Partially Informed Stochastic Optimization via Zeroth-Order Residual Tracking

Abstract: Many optimization problems provide direct access to only part of the gradient, while the remaining component must be inferred from function evaluations. This structure arises in performative prediction, performative reinforcement learning, and bilevel optimization with a black-box lower level. We introduce Partially Informed Stochastic Optimization (PISO), which combines the available gradient component with a running estimate of the residual. By centring each zeroth-order observation on this estimate, PISO makes the variance due to random probing depend on the tracking error rather than the residual's magnitude. This yields a randomized sketch iteration with an explicit contraction mechanism and a principled interpretation of momentum. For smooth nonconvex objectives, we establish convergence guarantees that depend on the residual's variation and initial magnitude, without requiring a uniform bound on the unavailable gradient component. With independently sampled function evaluations and Gaussian or spherical probes, PISO attains leading function-evaluation complexities of (O(d^{2}\varepsilon^{-6})) under Lipschitz gradients and (O(d^{2}\varepsilon^{-5})) under an additional Lipschitz-Hessian assumption. These rates require only one probe direction, two function samples, and one known-component sample per update. The analysis also quantifies how batching trades observations for fewer deployments. Across five settings, PISO variants improve on the tested baselines with comparable oracle access in three, reach a comparable solution faster in bilevel routing, and perform best among methods without response-model derivatives in performative reinforcement learning.