A Parameter-Free Stochastic LineseArch Method (SLAM) for Minimizing Expectation Residuals

Most existing rate and complexity guarantees for stochastic gradient methods in L-smooth settings mandates that such sequences be non-adaptive, non-increasing, and upper bounded by tfracaL for a > 0. This requires knowledge of L and may preclude larger steps. Motivated by these shortcomings, we present an Armijo-enabled stochastic linesearch framework with standard stochastic zeroth- and first-order oracles. The resulting steplength sequence is non-monotone and requires neither knowledge of L nor any other problem parameters. We then prove that the expected stationarity residual diminishes at a rate of O(1/√K), where K denotes the iteration budget. Furthermore, the resulting iteration and sample complexities for computing an ε-stationary point are O(ε-2) and Oleft(ε-4right). The proposed method allows for a simple nonsmooth convex component in the objective, addressed through proximal gradient updates. Analogous guarantees are provided in the Polyak-Lojasiewicz (PL) setting and convex regimes. Preliminary numerical experiments are seen to be promising.