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

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.

Machine Learning (cs.LG); Data Structures and Algorithms (cs.DS); Machine Learning (stat.ML)

arXiv:2609.26508 [cs.LG]

(or arXiv:2609.26508v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.26508 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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