Optimization of parameterized policies for reinforcement learning (RL) is an important and challenging problem in artificial intelligence.

Otherwise, we only need to seek the strongest one.

Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in $1/\epsilon$, where $\epsilon > 0$ is the duality gap. While it has been well-documented that linear convergence---an iteration complexity scaling as $\text{log}(1/\epsilon)$---can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game. To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\text{log}(1/\epsilon)$, and $1/\sigma$, where $\sigma$ measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity.

The long-run behavior of multi-agent online learning -- and, in particular, no-regret learning -- is relatively well-understood in potential games, where players have common interests. By contrast, in general harmonic games -- the strategic complement of potential games, where players have competing interests -- very little is known outside the narrow subclass of $2$-player zero-sum games with a fully-mixed equilibrium. Our paper seeks to partially fill this gap by focusing on the full class of (generalized) harmonic games and examining the convergence properties of follow-the-regularized-leader (FTRL), the most widely studied class of no-regret learning schemes. As a first result, we show that the continuous-time dynamics of FTRL are Poincaré recurrent, i.e., they return arbitrarily close to their starting point infinitely often, and hence fail to converge. In discrete time, the standard, vanilla implementation of FTRL may lead to even worse outcomes, eventually trapping the players in a perpetual cycle of best-responses. However, if FTRL is augmented with a suitable extrapolation step -- which includes as special cases the optimistic and mirror-prox variants of FTRL -- we show that learning converges to a Nash equilibrium from any initial condition, and all players are guaranteed at most $\mathcal{O}(1)$ regret. These results provide an in-depth understanding of no-regret learning in harmonic games, nesting prior work on $2$-player zero-sum games, and showing at a high level that potential and harmonic games are complementary not only from the strategic but also from the dynamic viewpoint.

In this paper we study the fundamental problems of maximizing a continuous non monotone submodular function over a hypercube, with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1/2 approximation algorithm for continuous submodular function maximization; this approximation factor of is the best possible for algorithms that use only polynomially many queries. For the special case of DR-submodular maximization, we provide a faster 1/2-approximation algorithm that runs in (almost) linear time. Both of these results improve upon prior work [Bian et al., 2017, Soma and Yoshida, 2017, Buchbinder et al., 2012]. Our first algorithm is a single-pass algorithm that uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm is a faster single-pass algorithm that exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

We propose the first online quantum algorithm for zero-sum games with $\widetilde O(1)$ regret under the game setting. Moreover, our quantum algorithm computes an $\varepsilon$-approximate Nash equilibrium of an $m \times n$ matrix zero-sum game in quantum time $\widetilde O(\sqrt{m+n}/\varepsilon^{2.5})$. Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. Technically, our online quantum algorithm quantizes classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.

In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy function to establish an $O(K^{-2/3})$ bound on its average regret after $K$ iterations. This quantitatively verifies the algorithm's better behavior than the simultaneous version of mirror descent algorithm, which is known to diverge and yields an $O(K^{-1/2})$ average regret bound. In the special case of an unconstrained setting, our results recover the behavior of alternating gradient descent algorithm for zero-sum games which was studied in (Bailey et al., COLT 2020).

We study gradient descent-ascent learning dynamics with timescale separation ($\tau$-GDA) in unconstrained continuous action zero-sum games where the minimizing player faces a nonconvex optimization problem and the maximizing player optimizes a Polyak-Lojasiewicz (PL) or strongly-concave (SC) objective. In contrast to past work on gradient-based learning in nonconvex-PL/SC zero-sum games, we assess convergence in relation to natural game-theoretic equilibria instead of only notions of stationarity. In pursuit of this goal, we prove that the only locally stable points of the $\tau$-GDA continuous-time limiting system correspond to strict local minmax equilibria in each class of games. For these classes of games, we exploit timescale separation to construct a potential function that when combined with the stability characterization and an asymptotic saddle avoidance result gives a global asymptotic almost-sure convergence guarantee for the discrete-time gradient descent-ascent update to a set of the strict local minmax equilibrium. Moreover, we provide convergence rates for the gradient descent-ascent dynamics with timescale separation to approximate stationary points.