Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes
This addresses a theoretical gap for practitioners using SGD on complex models, though it is incremental as it builds on existing kernel method frameworks.
The paper tackles the statistical optimality of stochastic gradient descent (SGD) for least-squares regression, showing that multiple passes over the data lead to optimal predictions on hard learning problems, while a single pass does not, with the optimal number of passes increasing with sample size.
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.