Convergence of Unregularized Online Learning Algorithms

In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess generalization errors in expectation. A sufficient condition for the almost sure convergence is also given. With high probability, we provide explicit convergence rates of the excess generalization errors for both averaged iterates and the last iterate, which in turn also imply convergence rates with probability one. To our best knowledge, this is the first high-probability convergence rate for the last iterate of online gradient descent algorithms without strong convexity. Without any boundedness assumptions on iterates, our results are derived by a novel use of two measures of the algorithm's one-step progress, respectively by generalization errors and by distances in RKHSs, where the variances of the involved martingales are cancelled out by the descent property of the algorithm

Convergence of Online Mirror Descent

In this paper we consider online mirror descent (OMD) algorithms, a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence $\{\eta_t\}_{t}$ for the convergence of an OMD algorithm with respect to the expected Bregman distance induced by the mirror map. The condition is $\lim_{t\to\infty}\eta_t=0, \sum_{t=1}^{\infty}\eta_t=\infty$ in the case of positive variances. It is reduced to $\sum_{t=1}^{\infty}\eta_t=\infty$ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of the OMD algorithm using smoothness and strong convexity of the mirror map and the loss function.Comment: Published in Applied and Computational Harmonic Analysis, 202

Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms

This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on $\ell_p$-norm regularization, where the parameter $p$ controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art

Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent

Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, strong smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introduction of a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting using stability approach. Secondly, the smoothness assumption is relaxed by considering loss functions with Holder continuous (sub)gradients for which we show that optimal bounds are still achieved by balancing computation and stability. To our best knowledge, this gives the first-ever-known stability and generalization bounds for SGD with even non-differentiable loss functions. Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.Comment: to appear in ICML 202