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Register a new userDifferential Privacy in Personalized Pricing with Nonparametric Demand Models
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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.
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The prevalence of e-commerce has made detailed customers personal information readily accessible to retailers, and this information has been widely used in pricing decisions. When involving personalized information, how to protect the privacy of such information becomes a critical issue in practice. In this paper, we consider a dynamic pricing problem over $T$ time periods with an emph{unknown} demand function of posted price and personalized information. At each time $t$, the retailer observes an arriving customers personal information and offers a price. The customer then makes the purchase decision, which will be utilized by the retailer to learn the underlying demand function. There is potentially a serious privacy concern during this process: a third party agent might infer the personalized information and purchase decisions from price changes from the pricing system. Using the fundamental framework of differential privacy from computer science, we develop a privacy-preserving dynamic pricing policy, which tries to maximize the retailer revenue while avoiding information leakage of individual customers information and purchasing decisions. To this end, we first introduce a notion of emph{anticipating} $(varepsilon, delta)$-differential privacy that is tailored to dynamic pricing problem. Our policy achieves both the privacy guarantee and the performance guarantee in terms of regret. Roughly speaking, for $d$-dimensional personalized information, our algorithm achieves the expected regret at the order of $tilde{O}(varepsilon^{-1} sqrt{d^3 T})$, when the customers information is adversarially chosen. For stochastic personalized information, the regret bound can be further improved to $tilde{O}(sqrt{d^2T} + varepsilon^{-2} d^2)$
Because learning sometimes involves sensitive data, machine learning algorithms have been extended to offer privacy for training data. In practice, this has been mostly an afterthought, with privacy-preserving models obtained by re-running training with a different optimizer, but using the model architectures that already performed well in a non-privacy-preserving setting. This approach leads to less than ideal privacy/utility tradeoffs, as we show here. Instead, we propose that model architectures are chosen ab initio explicitly for privacy-preserving training. To provide guarantees under the gold standard of differential privacy, one must bound as strictly as possible how individual training points can possibly affect model updates. In this paper, we are the first to observe that the choice of activation function is central to bounding the sensitivity of privacy-preserving deep learning. We demonstrate analytically and experimentally how a general family of bounded activation functions, the tempered sigmoids, consistently outperform unbounded activation functions like ReLU. Using this paradigm, we achieve new state-of-the-art accuracy on MNIST, FashionMNIST, and CIFAR10 without any modification of the learning procedure fundamentals or differential privacy analysis.
We consider a firm that sells products over $T$ periods without knowing the demand function. The firm sequentially sets prices to earn revenue and to learn the underlying demand function simultaneously. A natural heuristic for this problem, commonly used in practice, is greedy iterative least squares (GILS). At each time period, GILS estimates the demand as a linear function of the price by applying least squares to the set of prior prices and realized demands. Then a price that maximizes the revenue, given the estimated demand function, is used for the next time period. The performance is measured by the regret, which is the expected revenue loss from the optimal (oracle) pricing policy when the demand function is known. Recently, den Boer and Zwart (2014) and Keskin and Zeevi (2014) demonstrated that GILS is sub-optimal. They introduced algorithms which integrate forced price dispersion with GILS and achieve asymptotically optimal performance. In this paper, we consider this dynamic pricing problem in a data-rich environment. In particular, we assume that the firm knows the expected demand under a particular price from historical data, and in each period, before setting the price, the firm has access to extra information (demand covariates) which may be predictive of the demand. We prove that in this setting GILS achieves asymptotically optimal regret of order $log(T)$. We also show the following surprising result: in the original dynamic pricing problem of den Boer and Zwart (2014) and Keskin and Zeevi (2014), inclusion of any set of covariates in GILS as potential demand covariates (even though they could carry no information) would make GILS asymptotically optimal. We validate our results via extensive numerical simulations on synthetic and real data sets.
What is the information leakage of an iterative learning algorithm about its training data, when the internal state of the algorithm is emph{not} observable? How much is the contribution of each specific training epoch to the final leakage? We study this problem for noisy gradient descent algorithms, and model the emph{dynamics} of Renyi differential privacy loss throughout the training process. Our analysis traces a provably tight bound on the Renyi divergence between the pair of probability distributions over parameters of models with neighboring datasets. We prove that the privacy loss converges exponentially fast, for smooth and strongly convex loss functions, which is a significant improvement over composition theorems. For Lipschitz, smooth, and strongly convex loss functions, we prove optimal utility for differential privacy algorithms with a small gradient complexity.
Privacy concerns with sensitive data are receiving increasing attention. In this paper, we study local differential privacy (LDP) in interactive decentralized optimization. By constructing random local aggregators, we propose a framework to amplify LDP by a constant. We take Alternating Direction Method of Multipliers (ADMM), and decentralized gradient descent as two concrete examples, where experiments support our theory. In an asymptotic view, we address the following question: Under LDP, is it possible to design a distributed private minimizer for arbitrary closed convex constraints with utility loss not explicitly dependent on dimensionality? As an affiliated result, we also show that with merely linear secret sharing, information theoretic privacy is achievable for bounded colluding agents.
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