Learning While Dissipating Information: Understanding the Generalization Capability of SGLD
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Understanding the generalization capability of learning algorithms is at the heart of statistical learning theory. In this paper, we investigate the generalization gap of stochastic gradient Langevin dynamics (SGLD), a widely used optimizer for training deep neural networks (DNNs). We derive an algorithm-dependent generalization bound by analyzing SGLD through an information-theoretic lens. Our analysis reveals an intricate trade-off between learning and information dissipation: SGLD learns from data by updating parameters at each iteration while dissipating information from early training stages. Our bound also involves the variance of gradients which captures a particular kind of "sharpness" of the loss landscape. The main proof techniques in this paper rely on strong data processing inequalities – a fundamental concept in information theory – and Otto-Villani's HWI inequality. Finally, we demonstrate our bound through numerical experiments, showing that it can predict the behavior of the true generalization gap.
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