No-regret self-play learning dynamics have become one of the premier ways to solve large-scale games in practice. Accelerating their convergence via improving the regret of the players over the naive $O(\sqrt{T})$ bound after $T$ rounds has been extensively studied in recent years, but almost all studies assume access to exact gradient feedback. We address the question of whether acceleration is possible under bandit feedback only and provide an affirmative answer for two-player zero-sum normal-form games. Specifically, we show that if both players apply the Tsallis-INF algorithm of Zimmert and Seldin (2018, arXiv:1807.07623), then their regret is at most $O(c_1 \log T + \sqrt{c_2 T})$, where $c_1$ and $c_2$ are game-dependent constants that characterize the difficulty of learning -- $c_1$ resembles the complexity of learning a stochastic multi-armed bandit instance and depends inversely on some gap measures, while $c_2$ can be much smaller than the number of actions when the Nash equilibria have a small support or are close to the boundary. In particular, for the case when a pure strategy Nash equilibrium exists, $c_2$ becomes zero, leading to an optimal instance-dependent regret bound as we show. We additionally prove that in this case, our algorithm also enjoys last-iterate convergence and can identify the pure strategy Nash equilibrium with near-optimal sample complexity.

Understanding relationships between feature variables is one important way humans use to make decisions. However, state-of-the-art deep learning studies either focus on task-agnostic statistical dependency learning or do not model explicit feature dependencies during prediction. We propose a deep neural network framework, dGAP, to learn neural dependency Graph and optimize structure-Aware target Prediction simultaneously. dGAP trains towards a structure self-supervision loss and a target prediction loss jointly. Our method leads to an interpretable model that can disentangle sparse feature relationships, informing the user how relevant dependencies impact the target task. We empirically evaluate dGAP on multiple simulated and real datasets. dGAP is not only more accurate, but can also recover correct dependency structure.