We formalize and study the natural approach of designing convex surrogate loss functions via embeddings, for problems such as classification, ranking, or structured prediction. In this approach, one embeds each of the finitely many predictions (e.g. rankings) as a point in Rd, assigns the original loss values to these points, and "convexifies" the loss in some way to obtain a surrogate. We establish a strong connection between this approach and polyhedral (piecewise-linear convex) surrogate losses: every discrete loss is embedded by some polyhedral loss, and every polyhedral loss embeds some discrete loss. Moreover, an embedding gives rise to a consistent link function as well as linear surrogate regret bounds. Our results are constructive, as we illustrate with several examples. In particular, our framework gives succinct proofs of consistency or inconsistency for various polyhedral surrogates in the literature, and for inconsistent surrogates, it further reveals the discrete losses for which these surrogates are consistent. We go on to show additional structure of embeddings, such as the equivalence of embedding and matching Bayes risks, and the equivalence of various notions of non-redudancy. Using these results, we establish that indirect elicitation, a necessary condition for consistency, is also sufficient when working with polyhedral surrogates.

We initiate the study of proper losses for evaluating generative models in the discrete setting. Unlike traditional proper losses, we treat both the generative model and the target distribution as black-boxes, only assuming ability to draw i.i.d. samples. We define a loss to be black-box proper if the generative distribution that minimizes expected loss is equal to the target distribution. Using techniques from statistical estimation theory, we give a general construction and characterization of black-box proper losses: they must take a polynomial form, and the number of draws from the model and target distribution must exceed the degree of the polynomial. The characterization rules out a loss whose expectation is the cross-entropy between the target distribution and the model. By extending the construction to arbitrary sampling schemes such as Poisson sampling, however, we show that one can construct such a loss.

Top-k classification is a generalization of multiclass classification used widely in information retrieval, image classification, and other extreme classification settings. Several hinge-like (piecewise-linear) surrogates have been proposed for the problem, yet all are either non-convex or inconsistent. For the proposed hinge-like surrogates that are convex (i.e., polyhedral), we apply the recent embedding framework of Finocchiaro et al. (2019; 2022) to determine the prediction problem for which the surrogate is consistent. These problems can all be interpreted as variants of top-k classification, which may be better aligned with some applications. We leverage this analysis to derive constraints on the conditional label distributions under which these proposed surrogates become consistent for top-k. It has been further suggested that every convex hinge-like surrogate must be inconsistent for top-k. Yet, we use the same embedding framework to give the first consistent polyhedral surrogate for this problem.

The Lovász hinge is a convex surrogate recently proposed for structured binary classification, in which k binary predictions are made simultaneously and the error is judged by a submodular set function. Despite its wide usage in image segmentation and related problems, its consistency has remained open. We resolve this open question, showing that the Lovász hinge is inconsistent for its desired target unless the set function is modular. Leveraging a recent embedding framework, we instead derive the target loss for which the Lovász hinge is consistent. This target, which we call the structured abstain problem, allows one to abstain on any subset of the k predictions. We derive two link functions, each of which are consistent for all submodular set functions simultaneously.

We present a model of truthful elicitation which generalizes and extends mechanisms, scoring rules, and a number of related settings that do not qualify as one or the other. Our main result is a characterization theorem, yielding characterizations for all of these settings. This includes a new characterization of scoring rules for non-convex sets of distributions. We combine the characterization theorem with duality to give a simple construction to convert between scoring rules and randomized mechanisms. We also show how a generalization of this characterization gives a new proof of a mechanism design result due to Saks and Yu.

Surrogate risk minimization is an ubiquitous paradigm in supervised machine learning, wherein a target problem is solved by minimizing a surrogate loss on a dataset. Surrogate regret bounds, also called excess risk bounds, are a common tool to prove generalization rates for surrogate risk minimization. While surrogate regret bounds have been developed for certain classes of loss functions, such as proper losses, general results are relatively sparse. We provide two general results. The first gives a linear surrogate regret bound for any polyhedral (piecewise-linear convex) surrogate, meaning that surrogate generalization rates translate directly to target rates. The second shows that for sufficiently non-polyhedral surrogates, the regret bound is a square root, meaning fast surrogate generalization rates translate to slow rates for the target. Together, these results suggest polyhedral surrogates are optimal in many cases.

The convex consistency dimension of a supervised learning task is the lowest prediction dimension d such that there exists a convex surrogate L : \mathbb{R}^d \times \mathcal{Y} \to \mathbb{R} that is consistent for the given task. We present a new tool based on property elicitation, d-flats, for lower-bounding convex consistency dimension. This tool unifies approaches from a variety of domains, including continuous and discrete prediction problems. We use d-flats to obtain a new lower bound on the convex consistency dimension of risk measures, resolving an open question due to Frongillo and Kash (NeurIPS 2015). In discrete prediction settings, we show that the d-flats approach recovers and even tightens previous lower bounds using feasible subspace dimension.

We introduce a theoretical framework of elicitability and identifiability of set-valued functionals, such as quantiles, prediction intervals, and systemic risk measures. A functional is elicitable if it is the unique minimiser of an expected scoring function, and identifiable if it is the unique zero of an expected identification function; both notions are essential for forecast ranking and validation, and M- and Z-estimation. Our framework distinguishes between exhaustive forecasts, being set-valued and aiming at correctly specifying the entire functional, and selective forecasts, content with solely specifying a single point in the correct functional. We establish a mutual exclusivity result: A set-valued functional can be either selectively elicitable or exhaustively elicitable or not elicitable at all. Notably, since quantiles are well known to be selectively elicitable, they fail to be exhaustively elicitable. We further show that the class of prediction intervals and Vorob'ev quantiles turn out to be exhaustively elicitable and selectively identifiable. In particular, we provide a mixture representation of elementary exhaustive scores, leading the way to Murphy diagrams. We give possibility and impossibility results for the shortest prediction interval and prediction intervals specified by an endpoint or a midpoint. We end with a comprehensive literature review on common practice in forecast evaluation of set-valued functionals.

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, our main result gives tight complexity bounds for the broad class of Bayes risks. We apply these results to several properties of interest, including variance, entropy, norms, and several classes of financial risk measures. We conclude with discussion and open directions.

One way to define the randomness of a fixed individual sequence is to ask how hard it is to predict relative to a given loss function. A sequence is memoryless if, with respect to average loss, no continuous function can predict the next entry of the sequence from a finite window of previous entries better than a constant prediction. For squared loss, memoryless sequences are known to have stochastic attributes analogous to those of truly random sequences. In this paper, we address the question of how changing the loss function changes the set of memoryless sequences, and in particular, the stochastic attributes they possess. For convex differentiable losses we establish that the statistic or property elicited by the loss determines the identity and stochastic attributes of the corresponding memoryless sequences. We generalize these results to convex non-differentiable losses, under additional assumptions, and to non-convex Bregman divergences. In particular, our results show that any Bregman divergence has the same set of memoryless sequences as squared loss. We apply our results to price calibration in prediction markets.

A common technique in supervised learning with discrete losses, such as 0-1 loss, is to optimize a convex surrogate loss over R d , calibrated with respect to the original loss. In particular, recent work has investigated embedding the original predictions (e.g. labels) as points in R^d , showing an equivalence to using polyhedral surrogates. In this work, we study the notion of the embedding dimension of a given discrete loss: the minimum dimension d such that an embedding exists. We characterize d-embeddability for all d, with a particularly tight characterization for d = 1 (embedding into the real line), and useful necessary conditions for d > 1 in the form of a quadratic feasibility program. We illustrate our results with novel lower bounds for abstain loss.

Scoring functions are commonly used to evaluate a point forecast of a particular statistical functional. This scoring function should be calibrated, meaning the correct value of the functional is the Bayes optimal forecast, in which case we say the scoring function elicits the functional. Heinrich (2014) shows that the mode functional is not elicitable. In this work, we ask whether it is at least possible to indirectly elicit the mode, wherein one elicits a low-dimensional functional from which the mode can be computed. We show that this cannot be done: neither the mode nor modal interval are indirectly elicitable with respect to the class of identifiable functionals.

We formalize and study the natural approach of designing convex surrogate loss functions via embeddings for problems such as classification or ranking. In this approach, one embeds each of the finitely many predictions (e.g. classes) as a point in Rd, assigns the original loss values to these points, and convexifies the loss in between to obtain a surrogate. We prove that this approach is equivalent, in a strong sense, to working with polyhedral (piecewise linear convex) losses. Moreover, given any polyhedral loss L, we give a construction of a link function through which $L$ is a consistent surrogate for the loss it embeds. We go on to illustrate the power of this embedding framework with succinct proofs of consistency or inconsistency of various polyhedral surrogates in the literature.

Given a data set of (x,y) pairs, a common learning task is to fit a model predicting y (a label or dependent variable) conditioned on x. This paper considers the similar but much less-understood problem of modeling "higher-order" statistics of y's distribution conditioned on x. Such statistics are often challenging to estimate using traditional empirical risk minimization (ERM) approaches. We develop and theoretically analyze an ERM-like approach with multi-observation loss functions. We propose four algorithms formalizing the concept of empirical risk minimization for this problem, two of which have statistical guarantees in settings allowing both slow and fast convergence rates, but which are out-performed empirically by the other two. Empirical results illustrate potential practicality of these algorithms in low dimensions and significant improvement over standard approaches in some settings.

Power diagrams, a type of weighted Voronoi diagrams, have many applications throughout operations research. We study the problem of power diagram detection: determining whether a given finite partition of Rd takes the form of a power diagram. This detection problem is particularly prevalent in the field of information elicitation, where one wishes to design contracts to incentivize self-minded agents to provide honest information. We devise a simple linear program to decide whether a polyhedral cell decomposition can be described as a power diagram. Further, we discuss applications to property elicitation, peer prediction, and mechanism design, where this question arises. Our model is able to efficiently decide the question for decompositions of Rd or of a restricted domain in Rd. The approach is based on the use of an alternative representation of power diagrams, and invariance of a power diagram under uniform scaling of the parameters in this representation.

A property or statistic of a distribution is said to be elicitable if it can be expressed as the minimizer of some loss function in expectation. Recent work shows that continuous real-valued properties are elicitable if and only if they are identifiable, meaning the set of distributions with the same property value can be described by linear constraints. From a practical standpoint, one may ask for which such properties do there exist convex loss functions. In this paper, in a finite-outcome setting, we show that in fact every elicitable real-valued property can be elicited by a convex loss function. Our proof is constructive, and leads to convex loss functions for new properties.

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.

One way to define the "randomness" of a fixed individual sequence is to ask how hard it is to predict. When prediction error is measured via squared loss, it has been established that memoryless sequences (which are, in a precise sense, hard to predict) have some of the stochastic attributes of truly random sequences. In this paper, we ask how changing the loss function used changes the set of memoryless sequences, and in particular, the stochastic attributes they possess. We answer this question using tools from property elicitation, showing that the property elicited by the loss function determines the stochastic attributes of the corresponding memoryless sequences. We apply our results to the question of price calibration in prediction markets.

We study loss functions that measure the accuracy of a prediction based on multiple data points simultaneously. To our knowledge, such loss functions have not been studied before in the area of property elicitation or in machine learning more broadly. As compared to traditional loss functions that take only a single data point, these multi-observation loss functions can in some cases drastically reduce the dimensionality of the hypothesis required. In elicitation, this corresponds to requiring many fewer reports; in empirical risk minimization, it corresponds to algorithms on a hypothesis space of much smaller dimension. We explore some examples of the tradeoff between dimensionality and number of observations, give some geometric characterizations and intuition for relating loss functions and the properties that they elicit, and discuss some implications for both elicitation and machine-learning contexts.

The study of property elicitation is gaining ground in statistics and machine learning as a way to view and reason about the expressive power of emiprical risk minimization (ERM). Yet beyond a widening frontier of special cases, the two most fundamental questions in this area remain open: which statistics are elicitable (computable via ERM), and which loss functions elicit them? Moreover, recent work suggests a complementary line of questioning: given a statistic, how many ERM parameters are needed to compute it? We give concrete instantiations of these important questions, which have numerous applications to machine learning and related fields.

Minimal peer prediction mechanisms truthfully elicit private information (e.g., opinions or experiences) from rational agents without the requirement that ground truth is eventually revealed. In this paper, we use a geometric perspective to prove that minimal peer prediction mechanisms are equivalent to power diagrams, a type of weighted Voronoi diagram. Using this characterization and results from computational geometry, we show that many of the mechanisms in the literature are unique up to affine transformations, and introduce a general method to construct new truthful mechanisms.

Elicitation is the study of statistics or properties which are computable via empirical risk minimization. While several recent papers have approached the general question of which properties are elicitable, we suggest that this is the wrong question---all properties are elicitable by first eliciting the entire distribution or data set, and thus the important question is how elicitable. Specifically, what is the minimum number of regression parameters needed to compute the property?

Building on previous work, we introduce a new notion of elicitation complexity and lay the foundations for a calculus of elicitation. We establish several general results and techniques for proving upper and lower bounds on elicitation complexity. These results provide tight bounds for eliciting the Bayes risk of any loss, a large class of properties which includes spectral risk measures and several new properties of interest.

The elicitation of a statistic, or property of a distribution, is the task of devising proper scoring rules, equivalently proper losses, which incentivize an agent or algorithm to truthfully estimate the desired property of the underlying probability distribution.
Exploiting the connection of such elicitation to convex analysis, we address the vector-valued property case, which has received relatively little attention in the literature but is relevant for machine learning applications. We first provide a very general characterization of linear and ratio-of-linear properties, which resolves an open problem by unifying several previous characterizations in machine learning and statistics. We then ask which vectors of properties admit nonseparable scores, which cannot be expressed as a sum of scores for each coordinate separately, a natural desideratum for machine learning. We show that linear and ratio-of-linear do admit nonseparable scores, and provide evidence for the conjecture that these are the only such properties (up to link functions). Finally, we provide a general method for producing identification functions and address an open problem by showing that convex maximal level sets are insufficient for elicitability in general.

We study the problem of eliciting and aggregating probabilistic information from multiple agents. In order to successfully aggregate the predictions of agents, the principal needs to elicit some notion of confidence from agents, capturing how much experience or knowledge led to their predictions. To formalize this, we consider a principal who wishes to elicit predictions about a random variable from a group of Bayesian agents, each of whom have privately observed some independent samples of the random variable, and hopes to aggregate the predictions as if she had directly observed the samples of all agents. Leveraging techniques from Bayesian statistics, we represent confidence as the number of samples an agent has observed, which is quantified by a hyperparameter from a conjugate family of prior distributions. This then allows us to show that if the principal has access to a few samples, she can achieve her aggregation goal by eliciting predictions from agents using proper scoring rules. In particular, if she has access to one sample, she can successfully aggregate the agents' predictions if and only if every posterior predictive distribution corresponds to a unique value of the hyperparameter. Furthermore, this uniqueness holds for many common distributions of interest. When this uniqueness property does not hold, we construct a novel and intuitive mechanism where a principal with two samples can elicit and optimally aggregate the agents' predictions.

We present a model of truthful elicitation which generalizes and extends mechanisms, scoring rules, and a number of related settings that do not quite qualify as one or the other. Our main result is a characterization theorem, yielding characterizations for all of these settings, including a new characterization of scoring rules for non-convex sets of distributions. We generalize this model to eliciting some property of the agent's private information, and provide the first general characterization for this setting. We also show how this yields a new proof of a result in mechanism design due to Saks and Yu.

Ever since the Internet opened the floodgates to millions of users, each looking after their own interests, modern algorithm design has needed to be increasingly robust to strategic manipulation. Often, it is the inputs to these algorithms which are provided by strategic agents, who may lie for their own benefit, necessitating the design of algorithms which incentivize the truthful revelation of private information -- but how can this be done? This is a fundamental question with answers from many disciplines, from mechanism design to scoring rules and prediction markets. Each domain has its own model with its own assumptions, yet all seek schemes to gather private information from selfish agents, by leveraging incentives. Together, we refer to such models as elicitation.

This dissertation unifies and advances the theory of incentivized information elicitation. Using tools from convex analysis, we introduce a new model of elicitation with a matching characterization theorem which together encompass mechanism design, scoring rules, prediction markets, and other models. This lays a firm foundation on which the rest of the dissertation is built.

It is natural to consider a setting where agents report some alternate representation of their private information, called a property, rather than reporting it directly. We extend our model and characterization to this setting, revealing even deeper ties to convex analysis and convex duality, and we use these connections to prove new results for linear, smooth nonlinear, and finite-valued properties. Exploring the linear case further, we show a new four-fold equivalence between scoring rules, prediction markets, Bregman divergences, and generalized exponential families.

Applied to mechanism design, our framework offers a new perspective. By focusing on the (convex) consumer surplus function, we simplify a number of existing results, from the classic revenue equivalence theorem, to more recent characterizations of mechanism implementability.

Finally, we follow a line of research on the interpretation of prediction markets, relating a new stochastic framework to the classic Walrasian equilibrium and to stochastic mirror descent, thereby strengthening ties between prediction markets and machine learning.

We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.