Beyond Bounded Variance: Variance-Reduced Normalized Methods for Nonconvex Optimization under Blum-Gladyshev Noise
Authors: Antesh Upadhyay, Arda Fazla, Abolfazl Hashemi
Summary
arXiv:2605. 15314v1 Announce Type: new Abstract: We study nonconvex stochastic optimization under the Blum-Gladyshev ($\mathsf{BG}$-0) noise model, where the stochastic gradient variance grows quadratically with the distance from the initialization.
Relevance
Read next because Beyond Bounded Variance: Variance-Reduced Normalized Methods for Nonconvex Optimization under Blum-Gladyshev Noise overlaps with clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)", clean result "Only continuous soft prefixes hit both EM axes at once on Qwen-2.5-7B-Instruct: discrete prompt searches split between the alignment objective and the distributional objective, and both discretizations of the soft prefix collapse (MODERATE confidence)", clean result "The marker is a representational handle, not a behavioural one — sharing it between a villain persona and the assistant transfers no misalignment (HIGH confidence)". Matching terms: under, alpha, rate, without, model. Source: arxiv cs.LG (Machine Learning).
Abstract
arXiv:2605.15314v1 Announce Type: new Abstract: We study nonconvex stochastic optimization under the Blum-Gladyshev ($\mathsf{BG}$-0) noise model, where the stochastic gradient variance grows quadratically with the distance from the initialization. We consider this problem under both standard smoothness and the symmetric generalized-smoothness framework, which captures objectives whose local curvature can scale with the gradient norm. We prove that normalized stochastic gradient descent with momentum, using only one stochastic gradient per iteration, converges under $\mathsf{BG}$-0 noise with oracle complexity $O(\varepsilon^{-6})$. This rate holds both for standard smoothness and for $\alpha$-symmetric generalized smoothness, showing that generalized smoothness is rate-neutral for normalized momentum in this setting. We then study a variance-reduced normalized STORM method. Under mean-square smoothness and sharp initialization, the method achieves the minimax optimal $O(\varepsilon^{-4})$ complexity, matching the lower bound. Under expected $\alpha$-symmetric generalized smoothness, the STORM recursion couples gradient-dependent smoothness with distance-dependent noise, leading to complexity $O(\varepsilon^{-(4+\alpha)})$ for $\alpha\in(0,1)$ and $O(\varepsilon^{-5})$ for $\alpha=1$. When the distance-growth parameter in the noise model vanishes, our guarantees recover the standard bounded-variance rates: $O(\varepsilon^{-4})$ for momentum, $O(\varepsilon^{-3})$ for variance reduction, and $O(\varepsilon^{-2})$ in the deterministic case. To our knowledge, these are the first convergence guarantees for normalized methods in non-convex stochastic optimization under $\mathsf{BG}$-0 noise without bounded domains, increasing batch sizes, or explicit anchoring, covering both standard and generalized smoothness regimes.