Low-Precision Streaming PCA

TL;DR

This paper investigates the limits of low-precision quantization in streaming PCA, establishing bounds and proposing quantized variants of Oja's algorithm that nearly match these bounds under certain data assumptions.

Contribution

It provides an information-theoretic lower bound on quantization resolution and introduces nearly optimal low-precision streaming PCA algorithms with theoretical guarantees.

Findings

Quantization resolution bounds for streaming PCA.

Proposed quantized Oja's algorithms achieve near-optimal accuracy.

Empirical results confirm theoretical predictions and performance close to standard methods.

Abstract

Low-precision streaming PCA estimates the top principal component in a streaming setting under limited precision. We establish an information-theoretic lower bound on the quantization resolution required to achieve a target accuracy for the leading eigenvector. We study Oja's algorithm for streaming PCA under linear and nonlinear stochastic quantization. The quantized variants use unbiased stochastic quantization of the weight vector and the updates. Under mild moment and spectral-gap assumptions on the data distribution, we show that a batched version achieves the lower bound up to logarithmic factors under both schemes. This leads to a nearly dimension-free quantization error in the nonlinear quantization setting. Empirical evaluations on synthetic streams validate our theoretical findings and demonstrate that our low-precision methods closely track the performance of standard Oja's algorithm.