On Traceability in $\ell_p$ Stochastic Convex Optimization

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 \emph{$m$-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 \emph{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.

Authors

• Sasha Voitovych
• Mahdi Haghifam
• Idan Attias Ben-Gurion University
• Gintare Karolina Dziugaite
• Roi Livni
• Dan Roy

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