Stochastic Variance-Reduced Newton: Accelerating Finite-Sum Minimization with Large Batches
This work addresses optimization efficiency for large-scale machine learning tasks, offering a novel acceleration for stochastic Newton methods with practical benefits like parallelizability, though it is incremental in building on existing variance reduction and second-order techniques.
The paper tackles the problem of accelerating stochastic second-order methods for finite-sum minimization by proposing Stochastic Variance-Reduced Newton (SVRN), which reduces the number of data passes from O(α log(1/ε)) to O(log(1/ε)/log(n)), achieving a factor of O(α log(n)) improvement that increases with data size n.
Stochastic variance reduction has proven effective at accelerating first-order algorithms for solving convex finite-sum optimization tasks such as empirical risk minimization. Incorporating second-order information has proven helpful in further improving the performance of these first-order methods. Yet, comparatively little is known about the benefits of using variance reduction to accelerate popular stochastic second-order methods such as Subsampled Newton. To address this, we propose Stochastic Variance-Reduced Newton (SVRN), a finite-sum minimization algorithm that provably accelerates existing stochastic Newton methods from $O(α\log(1/ε))$ to $O\big(\frac{\log(1/ε)}{\log(n)}\big)$ passes over the data, i.e., by a factor of $O(α\log(n))$, where $n$ is the number of sum components and $α$ is the approximation factor in the Hessian estimate. Surprisingly, this acceleration gets more significant the larger the data size $n$, which is a unique property of SVRN. Our algorithm retains the key advantages of Newton-type methods, such as easily parallelizable large-batch operations and a simple unit step size. We use SVRN to accelerate Subsampled Newton and Iterative Hessian Sketch algorithms, and show that it compares favorably to popular first-order methods with variance~reduction.