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Conference on Learning Theory 2020 

Prediction markets are well-studied in the case where predictions are probabilities or expectations of future random variables. In 2008, Lambert, et al. proposed a generalization, which we call ``scoring rule markets'' (SRMs), in which traders predict the value of arbitrary statistics of the random variables, provided these statistics can be elicited by a scoring rule. Surprisingly, despite active recent work on prediction markets, there has not yet been any investigation into more general SRMs. To initiate such a study, we ask the following question: in what sense are SRMs ``markets''? We classify SRMs according to several axioms that capture potentially desirable qualities of a market, such as the ability to freely exchange goods (contracts) for money. Not all SRMs satisfy our axioms: once a contract is purchased in any market for prediction the median of some variable, there will not necessarily be any way to sell that contract back, even in a very weak sense. Our main result is a characterization showing that slight generalizations of cost-function-based markets are the only markets to satisfy all of our axioms for finite-outcome random variables. Nonetheless, we find that several SRMs satisfy weaker versions of our axioms, including a novel share-based market mechanism for ratios of expected values. 

There are now several large scale deployments of differential privacy used to collect statistical information about users. However, these deployments periodically recollect the data and recompute the statistics using algorithms designed for a single use. As a result, these systems do not provide meaningful privacy guarantees over long time scales. Moreover, existing techniques to mitigate this effect do not apply in the “local model” of differential privacy that these systems use. In this paper, we introduce a new technique for local differential privacy that makes it possible to maintain up-to-date statistics over time, with privacy guarantees that degrade only in the number of changes in the underlying distribution rather than the number of collection periods. We use our technique for tracking a changing statistic in the setting where users are partitioned into an unknown collection of groups, and at every time period each user draws a single bit from a common (but changing) group-specific distribution. We also provide an application to frequency and heavy-hitter estimation. 

We consider partially-specified optimization problems where the goal is to actively, but efficiently, acquire missing information about the problem in order to solve it. An algo- rithm designer wishes to solve a linear pro- gram (LP), maxcT x s.t. Ax ≤ b,x ≥ 0, but does not initially know some of the pa- rameters. The algorithm can iteratively choose an unknown parameter and gather information in the form of a noisy sample centered at the parameter’s (unknown) value. The goal is to find an approximately feasible and optimal so- lution to the underlying LP with high proba- bility while drawing a small number of sam- ples. We focus on two cases. (1) When the parameters b of the constraints are initially un- known, we propose an efficient algorithm com- bining techniques from the ellipsoid method for LP and confidence-bound approaches from bandit algorithms. The algorithm adaptively gathers information about constraints only as needed in order to make progress. We give sample complexity bounds for the algorithm and demonstrate its improvement over a naive approach via simulation. (2) When the param- eters c of the objective are initially unknown, we take an information-theoretic approach and give roughly matching upper and lower sam- ple complexity bounds, with an (inefficient) successive-elimination algorithm.