RanSOM uses randomized scaling in second-order momentum updates to obtain unbiased bias estimates via Stein identities, recovering optimal O(ε^{-3}) rates for bounded and heavy-tailed noise in constrained and unconstrained settings.
Taking the expectation and using(a+b) q ≤2 q−1(aq +b q)again: E[∥T2∥q 2]≤E[(|w t|+s t)q]κqρq (L0 +L 1∥∇f(x t+1)∥∗)q ≤M wsηq t κqρq2q−1 (Lq 0 +L q 1∥∇f(x t+1)∥q ∗)
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RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization
RanSOM uses randomized scaling in second-order momentum updates to obtain unbiased bias estimates via Stein identities, recovering optimal O(ε^{-3}) rates for bounded and heavy-tailed noise in constrained and unconstrained settings.