Time-Independent Information-Theoretic Generalization Bounds for SGLD
This work addresses generalization guarantees for SGLD in non-convex optimization, offering incremental improvements over prior bounds by removing step size dependence and enhancing risk analysis.
The paper tackles the problem of deriving generalization bounds for stochastic gradient Langevin dynamics (SGLD) by providing time-independent, information-theoretic bounds that decay with sample size, regardless of iterations or step size, and establishes the first such bound for identical training and test losses, leading to improved excess risk bounds.
We provide novel information-theoretic generalization bounds for stochastic gradient Langevin dynamics (SGLD) under the assumptions of smoothness and dissipativity, which are widely used in sampling and non-convex optimization studies. Our bounds are time-independent and decay to zero as the sample size increases, regardless of the number of iterations and whether the step size is fixed. Unlike previous studies, we derive the generalization error bounds by focusing on the time evolution of the Kullback--Leibler divergence, which is related to the stability of datasets and is the upper bound of the mutual information between output parameters and an input dataset. Additionally, we establish the first information-theoretic generalization bound when the training and test loss are the same by showing that a loss function of SGLD is sub-exponential. This bound is also time-independent and removes the problematic step size dependence in existing work, leading to an improved excess risk bound by combining our analysis with the existing non-convex optimization error bounds.