Finding a stationary point of a stochastic convex problem

This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers smooth losses with Lipschitz gradient and differences of such losses with convex functions. At each iteration the expected cost is replaced by a sample average that is progressively refined, and the proximal-subgradient stepsize is selected by an Armijo-type line search enforcing a sufficient-decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates substantially more general problem formulations than existing methods, in particular, it requires neither (weak) convexity of the regularizer nor a uniform bound on the variance of the stochastic oracle, and our analysis yields convergence guarantees that are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded, with no prescribed growth rate. Leveraging the Kurdyka-Lojasiewicz (KL) property, we further upgrade this subsequential guarantee to convergence of the whole trajectory to a single stationary point. Finally, for exponential-type KL desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to a logarithmic factor, for both the function values and the iterates.

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ sample complexity to find an $\varepsilon$-stationary point. Some works indicate that this complexity can be improved to $\mathcal{O}(\varepsilon^{-3})$ when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains open. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We introduce novel bias-corrected momentum algorithms utilizing efficient Hessian-vector products. We establish convergence conditions and demonstrate a lower iteration complexity of $\mathcal{O}(\varepsilon^{-3})$ for the proposed algorithms. The effectiveness of the proposed method is validated through applications to robust logistic regression and robust adaptive cruise control.