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Private Online Learning via Lazy Algorithms

Hilal Asi, Tomer Koren, Daogao Liu, Kunal Talwar

TL;DR

Private Online Learning via Lazy Algorithms studies privatizing online learning for DP-OPE and DP-OCO through a novel L2P transformation that converts lazy algorithms into DP ones with a small regret overhead. The method leverages a history-local switching rule and large-batch correlated sampling to bound privacy loss, enabling improved regret rates: $\sqrt{T \log d} + T^{1/3} \log d / \varepsilon^{2/3}$ for DP-OPE and $\sqrt{T} + T^{1/3} \sqrt{d} / \varepsilon^{2/3}$ for DP-OCO. A near-matching lower bound for low-switching private algorithms shows these rates are close to optimal within that family, motivating exploration beyond limited switching. The paper applies L2P to multiplicative weights and to a regularized MW scheme, delivering practical improvements in high-privacy regimes and suggesting avenues for future work on breaking the low-switching barrier.

Abstract

We study the problem of private online learning, specifically, online prediction from experts (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret, which significantly improves the regret in the high privacy regime $\varepsilon \ll 1$, obtaining $\sqrt{T \log d} + T^{1/3} \log(d)/\varepsilon^{2/3}$ for DP-OPE and $\sqrt{T} + T^{1/3} \sqrt{d}/\varepsilon^{2/3}$ for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.

Private Online Learning via Lazy Algorithms

TL;DR

Private Online Learning via Lazy Algorithms studies privatizing online learning for DP-OPE and DP-OCO through a novel L2P transformation that converts lazy algorithms into DP ones with a small regret overhead. The method leverages a history-local switching rule and large-batch correlated sampling to bound privacy loss, enabling improved regret rates: for DP-OPE and for DP-OCO. A near-matching lower bound for low-switching private algorithms shows these rates are close to optimal within that family, motivating exploration beyond limited switching. The paper applies L2P to multiplicative weights and to a regularized MW scheme, delivering practical improvements in high-privacy regimes and suggesting avenues for future work on breaking the low-switching barrier.

Abstract

We study the problem of private online learning, specifically, online prediction from experts (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret, which significantly improves the regret in the high privacy regime , obtaining for DP-OPE and for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.
Paper Structure (27 sections, 19 theorems, 68 equations, 1 figure, 1 table)

This paper contains 27 sections, 19 theorems, 68 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. $\mathsf{L2P}$: a transformation from lazy to private algorithms (Section \ref{['sec:ltp']}).
  4. Faster rates for DP-OPE (Section \ref{['sec:ope']}).
  5. Faster rates for DP-OCO (Section \ref{['sec:oco']}).
  6. Lower bounds for low-switching private algorithms (Section \ref{['sec:LB']}).
  7. Related work
  8. Lazy online learning.
  9. Differentially Private Stochastic Convex Optimization (DP-SCO).
  10. Preliminaries
  11. Problem setup
  12. Tools from differential privacy
  13. L2P: From Lazy to Private Algorithms for Online Learning
  14. Technique Overview:
  15. Case 1 $(s<s_0)$:
  16. ...and 12 more sections

Key Result

Lemma 2.4

Let $\varepsilon \le 1/10$. If $D_{\infty}^{\delta}(\mu, \nu) \le \varepsilon/2$ then we have

Figures (1)

  • Figure 1: Regret bounds for (a) DP-OCO with $d = \mathop{\mathrm{poly}}\nolimits\log(T)$, (b) DP-OCO with $d = T^{1/3}$ and (c) DP-OPE with $d=T$. We denote the privacy parameter $\varepsilon = T^{-\alpha}$ and regret $T^\beta$, and plot $\beta$ as a function of $\alpha$ (ignoring logarithmic factors).

Theorems & Definitions (35)

  • Definition 2.1: Differential privacy
  • Definition 2.2: $\delta$-Approximate Max Divergence
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6: Advanced Composition,kairouz2015composition
  • Definition 2.7: concentrated DP
  • Lemma 2.8: BunS16
  • Theorem 3.2
  • Proposition 3.3: Regret of Batched Lazy Algorithm
  • ...and 25 more