Rivalry of Two Families of Algorithms for Memory-Restricted Streaming PCA

TL;DR

This paper compares two families of algorithms for memory-limited streaming PCA, analyzing their convergence rates and proposing improvements, with empirical results highlighting their respective strengths and weaknesses.

Contribution

It provides a detailed convergence analysis of a stochastic gradient descent-based algorithm and introduces a new power method algorithm with automatic block size adjustment.

Findings

The stochastic gradient descent-based algorithm's convergence depends on the learning rate.

The new power method algorithm automatically sets block sizes for faster convergence.

Empirical studies compare the practical performance of both algorithm families.

Abstract

We study the problem of recovering the subspace spanned by the first $k$ principal components of $d$-dimensional data under the streaming setting, with a memory bound of $O(kd)$. Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheless, the convergence rate of the family can be seriously affected by the learning rate of the descent steps and deserves more serious study. The second family is based on the power method over blocks of data, but setting the block size for its existing algorithms is not an easy task. In this paper, we analyze the convergence rate of a representative algorithm with decayed learning rate (Oja and Karhunen, 1985) in the first family for the general $k>1$ case. Moreover, we propose a novel algorithm for the second family that sets the block sizes automatically and dynamically with faster convergence rate. We then conduct empirical studies that fairly compare the two families on real-world data. The studies reveal the advantages and disadvantages of these two families.