Cycles in adversarial regularized learning

Regularized learning is a fundamental technique in online optimization, machine learning and many other fields of computer science. A natural question that arises in these settings is how regularized learning algorithms behave when faced against each other. We study a natural formulation of this problem by coupling regularized learning dynamics in zero-sum games. We show that the system's behavior is Poincar\'e recurrent, implying that almost every trajectory revisits any (arbitrarily small) neighborhood of its starting point infinitely often. This cycling behavior is robust to the agents' choice of regularization mechanism (each agent could be using a different regularizer), to positive-affine transformations of the agents' utilities, and it also persists in the case of networked competition, i.e., for zero-sum polymatrix games.Comment: 22 pages, 4 figure

A General Framework for Computing Optimal Correlated Equilibria in Compact Games

We analyze the problem of computing a correlated equilibrium that optimizes some objective (e.g., social welfare). Papadimitriou and Roughgarden [2008] gave a sufficient condition for the tractability of this problem; however, this condition only applies to a subset of existing representations. We propose a different algorithmic approach for the optimal CE problem that applies to all compact representations, and give a sufficient condition that generalizes that of Papadimitriou and Roughgarden. In particular, we reduce the optimal CE problem to the deviation-adjusted social welfare problem, a combinatorial optimization problem closely related to the optimal social welfare problem. This framework allows us to identify new classes of games for which the optimal CE problem is tractable; we show that graphical polymatrix games on tree graphs are one example. We also study the problem of computing the optimal coarse correlated equilibrium, a solution concept closely related to CE. Using a similar approach we derive a sufficient condition for this problem, and use it to prove that the problem is tractable for singleton congestion games.Comment: 14 pages. Short version to appear in WINE 201

Tree Polymatrix Games Are PPAD-Hard.

We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an $\epsilon$-fixed point of a 2D LinearFIXP instance, when $\epsilon$ is any constant less than $(\sqrt{2} - 1)/2 \approx 0.2071$. This lifts the hardness regime from polynomially small approximations in $k$-dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of $0.5$

Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard

One of the most appealing aspects of the (coarse) correlated equilibrium concept is that natural dynamics quickly arrive at approximations of such equilibria, even in games with many players. In addition, there exist polynomial-time algorithms that compute exact (coarse) correlated equilibria. In light of these results, a natural question is how good are the (coarse) correlated equilibria that can arise from any efficient algorithm or dynamics. In this paper we address this question, and establish strong negative results. In particular, we show that in multiplayer games that have a succinct representation, it is NP-hard to compute any coarse correlated equilibrium (or approximate coarse correlated equilibrium) with welfare strictly better than the worst possible. The focus on succinct games ensures that the underlying complexity question is interesting; many multiplayer games of interest are in fact succinct. Our results imply that, while one can efficiently compute a coarse correlated equilibrium, one cannot provide any nontrivial welfare guarantee for the resulting equilibrium, unless P=NP. We show that analogous hardness results hold for correlated equilibria, and persist under the egalitarian objective or Pareto optimality. To complement the hardness results, we develop an algorithmic framework that identifies settings in which we can efficiently compute an approximate correlated equilibrium with near-optimal welfare. We use this framework to develop an efficient algorithm for computing an approximate correlated equilibrium with near-optimal welfare in aggregative games.Comment: 21 page

Evolutionary Game Theory Squared: Evolving Agents in Endogenously Evolving Zero-Sum Games

The predominant paradigm in evolutionary game theory and more generally online learning in games is based on a clear distinction between a population of dynamic agents that interact given a fixed, static game. In this paper, we move away from the artificial divide between dynamic agents and static games, to introduce and analyze a large class of competitive settings where both the agents and the games they play evolve strategically over time. We focus on arguably the most archetypal game-theoretic setting -- zero-sum games (as well as network generalizations) -- and the most studied evolutionary learning dynamic -- replicator, the continuous-time analogue of multiplicative weights. Populations of agents compete against each other in a zero-sum competition that itself evolves adversarially to the current population mixture. Remarkably, despite the chaotic coevolution of agents and games, we prove that the system exhibits a number of regularities. First, the system has conservation laws of an information-theoretic flavor that couple the behavior of all agents and games. Secondly, the system is Poincar\'{e} recurrent, with effectively all possible initializations of agents and games lying on recurrent orbits that come arbitrarily close to their initial conditions infinitely often. Thirdly, the time-average agent behavior and utility converge to the Nash equilibrium values of the time-average game. Finally, we provide a polynomial time algorithm to efficiently predict this time-average behavior for any such coevolving network game.Comment: To appear in AAAI 202

Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization

We propose the first, to our knowledge, loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches