If A is (",0)- di erentiallyprivate, thenA is " Yis (", )- di B : Y anyrandomB Ais (", )- di erA is thenthealgoB Ais - z CDP. Proof.Algorithm1 isthedesiredsamplerAPo.Itis (", ) - di erentiallyprivatesinceA is (", ) - di erentially private.

Wewill also writeζθ(x) = fθ(x) f?(x)to denote the fitting error. We use Gaussian initialization: if` {1,2,...,L}, the weights are initialized as

In the deterministic setting where the data is deterministically given without any probabilistic assumptions, significant advances inDP linear regression has been made [77,57,68, 16, 7, 83, 31, 67, 82, 71]. In the randomized settings where each example{xi,yi} is drawn i.i.d. We explain the closely related ones in Section 2.3, with analysis when the covariance matrixhasaspectralgap. The resulting utility guarantees are the same as those from [23], which are discussedinSection2.3. When privacy is not required, we know from Theorem 2.2 that under Assumptions A.1-A.3, we can achieve an error rate of O(κ p V/n).

Policy optimization is a widely-used method in reinforcement learning.

Thefollowing lemma provides moment bounds for a summation of weakly dependent and mean-zero random functions withbounded increments underachange ofanysinglecoordinate [1,10]. The stated bound then follows by combining the above two inequalities together. Note A(S0) is independent ofS and can be considered as a fixed model if we only consider the randomness induced fromS. In this section, we present the proof related to stability and generalization for pairwise learning with convex and smooth loss functions. For anyi [n], define Si as (3.3).

The expected improvement (EI) is a popular technique to handle the tradeoff between exploration andexploitation underuncertainty. Thistechnique hasbeen widely used in Bayesian optimization but it is not applicable for the contextual bandit problem which is a generalization of the standard bandit and Bayesian optimization.

First,theobtained high-probability regret bounds are data-dependent and could be much smaller thantheworst-case bounds, which resolvesanopenproblem askedbyNeu[31].

Recall thatx = argmina Ax>θ so x can be viewed as a deterministic functionθ . " log p(zn|θ) (1/|Nε|) P Since Rmax is the upper bound of maximum expected reward, the second term can be bounded 2Rmaxγ. We letΦ R|A| d as the feature matrix where each row ofΦrepresent each action inA. We summarize the procedure of estimating t,It inAlgorithm3. LetX denote the feasible set.

Weproveupper and lower bounds on the amount ofqueries that are needed for an auditor to succeed within this framework.

In Appendix A, we review some statistical results for sparse linear regression. We review some classical results in sparse linear regression. Let the design matrix beX = (x1,...,xn)> Rn d. Second, we derive a regret lower bound of alternative banditeθ.