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A variety of practical problems can be modeled by the decision-making processin multi-player games where a group of self-interested players aim atoptimizing their own local objectives, while the objectives depend on theactions taken by others. The local gradient information of each player,essential in implementing algorithms for finding game solutions, is all toooften unavailable. In this paper, we focus on designing solution algorithms formulti-player games using bandit feedback, i.e., the only available feedback ateach player's disposal is the realized objective values. To tackle the issue oflarge variances in the existing bandit learning algorithms with a single oraclecall, we propose two algorithms by integrating the residual feedback schemeinto single-call extra-gradient methods. Subsequently, we show that the actualsequences of play can converge almost surely to a critical point if the game ispseudo-monotone plus and characterize the convergence rate to the criticalpoint when the game is strongly pseudo-monotone. The ergodic convergence ratesof the generated sequences in monotone games are also investigated as asupplement. Finally, the validity of the proposed algorithms is furtherverified via numerical examples.

Zeroth-Order Learning in Continuous Games via Residual Pseudogradient Estimates