Convergence of a Normal Map-based Prox-SGD Method under the KL Inequality

05/10/2023

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by   Andre Milzarek, et al.

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In this paper, we present a novel stochastic normal map-based algorithm (𝗇𝗈𝗋𝖬-𝖲𝖦𝖣) for nonconvex composite-type optimization problems and discuss its convergence properties. Using a time window-based strategy, we first analyze the global convergence behavior of 𝗇𝗈𝗋𝖬-𝖲𝖦𝖣 and it is shown that every accumulation point of the generated sequence of iterates {x^k}_k corresponds to a stationary point almost surely and in an expectation sense. The obtained results hold under standard assumptions and extend the more limited convergence guarantees of the basic proximal stochastic gradient method. In addition, based on the well-known Kurdyka-Łojasiewicz (KL) analysis framework, we provide novel point-wise convergence results for the iterates {x^k}_k and derive convergence rates that depend on the underlying KL exponent θ and the step size dynamics {α_k}_k. Specifically, for the popular step size scheme α_k=𝒪(1/k^γ), γ∈ (2/3,1], (almost sure) rates of the form x^k-x^* = 𝒪(1/k^p), p ∈ (0,1/2), can be established. The obtained rates are faster than related and existing convergence rates for 𝖲𝖦𝖣 and improve on the non-asymptotic complexity bounds for 𝗇𝗈𝗋𝖬-𝖲𝖦𝖣.