Composition is a key feature of differential privacy. Well-known advanced composition theorems allow one to query a private database quadratically more times than basic privacy composition would permit. However, these results require that the privacy parameters of all algorithms be fixed before interacting with the data. To address this, Rogers et al. introduced fully adaptive composition, wherein both algorithms and their privacy parameters can be selected adaptively. They defined two probabilistic objects to measure privacy in adaptive composition: privacy filters, which provide differential privacy guarantees for composed interactions, and privacy odometers, time-uniform bounds on privacy loss. There are substantial gaps between advanced composition and existing filters and odometers. First, existing filters place stronger assumptions on the algorithms being composed. Second, these odometers and filters suffer from large constants, making them impractical. We construct filters that match the rates of advanced composition, including constants, despite allowing for adaptively chosen privacy parameters. En route we also derive a privacy filter for approximate zCDP. We also construct several general families of odometers. These odometers match the tightness of advanced composition at an arbitrary, preselected point in time, or at all points in time simultaneously, up to a doubly-logarithmic factor. We obtain our results by leveraging advances in martingale concentration. In sum, we show that fully adaptive privacy is obtainable at almost no loss.

Although there has been work to develop ex-post private mechanisms from Ligett et al. '17 and Whitehouse et al '22 that seeks to provide privacy guarantees subject to a target level of accuracy, there was not a way to use them in conjunction with differentially private mechanisms. Furthermore, there has yet to be work in developing a theory for how these ex-post privacy mechanisms compose, so that we can track the accumulated privacy over several mechanisms. We develop privacy filters that allow an analyst to adaptively switch between differentially private mechanisms and ex-post private mechanisms subject to an overall privacy loss guarantee. We show that using a particular ex-post private mechanism --- noise reduction mechanisms --- can substantially outperform baseline approaches that use existing privacy loss composition bounds. We use the common task of returning as many counts as possible subject to a relative error guarantee and an overall privacy budget as a motivating example.

We generalize the continuous observation privacy setting from Dwork et al. and Chan et al. by allowing each event in a stream to be a subset of some (possibly unknown) universe of items. We design differentially private (DP) algorithms for histograms in several settings, including top-k selection, with privacy loss that scales with polylog(T), where T is the maximum length of the input stream. We present a meta-algorithm that can use existing one-shot top-k private algorithms as a subroutine to continuously release DP histograms from a stream. Further, we present more practical DP algorithms for two settings: 1) continuously releasing the top-k counts from a histogram over a known domain when an event can consist of an arbitrary number of items, and 2) continuously releasing histograms over an unknown domain when an event has a limited number of items.

There is a disconnect between how researchers and practitioners handle privacy-utility tradeoffs. Researchers primarily operate from a privacy first perspective, setting strict privacy requirements and minimizing risk subject to these constraints. Practitioners often desire an accuracy first perspective, possibly satisfied with the greatest privacy they can get subject to obtaining sufficiently small error. Ligett et al. have introduced a `"noise reduction" algorithm to address the latter perspective. The authors show that by adding correlated Laplace noise and progressively reducing it on demand, it is possible to produce a sequence of increasingly accurate estimates of a private parameter and only pay a privacy cost for the least noisy iterate released. In this work, we generalize noise reduction to the setting of Gaussian noise, introducing the Brownian mechanism. The Brownian mechanism works by first adding Gaussian noise of high variance corresponding to the final point of a simulated Brownian motion. Then, at the practitioner's discretion, noise is gradually decreased by tracing back along the Brownian path to an earlier time. Our mechanism is more naturally applicable to the common setting of bounded $\ell_2$-sensitivity, empirically outperforms existing work on common statistical tasks, and provides customizable control of privacy loss over the entire interaction with the practitioner. We complement our Brownian mechanism with ReducedAboveThreshold, a generalization of the classical AboveThreshold algorithm that provides adaptive privacy guarantees. Overall, our results demonstrate that one can meet utility constraints while still maintaining strong levels of privacy.

We unify existing privacy loss composition bounds for special classes of differentially private (DP) algorithms along with general DP composition bounds. In particular, we provide strong privacy loss bounds when an analyst may select pure DP, bounded range (e.g. exponential mechanisms), or concentrated DP mechanisms in any adaptively selected order. We also provide optimal privacy loss bounds that apply when an analyst can select pure DP and bounded range mechanisms in a batch, i.e. non-adaptively. Further, when an analyst selects mechanisms within each class adaptively, we show a difference in privacy loss between different, predetermined orderings of pure DP and bounded range mechanisms. Lastly, we compare the composition bounds of Laplace and Gaussian mechanisms and provide new private mechanisms for top-$k$ using truncated Gaussian noise.

We design a general framework for answering adaptive statistical queries that focuses on providing explicit confidence intervals along with point estimates. Prior work in this area has either focused on providing tight confidence intervals for specific analyses, or providing general worst-case bounds for point estimates. Unfortunately, as we observe, these worst-case bounds are loose in many settings — often not even beating simple baselines like sample splitting. Our main contribution is to design a framework for providing valid, instance-specific confidence intervals for point estimates that can be generated by heuristics. When paired with good heuristics, this method gives guarantees that are orders of magnitude better than the best worst-case bounds. We provide a Python library implementing our method.

Composition is one of the most important properties of differential privacy (DP), as it allows algorithm designers to build complex private algorithms from DP primitives. We consider precise composition bounds of the overall privacy loss for exponential mechanisms, one of the fundamental classes of mechanisms in DP. Exponential mechanism has also become a fundamental building block in private machine learning, e.g. private PCA and hyper-parameter selection. We give explicit formulations of the optimal privacy loss for both the adaptive and non-adaptive composition of exponential mechanism. For the non-adaptive setting in which each mechanism has the same privacy parameter, we give an efficiently computable formulation of the optimal privacy loss. In the adaptive case, we derive a recursive formula and an efficiently computable upper bound. These precise understandings about the problem lead to a 40% saving of the privacy budget in a practical application. Furthermore, the algorithm-specific analysis shows a difference in privacy parameters of adaptive and non-adaptive composition, which was widely believed to not exist based on the evidence from general analysis.

This work provides tight upper- and lower-bounds for the problem of mean estimation under differential privacy in the local-model, when the input is composed of $n$ i.i.d. drawn samples from a Gaussian. Our algorithms result in a $(1-\beta)$-confidence interval for the underlying distribution’s mean of length $O(\sigma *sqrt(log(n/beta)log(1/\beta))/(\epsilon*sqrt(n))$. In addition, our algorithms leverage on binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, our algorithms have a matching lower-bound, where we prove that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $\Omega(\sigma*sqrt(\log(1/\beta))/(\epsilon*sqrt(n)))$.

We study the problem of top-k selection over a large domain universe subject to user-level differential privacy. Typically, the exponential mechanism or report noisy max are the algorithms used to solve this problem. However, these algorithms require querying the database for the count of each domain element. We focus on the setting where the data domain is unknown, which is different than the setting of frequent itemsets where an apriori type algorithm can help prune the space of domain elements to query. We design algorithms that ensures (approximate) differential privacy and only needs access to the true top-k' elements from the data for any chosen k' ≥ k. This is a highly desirable feature for making differential privacy practical, since the algorithms require no knowledge of the domain. We consider both the setting where a user's data can modify an arbitrary number of counts by at most 1, i.e. unrestricted sensitivity, and the setting where a user's data can modify at most some small, fixed number of counts by at most 1, i.e. restricted sensitivity. Additionally, we provide a pay-what-you-get privacy composition bound for our algorithms. That is, our algorithms might return fewer than k elements when the top-k elements are queried, but the overall privacy budget only decreases by the size of the outcome set.

The local model for differential privacy is emerging as the reference model for practical applications of collecting and sharing sensitive information while satisfying strong privacy guarantees. In the local model, there is no trusted entity which is allowed to have each individual’s raw data as is assumed in the traditional curator model. Individuals’ data are usually perturbed before sharing them. We explore the design of private hypothesis tests in the local model, where each data entry is perturbed to ensure the privacy of each participant. Specifically, we analyze locally private chi-square tests for goodness of fit and independence testing.

In this paper, we develop new test statistics for hypothesis testing over differentially private data. These statistics are designed specifically so that their asymptotic distributions, after accounting for privacy noise, match the asymptotics of the non-private chi-square tests for testing if the multinomial data parameters lie in lower dimensional manifolds (examples include goodness-of-fit and independence testing). Empirically, these new test statistics outperform prior work, which focused on noisy versions of existing statistics.

We analyze sources of error in prediction market forecasts in order to bound the difference between a security's price and the ground truth it estimates. We consider cost-function-based prediction markets in which an automated market maker adjusts security prices according to the history of trade. We decompose the forecasting error into three components: sampling error, arising because traders only possess noisy estimates of ground truth; market-maker bias, resulting from the use of a particular market maker (i.e., cost function) to facilitate trade; and convergence error, arising because, at any point in time, market prices may still be in flux. Our goal is to make explicit the tradeoffs between these error components, influenced by design decisions such as the functional form of the cost function and the amount of liquidity in the market. We consider a specific model in which traders have exponential utility and exponential-family beliefs representing noisy estimates of ground truth. In this setting, sampling error vanishes as the number of traders grows, but there is a tradeoff between the other two components. We provide both upper and lower bounds on market-maker bias and convergence error, and demonstrate via numerical simulations that these bounds are tight. Our results yield new insights into the question of how to set the market's liquidity parameter and into the forecasting benefits of enforcing coherent prices across securities.

In this paper, we initiate a principled study of how the generalization properties of approximate differential privacy can be used to perform adaptive hypothesis testing, while giving statistically valid p-value corrections. We do this by observing that the guarantees of algorithms with bounded approximate max-information are sufficient to correct the p-values of adaptively chosen hypotheses, and then by proving that algorithms that satisfy (∈,δ)-differential privacy have bounded approximate max information when their inputs are drawn from a product distribution. This substantially extends the known connection between differential privacy and max-information, which previously was only known to hold for (pure) (∈,0)-differential privacy. It also extends our understanding of max-information as a partially unifying measure controlling the generalization properties of adaptive data analyses. We also show a lower bound, proving that (despite the strong composition properties of max-information), when data is drawn from a product distribution, (∈,δ)-differentially private algorithms can come first in a composition with other algorithms satisfying max-information bounds, but not necessarily second if the composition is required to itself satisfy a nontrivial max-information bound. This, in particular, implies that the connection between (∈,δ)-differential privacy and max-information holds only for inputs drawn from product distributions, unlike the connection between (∈,0)-differential privacy and max-information.

Hypothesis testing is a useful statistical tool in determining whether a given model should be rejected based on a sample from the population. Sample data may contain sensitive information about individuals, such as medical information. Thus it is important to design statistical tests that guarantee the privacy of subjects in the data. In this work, we study hypothesis testing subject to differential privacy, specifically chi-squared tests for goodness of fit for multinomial data and independence between two categorical variables.

We define and study the problem of predicting the solution to a linear program (LP) given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, that has been studied under the name “Learning from Revealed Preferences". We give mistake bound learning algorithms in two settings: in the first, the objective of the LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown. In the second setting, the objective of the LP is unknown, and changing in a controlled way. The constraints of the LP may also change every day, but are known. An example is given by a set of constraints and partial information about the objective, and the task of the learner is again to predict the optimal solution of the partially known LP.

In this paper we initiate the study of adaptive composition in differential privacy when the length of the composition, and the privacy parameters themselves can be chosen adaptively, as a function of the outcome of previously run analyses. This case is much more delicate than the setting covered by existing composition theorems, in which the algorithms themselves can be chosen adaptively, but the privacy parameters must be fixed up front. Indeed, it isn't even clear how to define differential privacy in the adaptive parameter setting. We proceed by defining two objects which cover the two main use cases of composition theorems. A privacy filter is a stopping time rule that allows an analyst to halt a computation before his pre-specified privacy budget is exceeded. A privacy odometer allows the analyst to track realized privacy loss as he goes, without needing to pre-specify a privacy budget. We show that unlike the case in which privacy parameters are fixed, in the adaptive parameter setting, these two use cases are distinct. We show that there exist privacy filters with bounds comparable (up to constants) with existing privacy composition theorems. We also give a privacy odometer that nearly matches non-adaptive private composition theorems, but is sometimes worse by a small asymptotic factor. Moreover, we show that this is inherent, and that any valid privacy odometer in the adaptive parameter setting must lose this factor, which shows a formal separation between the filter and odometer use-cases.

Walrasian equilibrium prices have a remarkable property: they allow each buyer to purchase a bundle of goods that she finds the most desirable, while guaranteeing that the induced allocation over all buyers will globally maximize social welfare. However, this clean story has two caveats. * First, the prices may induce indifferences. In fact, the minimal equilibrium prices necessarily induce indifferences. Accordingly, buyers may need to coordinate with one another to arrive at a socially optimal outcome---the prices alone are not sufficient to coordinate the market. * Second, although natural procedures converge to Walrasian equilibrium prices on a fixed population, in practice buyers typically observe prices without participating in a price computation process. These prices cannot be perfect Walrasian equilibrium prices, but instead somehow reflect distributional information about the market. To better understand the performance of Walrasian prices when facing these two problems, we give two results. First, we propose a mild genericity condition on valuations under which the minimal Walrasian equilibrium prices induce allocations which result in low over-demand, no matter how the buyers break ties. In fact, under genericity the over-demand of any good can be bounded by 1, which is the best possible at the minimal prices. We demonstrate our results for unit demand valuations and give an extension to matroid based valuations (MBV), conjectured to be equivalent to gross substitute valuations (GS). Second, we use techniques from learning theory to argue that the over-demand and welfare induced by a price vector converge to their expectations uniformly over the class of all price vectors, with respective sample complexity linear and quadratic in the number of goods in the market. These results make no assumption on the form of the valuation functions. These two results imply that under a mild genericity condition, the exact Walrasian equilibrium prices computed in a market are guaranteed to induce both low over-demand and high welfare when used in a new market where agents are sampled independently from the same distribution, whenever the number of agents is larger than the number of commodities in the market.

We consider the problem of implementing an individually rational, asymptotically Pareto optimal allocation in a barter-exchange economy where agents are endowed with goods and preferences over the goods of others, but may not use money as a medium of exchange. Because one of the most important instantiations of such economies is kidney exchange -- where the "input" to the problem consists of sensitive patient medical records -- we ask to what extent such exchanges can be carried out while providing formal privacy guarantees to the participants. We show that individually rational allocations cannot achieve any non-trivial approximation to Pareto optimality if carried out under the constraint of differential privacy -- or even the relaxation of joint-differential privacy, under which it is known that asymptotically optimal allocations can be computed in two sided markets [Hsu et al. STOC 2014]. We therefore consider a further relaxation that we call marginal-differential privacy --which promises, informally, that the privacy of every agent i is protected from every other agent j ≠ i so long as j does not collude or share allocation information with other agents. We show that under marginal differential privacy, it is possible to compute an individually rational and asymptotically Pareto optimal allocation in such exchange economies.

We revisit a classic coordination problem from the perspective of mechanism design: how can we coordinate a social welfare maximizing flow in a network congestion game with selfish players? The classical approach, which computes tolls as a function of known demands, fails when the demands are unknown to the mechanism designer, and naively eliciting them does not necessarily yield a truthful mechanism. Instead, we introduce a weak mediator that can provide suggested routes to players and set tolls as a function of reported demands. However, players can choose to ignore or misreport their type to this mediator. Using techniques from differential privacy, we show how to design a weak mediator such that it is an asymptotic ex-post Nash equilibrium for all players to truthfully report their types to the mediator and faithfully follow its suggestion, and that when they do, they end up playing a nearly optimal flow. Notably, our solution works in settings of incomplete information even in the absence of a prior distribution on player types. Along the way, we develop new techniques for privately solving convex programs which may be of independent interest.

Studying games in the complete information model makes them analytically tractable. However, large n player interactions are more realistically modeled as games of incomplete information, where players may know little to nothing about the types of other players. Unfortunately, games in incomplete information settings lose many of the nice properties of complete information games: the quality of equilibria can become worse, the equilibria lose their ex-post properties, and coordinating on an equilibrium becomes even more difficult. Because of these problems, we would like to study games of incomplete information, but still implement equilibria of the complete information game induced by the (unknown) realized player types.