  We consider the problem of learning sparse polymatrix games from observations
of strategic interactions. We show that a polynomial time method based on
$\ell_{1,2}$-group regularized logistic regression recovers a game, whose Nash
equilibria are the $\epsilon$-Nash equilibria of the game from which the data
was generated (true game), in $\mathcal{O}(m^4 d^4 \log (pd))$ samples of
strategy profiles --- 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 separability conditions on the payoff
matrices of the true game, we show that our method learns a game with the exact
same Nash equilibria as the true game. We also show that $\Omega(d \log (pm))$
samples are necessary for any method to consistently recover a game, with the
same Nash-equilibria as the true game, from observations of strategic
interactions. We verify our theoretical results through simulation experiments.
