On Traceability in ℓp Stochastic Convex Optimization

In this paper, we investigate the necessity of traceability for accurate learning in stochastic convex optimization (SCO) under ℓp 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∈ [1,∞), 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∈ [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 ∈ (2,∞), 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.