On the Dichotomy Between Privacy and Traceability in $\ell_p$ Stochastic Convex Optimization

• URL: http://arxiv.org/abs/2502.17384v1
• Date: Mon, 24 Feb 2025 18:10:06 GMT
• Title: On the Dichotomy Between Privacy and Traceability in $\ell_p$ Stochastic Convex Optimization
• Authors: Sasha Voitovych, Mahdi Haghifam, Idan Attias, Gintare Karolina Dziugaite, Roi Livni, Daniel M. Roy,
• Abstract summary: We investigate the necessity of memorization in convex optimization (SCO) under $ell_p$ geometries.<n>Our main results uncover a fundamental tradeoff between traceability and excess risk in SCO.
• Score: 34.23960368886818
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we investigate the necessity of memorization in stochastic convex optimization (SCO) under $\ell_p$ geometries. Informally, we say a learning algorithm memorizes $m$ samples (or is $m$-traceable) if, by analyzing its output, it is possible to identify at least $m$ of its training samples. Our main results uncover a fundamental tradeoff between traceability and excess risk in SCO. For every $p\in [1,\infty)$, we establish the existence of a risk threshold below which any sample-efficient learner must memorize a \em{constant fraction} of its sample. For $p\in [1,2]$, this threshold coincides with best risk of differentially private (DP) algorithms, i.e., above this threshold, there are algorithms that do not memorize even a single sample. This establishes a sharp dichotomy between privacy and traceability for $p \in [1,2]$. For $p \in (2,\infty)$, this threshold instead gives novel lower bounds for DP learning, partially closing an open problem in this setup. En route of proving these results, we introduce a complexity notion we term \em{trace value} of a problem, which unifies privacy lower bounds and traceability results, and prove a sparse variant of the fingerprinting lemma.

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