Linear Lower Bounds and Conditioning of Differentiable Games

Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly, that research focuses on methods and upper bounds on their speed of convergence. In this work, we approach the question of fundamental iteration complexity by providing lower bounds to complement the linear (i.e. geometric) upper bounds observed in the literature on a wide class of problems. We cast saddle-point and minmax problems as 2-player games. We leverage tools from single-objective convex optimisation to propose new linear lower bounds for convexconcave games. Notably, we give a linear lower bound for n-player differentiable games, by using the spectral properties of the update operator. We then propose a new definition of the condition number arising from our lower bound analysis. Unlike past definitions, our condition number captures the fact that linear rates are possible in games, even in the absence of strong convexity or strong concavity in the variables. In this paper, we will denote the spectrum of a matrix M by σ(M ), and define the block spectral bounds µ 1 , µ 2 , µ 12 , L 1 , L 2 , L 12 as constants bounding the spectra of the blocks in the Jacobian of eq. 4: