dependent step-sizes.