A Study on Differential Private Online Learning

Online learning algorithms are very attractive, in which iterations are applied efficiently instead of solving some optimization problems. In this paper, online learning with protecting privacy is considered. A perturbation term is added into the classical online algorithms to obtain the differential privacy property. Firstly the distribution for the perturbation term is deduced, and then an error analysis for the new algorithms is performed, which shows the convergence and learning rate. From the error analysis, a choice for the parameters for differential privacy can be found theoretically.


Introduction
Online learning is widely used recently in computer sciences, due to its efficiency in calculation and well theoretical results.Compared with the classical batch learning in learning theory, online algorithms update the output only according to the last sample point.So such algorithms are very effective to handle the practical problems and have been studied in [1] [2] [3] [4] [5] and etc.However, as the technologic development of data analysis, there are risks for applying such algorithms on a big data set.A commonly used notion for measuring the risk is differential privacy [6].Little references on this topic can be found except for [7].There the authors conducted an analysis for online convex programming.Choice for the parameters of differential privacy and utilities analysis are presented for algorithms such as implicit gradient descent and generalized infinitesimal gradient descent.In this paper, the line of work begins with [8] is considered which can be thought as a kernel online learning algorithm.

Fundamental Principles
Our setting for online learning is introduced as follow.Let the input space X be a compact metric space, and output [ ] for some 0 M > as a regression problem.Denote : Z X Y = × as the sample space.Assume there is a probability measure ρ on Z , which can be decomposed to marginal dis- tribution X ρ on X and conditional distribution ( ) Then the regression function is defined by ( ) which is indeed the conditional expectation of y given x .The regression fun- ction minimizes the least square generalization error (see [9] for more details) So learning algorithms always aim to approximate the regression function based on samples be a Mercer Kernel, and K  is the in- duced reproducing kernel Hilbert space (RKHS, [10]), i.e., the completion of { } span : x K x X ∈ where ( ) ( ) . The corresponding norm in K  is denoted as K ⋅ .Now our online learning algorithm as ( with 0 0 f = .Here 0 t η > is the step size and 0 t λ > is the regularization parameter.
When applying this online algorithm on private data set, it may leak some sensitive information.To deal with this privacy problem, Dwork et al. introduced differential privacy in [11].Which can be described as follow.For the sample space Z introduced above, the Hamming distance between two sample sets vate if for every two data sets 1 2 , z z satisfying ( ) , there holds To endow our online algorithm the differential privacy property, a perturbation term is added into the output of (3), that is, where t b takes value in R with distribution to be determined in following analysis.
Differential private online learning has already been studied in [7], there the authors consider a differentially private online convex programming problem.
Here our algorithm is different, which is based on the Mercer kernels.Our purpose in this paper is to firstly provide the explicit density function for b and then conduct an error analysis for (6), which reveals the learning rate.

Differential Privacy Analysis
In this section, a detail analysis for the perturbation term t b in algorithm (6)   will be conducted.Firstly recall the useful definition of sensitivity and lemma proposed in [11].
Definition 2 denote t f ∆ as the maximum infinite norm of difference between the outputs when changing the last sample point in z .Let Then a similar result to [11] is: , then algorithm (6) provides  -differential privacy.
Proof.For all possible output function r , and , and So by triangle inequality,

Pr
Pr Pr Then the lemma is proved by a union bound.


It is obvious that if finding the upper bound for t f ∆ , the distribution for t b can be derived.Set ( ) ( ) ≥ + , then for all t ∈  , there holds Now the main result for differential privacy for algorithm (6) follows.
Theorem 1 When choosing then the algorithm (6) provides  -differential privacy.Proof.From (3) there holds ( ) ( ) ( ) ( ) From the above lemma ≤ for all t .By the reproducing property that ( ) Set t B to be the right hand side in lemma 1 then the theorem is proved.

Error Analysis
In this section, K f ρ ∈  is assumed for simplicity.It will be shown that t f obtained from (6) still converge to regression function f ρ by choosing appropriate parameter  under the choice of t η and t λ as in the theorem in the last section.To this end, an error decomposition is needed.Denote operators :  , and I as the identity operator.It is easy to verify that decomposition can be deduced: ] Here In the following the first term is called initial error and second one is sample error.The initial error is easy to bound from the analysis above.Since 0 t is such that A is a positive operator with ( ) ( ) For the sample error, it is more difficult and the Pinelis-Bernstein inequality [4] will be applied.
Lemma 3 Let i ξ be a martingale difference sequence in a Hilbert space.Sup- pose that almost surely i B ξ ≤ and Then for any 0 1 δ < < , with probability at least 1 δ − , there holds Now the error bounds for sample error can be derived.Notice that And ( ) Note that ( ) Combining the initial error, sample error bounds and applying Markov inequality for the fact that

Conclusion
In this paper, analysis is performed for the differential privacy (Theorem 1) and generalization property (Theorem 2) for the online differential private learning algorithm (6).Under the choice of parameters in our theorems, the algorithm ≥ + , let the density function of t b is total error estimation is obtained.Theorem 2 Choose , t t η λ and t b as in the theorem in the last section, with confidence

( 6 )
can provide  -differential privacy and keep learning rate close to 1 2 , for any 0 >  .However, this error bound is not satisfactory enough.It might be an interesting problem to promote the error bound from 2 δ to ( ) ln 2 δ in our