Do you pay for Privacy in Online learning Giorgia Ramponi GIORGIA .RAMPONI AI.ETHZ .CH

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arXiv:2210.04817v1 [cs.LG] 10 Oct 2022

Proceedings of Machine Learning Research vol xxx:1–5, 2022

Do you pay for Privacy in Online learning?

Giorgia Ramponi GIORGIA.RAMPONI@AI.ETHZ.CH

ETH AI Center, Zurich, Switzerland

Amartya Sanyal AMARTYA.SANYAL@AI.ETHZ.CH

ETH AI Center, Zurich, Switzerland

Online learning, in the mistake bound model, is one of the most fundamental concepts in learn-

ing theory. Differential privacy, instead, is the most widely used statistical concept of privacy in

the machine learning community. It is thus clear that deﬁning learning problems that are online

differentially privately learnable is of great interest. In this paper, we pose the question on if the two

problems are equivalent from a learning perspective, i.e., is privacy for free in the online learning

framework?

Keywords: Online Learning, Differential Privacy, Mistake Bound Model

1. Introduction

Online learning, in the mistake bound model, is one of the most fundamental concepts in learning

theory. Let X=SXnbe the instance space. The learner, A, in this model, receives at each

timestep tan unlabelled example xt∈X, predicts a label bytcorresponding to xt, and then re-

ceives the true label ytfor xt. During this interaction, the learner maintains a working hypothesis

ht=A{(xτ, yτ)}t−1

τ=1, which it uses to predict byt=ht(xt), and then uses the true label ytto up-

date the working hypothesis to ht+1. The performance of the learner Ais measured by the number

of mistakes it makes, i.e.,:

Mistakes (A, T, (xt, yt)∞

t=1) :=

t=1

(ht(xt)6=yt).(1)

Given this deﬁnition of performance, a hypothesis class Con the instance space X=SXnis said

to be online learnable in the mistake bound model if there exists a learner Lthat makes at most

poly (n, size (c)) mistakes on any sequence of samples consistent with a concept c∈ C, where pis

some polynomial. This is also known as the realisable setting.

Another relevant concept in learning theory is privacy. The most widely used statistical notion

of privacy in the machine learning literature is differential privacy. An (ǫ, δ)-differentially pri-

vate (randomised) algorithm is guaranteed to output similar distributions over the output space of

the algorithm when presented with inputs that only differ in one element. More formally, in the of-

ﬂine setting, a learning algorithm A:X → Y is said to be (ǫ, δ)-differentially private if, for any two

datasets S1, S2that differ in just one element, we have that P[A(S1)∈Q]≤eǫP[A(S2)∈Q] + δ

where Q⊆ Y is any subset of the output space of the algorithm. We deﬁne differential privacy in

the online setting in Deﬁnition 3.

Some previous works (Jain et al.,2012;Agarwal and Singh,2017;Abernethy et al.,2019) treat

the problem of constructing online learning algorithms (mostly in the regret minimization setting

(Shalev-Shwartz and Singer,2007)) maintaining the differential-privacy properties. However, it is

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摘要：

arXiv:2210.04817v1[cs.LG]10Oct2022ProceedingsofMachineLearningResearchvolxxx:1–5,2022DoyoupayforPrivacyinOnlinelearning?GiorgiaRamponiGIORGIA.RAMPONI@AI.ETHZ.CHETHAICenter,Zurich,SwitzerlandAmartyaSanyalAMARTYA.SANYAL@AI.ETHZ.CHETHAICenter,Zurich,SwitzerlandOnlinelearning,inthemistakeboundmodel,ison...

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Do you pay for Privacy in Online learning Giorgia Ramponi GIORGIA .RAMPONI AI.ETHZ .CH

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