We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration.

This is meaningful because in most applications, the time complexities for evaluating gradients at different points are of the same magnitude.

SVR methods use control variates to reduce the variance of the traditional stochastic gradient descent (SGD) estimate f0i(w) of the full gradient f0(w). Control variates are a classical technique for reducing the variance of a stochastic quantity without introducing bias. Say we have some random variable X.

Stochastic Variance Reduced Gradient (SVRG) and its variants aim to speed-up training by using gradient corrections, but have seen limited success in deep learning. Here, we show surprising new foundational connections of SVRG to a recently proposed Bayesian method called posterior correction. Specifically, we show that SVRG is recovered as a special case of posterior correction over the isotropic-Gaussian family, while novel extensions are automatically obtained by using more flexible exponential families. We derive two new SVRG variants by using Gaussian families: First, a Newton-like variant that employs novel Hessian corrections, and second, an Adam-like extension that improves pretraining and finetuning of Transformer language models. This is the first work to connect SVRG to Bayes and use it to boost variational training for deep networks.

This paper studies empirical risk minimization (ERM) problems for large-scale datasets and incorporates the idea of adaptive sample size methods to improve the guaranteed convergence bounds for first-order stochastic and deterministic methods. In contrast to traditional methods that attempt to solve the ERM problem corresponding to the full dataset directly, adaptive sample size schemes start with a small number of samples and solve the corresponding ERM problem to its statistical accuracy. The sample size is then grown geometrically - e.g., scaling by a factor of two - and use the solution of the previous ERM as a warm start for the new ERM. Theoretical analyses show that the use of adaptive sample size methods reduces the overall computational cost of achieving the statistical accuracy of the whole dataset for a broad range of deterministic and stochastic first-order methods. The gains are specific to the choice of method. When particularized to, e.g., accelerated gradient descent and stochastic variance reduce gradient, the computational cost advantage is a logarithm of the number of training samples. Numerical experiments on various datasets confirm theoretical claims and showcase the gains of using the proposed adaptive sample size scheme.

We consider a composite convex optimization problem associated with regularized empirical risk minimization, which often arises in machine learning.

In contrast, sampling without replacement is far less understood, yet in practice it is very common, often easier to implement, and usually performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling under several scenarios, focusing on the natural regime of few passes over the data.

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