Convergence rate of stochastic gradient descent under Łojasiewicz type condition and polynomially controlled noise

Abstract: Stochastic gradient methods are widely used for solving optimization problems in many scientific fields. Recently, their popularity has risen due to the increase in machine learning's prominence. Their convergence is often studied under the convexity assumption, which is too restrictive in modern applications. In this paper, we investigate the almost sure convergence rate and expectation under a Łojasiewicz-type gradient domination condition. This framework includes both bounded and decaying noise, as well as noise with controlled polynomial growth. The obtained results extend existing convergence by allowing a broader class of noise sequences and explicitly capturing the effect of their asymptotic behavior on the convergence rate and applicability.