SGD with momentum has a provably worse generalization bound than plain SGD under heavy-tailed noise for quadratic losses, according to a comparison that is flawed as written.
Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical L\'{e}vy processes
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abstract
Via a Bismut-Elworthy-Li formula from [KPP23], we derive uniform gradient estimates for transition semigroups associated with stochastic differential equations driven by a large class of cylindrical L\'{e}vy processes which includes the important case of cylindrical $\alpha$-stable processes. As the first application, we formulate a Stein's method for quantitative approximation of the invariant measure of these stochastic differential equations in Wasserstein distance. As the second and main application, we study Euler-Maruyama numerical schemes of stochastic differential equations driven by stable L\'{e}vy processes with i.i.d. stable components and obtain a uniform-in-time approximation error in Wasserstein distance. Our approximation error has a linear dependence on the stepsize, which is expected to be tight, as can be seen from an explicit calculation for the case of an Ornstein-Uhlenbeck process.
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Algorithmic Stability of Stochastic Gradient Descent with Momentum under Heavy-Tailed Noise
SGD with momentum has a provably worse generalization bound than plain SGD under heavy-tailed noise for quadratic losses, according to a comparison that is flawed as written.