A Quadratic-Approximation-Based Stochastic Approximation Method for Weakly Convex Stochastic Programming

We propose a novel stochastic approximation algorithm, termed PMQSopt, for solving weakly convex stochastic optimization problems involving expectation-valued functions. The algorithm is constructed by integrating the proximal method of multipliers with quadratic approximations of the original stochastic problem. We analyze the sample complexity of PMQSopt in terms of the total number of stochastic gradient evaluations required. The convergence of the algorithm is characterized by three metrics associated with the $\epsilon$-KKT conditions: the average squared norm of the gradient of the Moreau envelope of the Lagrangian, the average constraint violation, and the average complementarity violation. For each of these metrics, we establish an expected convergence rate of $\mathcal{O}(T^{-1/4})$ after $T$ iterations. Furthermore, we show that with probability at least $1-1/T^{2/3}$, the gradient of the Lagrangian satisfies an $\mathcal{O}(T^{-1/8})$ bound; with probability at least $1-2/T^{2/3}$, the constraint violation achieves an $\mathcal{O}(T^{-1/4})$ bound; and with probability at least $1-3/T^{2/3}$, the complementarity violation attains an $\mathcal{O}(T^{-1/4})$ bound. All results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point. The proposed PMQSopt algorithm is a sequentially strongly convex programming method that is readily implementable. Numerical experiments illustrate its practical performance.