Carathéodory Sampling for Stochastic Gradient Descent
This addresses the problem of optimizing empirical risk functions over large datasets for machine learning practitioners, offering a novel approach to reduce variance in SGD.
The paper tackles the high variance issue in Stochastic Gradient Descent (SGD) by introducing a method based on Carathéodory measure reduction to replace empirical measures with smaller, carefully constructed probability measures that preserve expected gradients, resulting in scalable algorithms with experimental validation.
Many problems require to optimize empirical risk functions over large data sets. Gradient descent methods that calculate the full gradient in every descent step do not scale to such datasets. Various flavours of Stochastic Gradient Descent (SGD) replace the expensive summation that computes the full gradient by approximating it with a small sum over a randomly selected subsample of the data set that in turn suffers from a high variance. We present a different approach that is inspired by classical results of Tchakaloff and Carathéodory about measure reduction. These results allow to replace an empirical measure with another, carefully constructed probability measure that has a much smaller support, but can preserve certain statistics such as the expected gradient. To turn this into scalable algorithms we firstly, adaptively select the descent steps where the measure reduction is carried out; secondly, we combine this with Block Coordinate Descent so that measure reduction can be done very cheaply. This makes the resulting methods scalable to high-dimensional spaces. Finally, we provide an experimental validation and comparison.