Improved Regret in Stochastic Decision-Theoretic Online Learning under Differential Privacy

TL;DR

This paper advances understanding of the optimal rates in differentially private stochastic online learning by providing improved bounds and analyzing a deterministic setting, narrowing the gap between upper and lower bounds.

Contribution

It offers an improved upper bound for the problem and introduces a deterministic setting to analyze the fundamental gap in the bounds.

Findings

New upper bound: $O(rac{ extlog K}{ extDelta_{min}} + rac{ extlog^2 K}{ extvarepsilon})$

Matching bounds in the deterministic setting: $ heta(rac{ extlog K}{ extvarepsilon})$

Insights into the gap between existing bounds and fundamental limits.

Abstract

Hu and Mehta (2024) posed an open problem: what is the optimal instance-dependent rate for the stochastic decision-theoretic online learning (with $K$ actions and $T$ rounds) under $\varepsilon$-differential privacy? Before, the best known upper bound and lower bound are $O\left(\frac{\log K}{\Delta_{\min}} + \frac{\log K\log T}{\varepsilon}\right)$ and $\Omega\left(\frac{\log K}{\Delta_{\min}} + \frac{\log K}{\varepsilon}\right)$ (where $\Delta_{\min}$ is the gap between the optimal and the second actions). In this paper, we partially address this open problem by having two new results. First, we provide an improved upper bound for this problem $O\left(\frac{\log K}{\Delta_{\min}} + \frac{\log^2K}{\varepsilon}\right)$, which is $T$-independent and only has a log dependency in $K$. Second, to further understand the gap, we introduce the \textit{deterministic setting}, a weaker setting of this open problem, where the received loss vector is deterministic. At this weaker setting, a direct application of the analysis and algorithms from the original setting still leads to an extra log factor. We conduct a novel analysis which proves upper and lower bounds that match at $\Theta(\frac{\log K}{\varepsilon})$.