Producing statistics that respect the privacy of the samples while still maintaining their accuracy is an important topic of research. We study minimax lower bounds when the class of estimators is restricted to the differentially private ones. In particular, we show that characterizing the power of a distributional test under differential privacy can be done by solving a transport problem. With specific coupling constructions, this observation allows us to derivate Le Cam-type and Fano-type inequalities for both regular definitions of differential privacy and for divergence-based ones (based on Renyi divergence). We then proceed to illustrate our results on three simple, fully worked out examples. In particular, we show that the problem class has a huge importance on the provable degradation of utility due to privacy. For some problems, privacy leads to a provable degradation only when the rate of the privacy parameters is small enough whereas for other problem, the degradation systematically occurs under much looser hypotheses on the privacy parametters. Finally, we show that the known privacy guarantees of DP-SGLD, a private convex solver, when used to perform maximum likelihood, leads to an algorithm that is near-minimax optimal in both the sample size and the privacy tuning parameters of the problem for a broad class of parametric estimation procedures that includes exponential families.
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We study the hidden-action principal-agent problem in an online setting. In each round, the principal posts a contract that specifies the payment to the agent based on each outcome. The agent then makes a strategic choice of action that maximizes her own utility, but the action is not directly observable by the principal. The principal observes the outcome and receives utility from the agent's choice of action. Based on past observations, the principal dynamically adjusts the contracts with the goal of maximizing her utility. We introduce an online learning algorithm and provide an upper bound on its Stackelberg regret. We show that when the contract space is $[0,1]^m$, the Stackelberg regret is upper bounded by $\widetilde O(\sqrt{m} \cdot T^{1-C/m})$, and lower bounded by $\Omega(T^{1-1/(m+2)})$. This result shows that exponential-in-$m$ samples are both sufficient and necessary to learn a near-optimal contract, resolving an open problem on the hardness of online contract design. When contracts are restricted to some subset $\mathcal{F} \subset [0,1]^m$, we define an intrinsic dimension of $\mathcal{F}$ that depends on the covering number of the spherical code in the space and bound the regret in terms of this intrinsic dimension. When $\mathcal{F}$ is the family of linear contracts, the Stackelberg regret grows exactly as $\Theta(T^{2/3})$. The contract design problem is challenging because the utility function is discontinuous. Bounding the discretization error in this setting has been an open problem. In this paper, we identify a limited set of directions in which the utility function is continuous, allowing us to design a new discretization method and bound its error. This approach enables the first upper bound with no restrictions on the contract and action space.
The US Census Bureau will deliberately corrupt data sets derived from the 2020 US Census in an effort to maintain privacy, suggesting a painful trade-off between the privacy of respondents and the precision of economic analysis. To investigate whether this trade-off is inevitable, we formulate a semiparametric model of causal inference with high dimensional corrupted data. We propose a procedure for data cleaning, estimation, and inference with data cleaning-adjusted confidence intervals. We prove consistency, Gaussian approximation, and semiparametric efficiency by finite sample arguments, with a rate of $n^{-1/2}$ for semiparametric estimands that degrades gracefully for nonparametric estimands. Our key assumption is that the true covariates are approximately low rank, which we interpret as approximate repeated measurements and validate in the Census. In our analysis, we provide nonasymptotic theoretical contributions to matrix completion, statistical learning, and semiparametric statistics. Calibrated simulations verify the coverage of our data cleaning-adjusted confidence intervals and demonstrate the relevance of our results for 2020 Census data.
This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function (Stein, 1999, Section 6.7) with Fourier coefficients $\phi$($\alpha$^2 + j^2)^(--$\nu$--1/2). Convergence rates are studied for the joint maximum likelihood estimation of $\nu$ and $\phi$ when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a ''deterministic'' element of a continuous Sobolev space is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.
There are two methods for counting the number of occurrences of a string in another large string. One is to count the number of places where the string is found. The other is to determine how many pieces of string can be extracted without overlapping. The difference between the two becomes apparent when the string is part of a periodic pattern. This research reports that the difference is significant in estimating the occurrence probability of a pattern. In this study, the strings used in the experiments are approximated from time-series data. The task involves classifying strings by estimating the probability or computing the information quantity. First, the frequencies of all substrings of a string are computed. Each counting method may sometimes produce different frequencies for an identical string. Second, the probability of the most probable segmentation is selected. The probability of the string is the product of all probabilities of substrings in the selected segmentation. The classification results demonstrate that the difference in counting methods is statistically significant, and that the method without overlapping is better.
Naively storing a counter up to value $n$ would require $\Omega(\log n)$ bits of memory. Nelson and Yu [NY22], following work of [Morris78], showed that if the query answers need only be $(1+\epsilon)$-approximate with probability at least $1 - \delta$, then $O(\log\log n + \log\log(1/\delta) + \log(1/\epsilon))$ bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather $k$ counters for $k$ large. This motivates the following question: for $k$ large, can $k$ counters be simultaneously maintained using asymptotically less memory than $k$ times the cost of an individual counter? That is to say, does this problem benefit from an improved {\it amortized} space complexity bound? We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff in FOCS 2020 to prove lower bounds for streaming algorithms.
We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $\varepsilon$-LDP, each user has $m\ge1$ samples and the privacy of all $m$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing the privacy of all $m$ samples should make the estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $m$ samples per user is equivalent to having $m$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $m$ and the privacy parameter $\varepsilon$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings.
In diverse fields ranging from finance to omics, it is increasingly common that data is distributed and with multiple individual sources (referred to as ``clients'' in some studies). Integrating raw data, although powerful, is often not feasible, for example, when there are considerations on privacy protection. Distributed learning techniques have been developed to integrate summary statistics as opposed to raw data. In many of the existing distributed learning studies, it is stringently assumed that all the clients have the same model. To accommodate data heterogeneity, some federated learning methods allow for client-specific models. In this article, we consider the scenario that clients form clusters, those in the same cluster have the same model, and different clusters have different models. Further considering the clustering structure can lead to a better understanding of the ``interconnections'' among clients and reduce the number of parameters. To this end, we develop a novel penalization approach. Specifically, group penalization is imposed for regularized estimation and selection of important variables, and fusion penalization is imposed to automatically cluster clients. An effective ADMM algorithm is developed, and the estimation, selection, and clustering consistency properties are established under mild conditions. Simulation and data analysis further demonstrate the practical utility and superiority of the proposed approach.
In this paper, we present several new results on minimizing a nonsmooth and nonconvex function under a Lipschitz condition. Recent work shows that while the classical notion of Clarke stationarity is computationally intractable up to some sufficiently small constant tolerance, the randomized first-order algorithms find a $(\delta, \epsilon)$-Goldstein stationary point with the complexity bound of $\tilde{O}(\delta^{-1}\epsilon^{-3})$, which is independent of dimension $d \geq 1$~\citep{Zhang-2020-Complexity, Davis-2022-Gradient, Tian-2022-Finite}. However, the deterministic algorithms have not been fully explored, leaving open several problems in nonsmooth nonconvex optimization. Our first contribution is to demonstrate that the randomization is \textit{necessary} to obtain a dimension-independent guarantee, by proving a lower bound of $\Omega(d)$ for any deterministic algorithm that has access to both $1^{st}$ and $0^{th}$ oracles. Furthermore, we show that the $0^{th}$ oracle is \textit{essential} to obtain a finite-time convergence guarantee, by showing that any deterministic algorithm with only the $1^{st}$ oracle is not able to find an approximate Goldstein stationary point within a finite number of iterations up to sufficiently small constant parameter and tolerance. Finally, we propose a deterministic smoothing approach under the \textit{arithmetic circuit} model where the resulting smoothness parameter is exponential in a certain parameter $M > 0$ (e.g., the number of nodes in the representation of the function), and design a new deterministic first-order algorithm that achieves a dimension-independent complexity bound of $\tilde{O}(M\delta^{-1}\epsilon^{-3})$.
The ongoing transition from a linear (produce-use-dispose) to a circular economy poses significant challenges to current state-of-the-art information and communication technologies. In particular, the derivation of integrated, high-level views on material, process, and product streams from (real-time) data produced along value chains is challenging for several reasons. Most importantly, sufficiently rich data is often available yet not shared across company borders because of privacy concerns which make it impossible to build integrated process models that capture the interrelations between input materials, process parameters, and key performance indicators along value chains. In the current contribution, we propose a privacy-preserving, federated multivariate statistical process control (FedMSPC) framework based on Federated Principal Component Analysis (PCA) and Secure Multiparty Computation to foster the incentive for closer collaboration of stakeholders along value chains. We tested our approach on two industrial benchmark data sets - SECOM and ST-AWFD. Our empirical results demonstrate the superior fault detection capability of the proposed approach compared to standard, single-party (multiway) PCA. Furthermore, we showcase the possibility of our framework to provide privacy-preserving fault diagnosis to each data holder in the value chain to underpin the benefits of secure data sharing and federated process modeling.
Firms and statistical agencies that publish or collect data face practical and legal requirements to protect the privacy of individuals. Increasingly, these organizations meet these standards by using publication mechanisms that satisfy differential privacy. We consider the problem of choosing such a mechanism so as to maximize the value of its output to end users. We show that this is a constrained information design problem, and characterize its solution. When the underlying database is drawn from a symmetric distribution -- for instance, if individuals' data are i.i.d. -- we show that the problem's dimensionality can be reduced, and that its solution belongs to a simpler class of mechanisms. When, in addition, data users have supermodular payoffs, we show that the simple geometric mechanism is always optimal by using a novel comparative static that ranks information structures according to their usefulness in supermodular decision problems.
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