Convergence of Gradient Algorithms for Nonconvex C^{1+alpha} Cost Functions
This work provides a more general theoretical understanding of stochastic momentum methods for researchers and practitioners working with nonconvex optimization, relaxing a common assumption.
This paper analyzes the convergence of stochastic gradient algorithms with momentum terms in a nonconvex setting, demonstrating almost sure convergence for methods like SGD, heavy ball, and Nesterov's accelerated gradient. The analysis relaxes the traditional Lipschitz condition on the gradient to Holder continuity.
This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball, and Nesterov's accelerated gradient, is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, we prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize. Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of Holder continuity. As a byproduct, we apply a localization procedure to extend our results to stochastic stepsizes.