Yuanhao Wang

Fictitious play (FP) is a natural learning dynamic in two-player zero-sum games. Samuel Karlin conjectured in 1959 that FP converges at a rate of $O(t^{-1/2})$ to Nash equilibrium, where $t$ is the number of steps played. However, Daskalakis and Pan disproved the stronger form of this conjecture in 2014, where \emph{adversarial} tie-breaking is allowed. This paper disproves Karlin's conjecture in its weaker form. In particular, there exists a 10-by-10 zero-sum matrix game, in which FP converges at a rate of $Ω(t^{-1/3})$, and no ties occur except for the first step.

A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolved this question in the negative, providing an exponential (in $d$) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that this information theoretic barrier for RL can be circumvented by further supposing an even more favorable assumption: there exists a \emph{constant suboptimality gap} between the optimal $Q$-value of the best action and that of the second-best action (for all states). The hope is that having a large suboptimality gap would permit easier identification of optimal actions themselves, thus making the problem tractable; indeed, provided the agent has access to a generative model, sample-efficient RL is in fact possible with the addition of this more favorable assumption. This work focuses on this question in the standard online reinforcement learning setting, where our main result resolves this question in the negative: our hardness result shows that an exponential sample complexity lower bound still holds even if a constant suboptimality gap is assumed in addition to having a linearly realizable optimal $Q$-function. Perhaps surprisingly, this implies an exponential separation between the online RL setting and the generative model setting. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption (both implicitly place stronger conditions on the underlying dynamics model).

We study online learning in unknown Markov games, a problem that arises in episodic multi-agent reinforcement learning where the actions of the opponents are unobservable. We show that in this challenging setting, achieving sublinear regret against the best response in hindsight is statistically hard. We then consider a weaker notion of regret by competing with the \emph{minimax value} of the game, and present an algorithm that achieves a sublinear $\tilde{\mathcal{O}}(K^{2/3})$ regret after $K$ episodes. This is the first sublinear regret bound (to our knowledge) for online learning in unknown Markov games. Importantly, our regret bound is independent of the size of the opponents' action spaces. As a result, even when the opponents' actions are fully observable, our regret bound improves upon existing analysis (e.g., (Xie et al., 2020)) by an exponential factor in the number of opponents.

This paper studies minimax optimization problems $\min_x \max_y f(x,y)$, where $f(x,y)$ is $m_x$-strongly convex with respect to $x$, $m_y$-strongly concave with respect to $y$ and $(L_x,L_{xy},L_y)$-smooth. Zhang et al. provided the following lower bound of the gradient complexity for any first-order method: $\Omega\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L_{xy}^2}{m_x m_y}+\frac{L_y}{m_y}}\ln(1/\epsilon)\Bigr).$ This paper proposes a new algorithm with gradient complexity upper bound $\tilde{O}\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L\cdot L_{xy}}{m_x m_y}+\frac{L_y}{m_y}}\ln\left(1/\epsilon\right)\Bigr),$ where $L=\max\{L_x,L_{xy},L_y\}$. This improves over the best known upper bound $\tilde{O}\left(\sqrt{\frac{L^2}{m_x m_y}} \ln^3\left(1/\epsilon\right)\right)$ by Lin et al. Our bound achieves linear convergence rate and tighter dependency on condition numbers, especially when $L_{xy}\ll L$ (i.e., when the interaction between $x$ and $y$ is weak). Via reduction, our new bound also implies improved bounds for strongly convex-concave and convex-concave minimax optimization problems. When $f$ is quadratic, we can further improve the upper bound, which matches the lower bound up to a small sub-polynomial factor.