Gradient Noise: Smooth and Nonsmooth Risks

This lecture delves into the study of gradient noise in the context of smooth and nonsmooth risk functions, exploring its impact on optimization algorithms. The instructor explains the derivation of expressions for the first and second-order moments of gradient noise, highlighting its effects on stochastic optimization methods. The lecture covers the conditions for strongly convex risks and Lipschitz loss gradients, emphasizing the importance of understanding the dynamics of gradient noise in empirical and stochastic risk minimization. Various sampling procedures, such as with or without replacement and importance sampling, are discussed to analyze the behavior of gradient noise. The lecture concludes by discussing the implications of gradient noise on the convergence of optimization algorithms.