Generalization Bounds for Gradient Methods via Discrete and Continuous Prior
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Proving algorithm-dependent generalization error bounds for gradient-type optimization methods has attracted significant attention recently in learning theory. However, most existing trajectory-based analyses require either restrictive assumptions on the learning rate (e.g., fast decreasing learning rate), or continuous injected noise (such as the Gaussian noise in Langevin dynamics). In this paper, we introduce a new discrete data-dependent prior to the PAC-Bayesian framework, and prove a high probability generalization bound of order O(1/n·∑_t=1^T(γ_t/ε_t)^2𝐠_t^2) for Floored GD (i.e. a version of gradient descent with precision level ε_t), where n is the number of training samples, γ_t is the learning rate at step t, 𝐠_t is roughly the difference of the gradient computed using all samples and that using only prior samples. 𝐠_t is upper bounded by and and typical much smaller than the gradient norm ∇ f(W_t). We remark that our bound holds for nonconvex and nonsmooth scenarios. Moreover, our theoretical results provide numerically favorable upper bounds of testing errors (e.g., 0.037 on MNIST). Using a similar technique, we can also obtain new generalization bounds for certain variants of SGD. Furthermore, we study the generalization bounds for gradient Langevin Dynamics (GLD). Using the same framework with a carefully constructed continuous prior, we show a new high probability generalization bound of order O(1/n + L^2/n^2∑_t=1^T(γ_t/σ_t)^2) for GLD. The new 1/n^2 rate is due to the concentration of the difference between the gradient of training samples and that of the prior.
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