Adaptive Lipschitz-Free Proximal Gradient Methods for Stochastic Composite Nonconvex Optimization

We propose ALFPG, an Adaptive Lipschitz-Free Proximal Gradient framework for stochastic composite nonconvex minimization. The framework adapts its stepsizes using information collected along the optimization trajectory, without requiring prior knowledge of smoothness constants or performing line search. For finite-sum problems, ALFPG-FS combines coordinate-wise stepsizes, Nesterov's extrapolation, and the SPIDER estimator, with ALFPG-D as its full-batch specialization. Under component-wise smoothness and a bounded effective domain, ALFPG-FS achieves ε-stationarity in expectation using O(N+ sqrt N ε-2) component-gradient evaluations. For expectation problems, we develop ALFPG-MVR1 and ALFPG-MVR2, where MVR stands for momentum-based variance reduction. Both use one fresh sample per iteration: ALFPG-MVR1 uses one stochastic-gradient evaluation, whereas ALFPG-MVR2 uses at most two evaluations at different points with the same sample. ALFPG-MVR1 requires smoothness of the expected loss; ALFPG-MVR2 additionally requires uniformly smooth sample losses. When the regularizer is convex or Lipschitz continuous on its effective domain, and stochastic gradients are almost surely bounded, the two methods achieve ε-stationarity in expectation using (σ 2ε-4+ε-2) and (σε-3+ε-2) stochastic-gradient evaluations, respectively. Here, N is the number of component functions, σ squared bounds the gradient noise variance, and hides logarithmic factors. Neither expectation variant requires prior knowledge of σ, and both recover (ε-2) complexity in the zero-noise regime.