Enhancing Stochastic Gradient Descent: A Unified Framework and Novel Acceleration Methods for Faster Convergence

Based on SGD, previous works have proposed many algorithms that have improvedconvergence speed and generalization in stochastic optimization, such as SGDm,AdaGrad, Adam, etc. However, their convergence analysis under non-convexconditions is challenging. In this work, we propose a unified framework toaddress this issue. For any first-order methods, we interpret the updateddirection g_t as the sum of the stochastic subgradient ∇ f_t(x_t) andan additional acceleration term 2|⟨ v_t, ∇ f_t(x_t)⟩|/v_t_2^2 v_t, thus we can discuss the convergence by analyzing⟨ v_t, ∇ f_t(x_t) ⟩. Through our framework, we havediscovered two plug-and-play acceleration methods: Reject Acceleratingand Random Vector Accelerating, we theoretically demonstrate thatthese two methods can directly lead to an improvement in convergence rate.