Learning Sparse Potential Games in Polynomial Time and Sample Complexity

We consider the problem of learning sparse potential games --- a non-parametric class of graphical games where the payoffs are given by potential functions --- from observations of strategic interactions. We show that a polynomial time method based on $\ell_{1,2}$-group regularized logistic regression recovers the $\epsilon$-Nash equilibria set of the true game in $O(m^4 d^4 \log (pd))$ samples, where $m$ is the maximum number of pure strategies of a player, $p$ is the number of players and $d$ is the maximum degree of the game graph. Under slightly more stringent conditions on the payoff functions of the true game, we show that our method recovers the pure-strategy Nash equilibria (PSNE) set of the true game exactly. We also show that $\Omega(\log m + d \log p)$ samples are necessary for any method to recover the PSNE set of the true game.