First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities

• URL: http://arxiv.org/abs/2305.15938v2
• Date: Sat, 30 Mar 2024 13:50:06 GMT
• Title: First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities
• Authors: Aleksandr Beznosikov, Sergey Samsonov, Marina Sheshukova, Alexander Gasnikov, Alexey Naumov, Eric Moulines,
• Abstract summary: We present a unified approach for the theoretical analysis of first-order variation methods. Our approach covers both non-linear gradient and strongly Monte Carlo problems. We provide bounds that match the oracle strongly in the case of convex method optimization problems.
• Score: 91.46841922915418
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities. Our approach covers scenarios for both non-convex and strongly convex minimization problems. To achieve an optimal (linear) dependence on the mixing time of the underlying noise sequence, we use the randomized batching scheme, which is based on the multilevel Monte Carlo method. Moreover, our technique allows us to eliminate the limiting assumptions of previous research on Markov noise, such as the need for a bounded domain and uniformly bounded stochastic gradients. Our extension to variational inequalities under Markovian noise is original. Additionally, we provide lower bounds that match the oracle complexity of our method in the case of strongly convex optimization problems.

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