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The rate transfer theorem states that if the Lyapunov generator inequality holds for the coefficients $(b,\sigma)$, and the SGD update operator and the SDE coefficients satisfy certain moment and smoothness matching conditions, then the expected Lyapunov function of the discrete iteration also decays at a similar rate, and an explicit finite-time bound is given.
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10/14/2026
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UnverifiedA recent study proposed a 'rate transfer' method that, through Lyapunov analysis of stochastic differential equations (SDEs), directly converts continuous-time stability estimates into finite-time convergence bounds for discrete algorithms.75% similarUnverifiedThrough refined moment conditions (such as bounded variance or sub-Gaussianity of the gradient noise) and smoothness assumptions on the drift term (such as Lipschitz continuity), the rate transfer theorem proves that the deviation between the discrete trajectory and the SDE trajectory can be bounded within a finite time by a power of the step size.75% similarUnverifiedIn nonlinear systems, errors may grow at an exponential rate exp(λN), where λ is the maximum Lyapunov exponent.71% similarUnverifiedThe rate transfer proposed in the paper does not require an extremely small step size, but instead quantifies the deviation between the SDE and SGD at a finite step size, thereby obtaining non-asymptotic convergence bounds.70% similarUnverifiedThe key tools for propagating score estimation error into a KL divergence bound on the path distribution are Girsanov's theorem and the Fokker-Planck equation, with compensated Poisson random measures additionally required for processes with jumps69% similar
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