We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.

This paper identifies and separates (kernel) linear least-squares regression problems wherein carrying out multiple passes of stochastic gradient descent (SGD) over a training set can yield better statistical error than only a single pass. This is relevant to the core of machine learning theory, and relates to a line of work published at NIPS, ICML, COLT, and similar conferences in the past several years about the statistical error of one-pass, many-pass, and ERM-based learning. The authors focus on regression problems captured, by assumption, by two parameters: alpha, which governs the exponent of a power-law eigenvalue decay, and r, which governs a transformation under which the Hilbert norm of the optimal predictor is bounded. They refer to problems where r (alpha - 1) / (2 * alpha) as "hard". The main result of the paper is to show that for these "hard" problems, multiple SGD passes either achieve (minimax) optimal rates of statistical estimation, or at least improve the rate relative to a single pass. The results are interesting and might address an unanswered core question in machine learning, and the mathematical presentation is clear, with assumptions upfront.

We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.