Alternating proximal-gradient steps for (stochastic) nonconvex-concave minimax problems

Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions

“…Since there is an extensive literature on convergence rates in terms of a gap function or distance to a solution for monotone problems as well as generalizations such as nonconvex-concave [9,37], convex-nonconcave [58] or under the Polyak-Łojasiewicz assumption, see [59], we will only focus on the nonconvex-nonconcave setting and rates in terms of the gradient norm for the monotone setting.…”

“…While this problem was cast as a stochastic one in [3], we will look at a finite sum (sample average) approximation, which we then solve deterministically in a full batch fashion. (9). Figure 2a plots the squared gradient norm along the iterates.…”

“…Figure 2.: Covariance matrix learning. Solving a deterministic formulation of(9). Figure2aplots the squared gradient norm along the iterates.…”

AUC Maximization in the Era of Big Data and AI: A Survey

“…It is worth to point out that the complexity for the former case is worse than that established in [118] and the complexity for the latter case matches that in [118] but requires a large mini-batch size, which is not required in [118]. Recently, Boţ and Böhm [15] extend the analysis to stochastic alternating (proximal) gradient descent ascent method which uses w 𝑡 = w 𝑡 +1 to compute the estimator v 𝑡 +1 . However, this algorithm suffers from the same issue of requiring a large mini-batch size and the worse complexity for non-convex concave min-max problems.…”

“…Recently, Guo et al [52] develop a new stochastic primal-dual method for solving non-convex strongly concave min-max problems under the smoothness assumption of 𝐹 (w, 𝛼). They address the issue of large mini-batch size requirement in [15,88]. The key improvement lies at using moving average to compute the estimator u 𝑡 +1 , i.e., u 𝑡 +1 = (1 − 𝛽 1,𝑡 )u 𝑡 + 𝛽 1,𝑡 O w (w 𝑡 , 𝛼 𝑡 ), and simply use v 𝑡 +1 = O 𝛼 (w 𝑡 , 𝛼 𝑡 ; z 𝑡 ), where O w and O 𝛼 denote an unbiased stochastic estimator of ∇ w 𝐹 (w, 𝛼) and ∇ 𝛼 𝐹 (w, 𝛼), respectively.…”