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arXiv · 2502.17384

On Traceability in $\ell_p$ Stochastic Convex Optimization

Sasha Voitovych·Mahdi Haghifam·Idan Attias·Gintare Karolina Dziugaite·Roi Livni·Daniel M. Roy
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Abstract

In this paper, we investigate the necessity of traceability for accurate learning in stochastic convex optimization (SCO) under $\ell_p$ geometries. Informally, we say a learning algorithm 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 an excess risk threshold below which every sample-efficient learner is traceable with the number of samples which is a constant fraction of its training sample. For $p\in [1,2]$, this threshold coincides with the best excess risk of differentially private (DP) algorithms, i.e., above this threshold, there exist algorithms that are not traceable, which corresponds to a sharp phase transition. 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 to establishing these results, we prove a sparse variant of the fingerprinting lemma, which is of independent interest to the community.

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Sasha Voitovych, Mahdi Haghifam, Idan Attias, Gintare Karolina Dziugaite, Roi Livni, Daniel M. Roy. 2025-02-24. On Traceability in $\ell_p$ Stochastic Convex Optimization. https://arxiv.org/abs/2502.17384

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