Year: 2026
Type: preprint
Abstract: We study mistake bounds for differentially private online learning and online prediction under oblivious realisable adversaries. Online learning requires the learner to release a hypothesis at each time step whereas in online prediction, the learner only needs to make predictions without releasing a hypothesis. Using a novel lower bound for private online learning and an upper bound for private prediction, we show that the sample complexity of these two problems are separated by a factor that grows with the time horizon for every class of finite Littlestone dimension $d$. First, we prove that every $(ε,δ)$-private online learner has a deterministic realisable stream of length $T$ on which the mistake bound is at least $\mathbb E [M_T]= O\left(\frac dε\log^{2/3} T\right)$. In particular, this is the first non-trivial lower in the range $1/T<δ<1/\log T)$ left open in earlier works[SR22,DSS24,LWY24]. Second, we prove that for every class of of Littlestone dimension $d$, there exists an $(ε,δ)$-jointly private predictor with at most $2^{2^{cd^2}}ε^{-2}\log^2(2/(εδ))$ expected mistakes, independently of $T$, for some absolute constant $c>0$. Thus, for every fixed class of finite Littlestone dimension when $δ=Θ(1/\log T)$, private learning requires $O\left(\log^{2/3} T\right)$ expected mistakes, whereas private prediction admits $O(\log\log T^2)$.
Source: arXiv (Cornell University)
Author: Sanyal, Amartya
Institution: University of Copenhagen
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220654434