Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex nonconcave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the "hidden" convex-concave game. We provide conditions under which vanilla GDA provably converges not merely to local Nash, but the actual von-Neumann solution. If the hidden game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence results are non-local despite working in the setting of non-convex non-concave games. Critically, under proper assumptions we combine the Center-Stable Manifold Theorem along with novel type of initialization dependent Lyapunov functions to prove that almost all initial conditions converge to the solution. Finally, we discuss diverse applications of our framework ranging from generative adversarial networks to evolutionary biology.

When e 6 WΦ, we have E = Rd and WΦ,E = WΦ. By Theorem 1 in [10], we know that the projected Bellman equation (3.4) has a unique fixed point θ . Thus, L= {θ }. 2. When e WΦ, θe is a unique solution to Φθ = eas Φ is full column rank. We first show that the set of solutions to the projected Bellman equation (3.4) takes the form { θ+ cθe|c R}, where θis any solution to (3.4). On the other hand, suppose that θis not of the form θ+ cθe.

The focus of this paper is on sample complexity guarantees of average-reward reinforcement learning algorithms, which are known to be more challenging to study than their discounted-reward counterparts. To the best of our knowledge, we provide the first known finite sample guarantees using both constant and diminishing step sizes of (i) average-reward TD(λ) with linear function approximation for policy evaluation and (ii) average-reward Q-learning in the tabular setting to find the optimal policy. A major challenge is that since the value functions are agnostic to an additive constant, the corresponding Bellman operators are no longer contraction mappings under any norm. We obtain the results for TD(λ) by working in an appropriately defined subspace that ensures uniqueness of the solution. For Q-learning, we exploit the span seminorm contractive property of the Bellman operator, and construct a novel Lyapunov function obtained by infimal convolution of a generalized Moreau envelope and the indicator function of a set.

While ensuring stability for linear systems is well understood, it remains a major challenge for nonlinear systems. A general approach in such cases is to compute a combination of a Lyapunov function and an associated control policy. However, finding Lyapunov functions for general nonlinear systems is a challenging task. To address this challenge, several methods have been proposed that represent Lyapunov functions using neural networks. However, such approaches either focus on continuous-time systems, or highly restricted classes of nonlinear dynamics.

We study accelerated descent dynamics for constrained convex optimization. This dynamics can be described naturally as a coupling of a dual variable accumulating gradients at a given rate η(t), and a primal variable obtained as the weighted average of the mirrored dual trajectory, with weights w(t). Using a Lyapunov argument, we give sufficient conditions on η and wto achieve a desired convergence rate. As an example, we show that the replicator dynamics (an example of mirror descent on the simplex) can be accelerated using a simple averaging scheme. We then propose an adaptive averaging heuristic which adaptively computes the weights to speed up the decrease of the Lyapunov function. We provide guarantees on adaptive averaging in continuous-time, prove that it preserves the quadratic convergence rate of accelerated first-order methods in discrete-time, and give numerical experiments to compare it with existing heuristics, such as adaptive restarting. The experiments indicate that adaptive averaging performs at least as well as adaptive restarting, with significant improvements in some cases.

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics.We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems.We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

In this paper, we analyze the two time-scale stochastic approximation (TTSSA) algorithm introduced in Borkar (1997) using a martingale approach. This approach leads to simple sufficient conditions for the iterations to be bounded almost surely, as well as estimates on the rate of convergence of the mean-squared error of the TTSSA algorithm to zero. Our theory is applicable to nonlinear equations, in contrast to many papers in the TTSSA literature which assume that the equations are linear. The convergence of TTSSA is proved in the "almost sure" sense, in contrast to earlier papers on TTSSA that establish convergence in distribution, convergence in the mean, and the like. Moreover, in this paper we establish different rates of convergence for the fast and the slow subsystems, perhaps for the first time. Finally, all of the above results to continue to hold in the case where the two measurement errors have nonzero conditional mean, and/or have conditional variances that grow without bound as the iterations proceed. This is in contrast to previous papers which assumed that the errors form a martingale difference sequence with uniformly bounded conditional variance. It is shown that when the measurement errors have zero conditional mean and the conditional variance remains bounded, the mean-squared error of the iterations converges to zero at a rate of $o(t^{-η})$ for all $η\in (0,1)$. This improves upon the rate of $O(t^{-2/3})$ proved in Doan (2023) (which is the best bound available to date). Our bound is virtually the same as the rate of $O(t^{-1})$ proved in Doan (2024), but for a Polyak-Ruppert averaged version of TTSSA, and not directly. Rates of convergence are also established for the case where the errors have nonzero conditional mean and/or unbounded conditional variance.

Neural Information Processing Systems http://nips.cc/

We introduce a method for learning provably stable deep neural network based dynamic models from observed data.