Gradient-Free Methods with Inexact Oracle for Convex-Concave Stochastic Saddle-Point Problem

Abstract

In the paper, we generalize the approach Gasnikov et. al, 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle, to the convex-concave saddle-point problem. The proposed approach works, at least, like the best existing approaches. But for a special set-up (simplex type constraints and closeness of Lipschitz constants in 1 and 2 norms) our approach reduces n /log n times the required number of oracle calls (function calculations). Our method uses a stochastic approximation of the gradient via finite differences. In this case, the function must be specified not only on the optimization set itself, but in a certain neighbourhood of it. In the second part of the paper, we analyze the case when such an assumption cannot be made, we propose a general approach on how to modernize the method to solve this problem, and also we apply this approach to particular cases of some classical sets.

Accelerated gradient methods with absolute and relative noise in the gradient

“…This problem can be solved in a two different ways. The first one is "margins inward approach" [8]. The second one is "continuation" f to R n with preserving of convexity and Lipschitz continuity [44]:…”

Recent theoretical advances in decentralized distributed convex optimization

“…Assume that we have an access to the first-order oracle for g, i.e. gradient ∇g(x) is available, and to the biased stochastic zeroth-order oracle for f (see also [40,13]) that for a given point x returns noisy value f…”

“…Therefore, instead of solving (123) directly one can focus on the problem min x∈Q Ψ (x) = F(x) + g(x) (129) with small enough r. As mentioned earlier, this approach is universal. In particular, the analysis of gradient-free methods for non-smooth saddle-point problems can be carried out in a similar way [13]. Now we will give the main facts from [11] for zoSA algorithm itself.…”

Gradient-Free Optimization for Non-Smooth Saddle Point Problems under Adversarial Noise

“…Related Work and Contribution. Zeroth-order methods in the non-smooth setup were developed in a wide range of works (Polyak, 1987;Spall, 2003;Conn et al, 2009;Duchi et al, 2015;Shamir, 2017;Nesterov and Spokoiny, 2017;Gasnikov et al, 2017;Bayandina, 2017;Beznosikov et al, 2020;Gasnikov, 2022). Particularly, in (Shamir, 2017), an optimal algorithm was provided as an improvement to the work (Duchi et al, 2015) for a non-smooth case but Lipschitz continuous in stochastic convex optimization problems.…”