Gap-Free Streaming PCA Beyond Rank-One Updates: Near-Optimal Rates and Applications to Differential Privacy

2026-09-22

AI Summary

本文解决了无间隙流式PCA问题，通过改进Oja算法分析方法，在无需特征间隙假设下实现近最优速率，并应用于差分隐私PCA。

Abstract

Streaming principal component analysis (PCA) seeks to recover a leading spectral subspace in a single pass over a data stream. We give a new analysis of the ubiquitous Oja's algorithm [Oja82] for the most general, gap-free variant of this problem, where no eigengap assumptions are made on the underlying mean matrix, complemented by a nearly-matching lower bound. Prior works achieving near-optimal rates for streaming PCA either required gap assumptions [JJK+16, HNWW21], or were limited to rank-one updates [AZL17, Lia23]. Our proof only uses a second moment bound on the individual stochastic updates, bypassing the almost sure bounds needed by prior near-optimal analyses, and the analogous offline matrix Bernstein bound. We also extend our result to a Rayleigh quotient notion of approximate PCA, addressing an open question of [JJK+16]. As our main application, we give gap-free differentially private PCA guarantees for sub-Gaussian data, settling Conjecture 1.1 of [Bro26] up to logarithmic factors.

Problem

• Streaming PCA
• Gap-Free
• Differential Privacy

Innovation

• Gap-Free Streaming PCA
• Oja's Algorithm
• Second Moment Bound
• Differential Privacy