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Differential Privacy in Personalized Pricing with Nonparametric Demand Models

arXiv.org Machine Learning

In the recent decades, the advance of information technology and abundant personal data facilitate the application of algorithmic personalized pricing. However, this leads to the growing concern of potential violation of privacy due to adversarial attack. To address the privacy issue, this paper studies a dynamic personalized pricing problem with \textit{unknown} nonparametric demand models under data privacy protection. Two concepts of data privacy, which have been widely applied in practices, are introduced: \textit{central differential privacy (CDP)} and \textit{local differential privacy (LDP)}, which is proved to be stronger than CDP in many cases. We develop two algorithms which make pricing decisions and learn the unknown demand on the fly, while satisfying the CDP and LDP gurantees respectively. In particular, for the algorithm with CDP guarantee, the regret is proved to be at most $\tilde O(T^{(d+2)/(d+4)}+\varepsilon^{-1}T^{d/(d+4)})$. Here, the parameter $T$ denotes the length of the time horizon, $d$ is the dimension of the personalized information vector, and the key parameter $\varepsilon>0$ measures the strength of privacy (smaller $\varepsilon$ indicates a stronger privacy protection). On the other hand, for the algorithm with LDP guarantee, its regret is proved to be at most $\tilde O(\varepsilon^{-2/(d+2)}T^{(d+1)/(d+2)})$, which is near-optimal as we prove a lower bound of $\Omega(\varepsilon^{-2/(d+2)}T^{(d+1)/(d+2)})$ for any algorithm with LDP guarantee.


The Price of Differential Privacy For Online Learning

arXiv.org Machine Learning

We design differentially private algorithms for the problem of online linear optimization in the full information and bandit settings with optimal $\tilde{O}(\sqrt{T})$ regret bounds. In the full-information setting, our results demonstrate that $\epsilon$-differential privacy may be ensured for free -- in particular, the regret bounds scale as $O(\sqrt{T})+\tilde{O}\left(\frac{1}{\epsilon}\right)$. For bandit linear optimization, and as a special case, for non-stochastic multi-armed bandits, the proposed algorithm achieves a regret of $\tilde{O}\left(\frac{1}{\epsilon}\sqrt{T}\right)$, while the previously known best regret bound was $\tilde{O}\left(\frac{1}{\epsilon}T^{\frac{2}{3}}\right)$.


Locally Differentially Private (Contextual) Bandits Learning

arXiv.org Machine Learning

We study locally differentially private (LDP) bandits learning in this paper. First, we propose simple black-box reduction frameworks that can solve a large family of context-free bandits learning problems with LDP guarantee. Based on our frameworks, we can improve previous best results for private bandits learning with one-point feedback, such as private Bandits Convex Optimization, and obtain the first result for Bandits Convex Optimization (BCO) with multi-point feedback under LDP. LDP guarantee and black-box nature make our frameworks more attractive in real applications compared with previous specifically designed and relatively weaker differentially private (DP) context-free bandits algorithms. Further, we extend our $(\varepsilon, \delta)$-LDP algorithm to Generalized Linear Bandits, which enjoys a sub-linear regret $\tilde{O}(T^{3/4}/\varepsilon)$ and is conjectured to be nearly optimal. Note that given the existing $\Omega(T)$ lower bound for DP contextual linear bandits (Shariff&Sheffe, 2018), our result shows a fundamental difference between LDP and DP contextual bandits learning.


Differentially Private Contextual Linear Bandits

Neural Information Processing Systems

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day t is only revealed to user t and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.


Differentially Private Contextual Linear Bandits

Neural Information Processing Systems

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day t is only revealed to user t and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.