Bo Waggoner

Robert Kleinberg, Bo Waggoner, E. Glen Weyl

Investigating potential purchases is often a substantial investment under uncertainty. Standard market designs, such as simultaneous or English auctions, compound this with uncertainty about the price a bidder will have to pay in order to win. As a result they tend to confuse the process of search both by leading to wasteful information acquisition on goods that have already found a good purchaser and by discouraging needed investigations of objects, potentially eliminating all gains from trade. In contrast, we show that the Dutch auction preserves all of its properties from a standard setting without information costs because it guarantees, at the time of information acquisition, a price at which the good can be purchased. Calibrations to start-up acquisition and timber auctions suggest that in practice the social losses through poor search coordination in standard formats are an order of magnitude or two larger than the (negligible) inefficiencies arising from ex-ante bidder asymmetries.

Jessie Finocchiaro, Rafael M. Frongillo, Bo Waggoner

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 $R^d$, 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.

Rafael Frongillo, Dhamma Kimpara, Bo Waggoner

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.

Rafael Frongillo, Mary Monroe, Eric Neyman et al.

We study the problem of robust forecast aggregation: combining expert forecasts with provable accuracy guarantees compared to the best possible aggregation of the underlying information. Prior work shows strong impossibility results, e.g. that even under natural assumptions, no aggregation of the experts' individual forecasts can outperform simply following a random expert (Neyman and Roughgarden, 2022). In this paper, we introduce a more general framework that allows the principal to elicit richer information from experts through structured queries. Our framework ensures that experts will truthfully report their underlying beliefs, and also enables us to define notions of complexity over the difficulty of asking these queries. Under a general model of independent but overlapping expert signals, we show that optimal aggregation is achievable in the worst case with each complexity measure bounded above by the number of agents $n$. We further establish tight tradeoffs between accuracy and query complexity: aggregation error decreases linearly with the number of queries, and vanishes when the "order of reasoning" and number of agents relevant to a query is $ω(\sqrt{n})$. These results demonstrate that modest extensions to the space of expert queries dramatically strengthen the power of robust forecast aggregation. We therefore expect that our new query framework will open up a fruitful line of research in this area.

Rafael Frongillo, Manuel Lladser, Anish Thilagar et al.

Beginning with Witkowski et al. [2022], recent work on forecasting competitions has addressed incentive problems with the common winner-take-all mechanism. Frongillo et al. [2021] propose a competition mechanism based on follow-the-regularized-leader (FTRL), an online learning framework. They show that their mechanism selects an $ε$-optimal forecaster with high probability using only $O(\log(n)/ε^2)$ events. These works, together with all prior work on this problem thus far, assume that events are independent. We initiate the study of forecasting competitions for correlated events. To quantify correlation, we introduce a notion of block correlation, which allows each event to be strongly correlated with up to $b$ others. We show that under distributions with this correlation, the FTRL mechanism retains its $ε$-optimal guarantee using $O(b^2 \log(n)/ε^2)$ events. Our proof involves a novel concentration bound for correlated random variables which may be of broader interest.

Justin D. Harris, Bo Waggoner

Machine learning has recently enabled large advances in artificial intelligence, but these tend to be highly centralized. The large datasets required are generally proprietary; predictions are often sold on a per-query basis; and published models can quickly become out of date without effort to acquire more data and re-train them. We propose a framework for participants to collaboratively build a dataset and use smart contracts to host a continuously updated model. This model will be shared publicly on a blockchain where it can be free to use for inference. Ideal learning problems include scenarios where a model is used many times for similar input such as personal assistants, playing games, recommender systems, etc. In order to maintain the model's accuracy with respect to some test set we propose both financial and non-financial (gamified) incentive structures for providing good data. A free and open source implementation for the Ethereum blockchain is provided at https://github.com/microsoft/0xDeCA10B.

Enrique Nueve, Bao Nguyen, Rafael Frongillo et al.

Prediction markets rely on liquidity to convert trades into informative prices, yet existing mechanisms fix liquidity ex ante. This restriction enforces a static trade-off between price responsiveness and worst-case loss despite inherently nonstationary trading conditions. We propose a fundamentally different approach that treats liquidity selection itself as an online learning problem. Our mechanism mixes a family of cost-function markets via learnable weights, yielding a single adaptive market that preserves no-arbitrage, bounded worst-case loss, expressiveness, and positive upside. We introduce a hybrid structural risk signal, a per-round objective that quantifies the trade-off between price impact and inventory risk, and show that standard online learning algorithms achieve switching-regret guarantees relative to the best sequence of liquidity regimes in hindsight. Simulations demonstrate that the mechanism adaptively shifts liquidity across regimes in response to both order flow and inventory dynamics. Our results establish a principled framework for adaptive liquidity, connecting prediction market design with online learning.

Robin Bowers, Elias Lindgren, Bo Waggoner

A decisionmaker faces $n$ alternatives, each of which represents a potential reward. After investing costly resources into investigating the alternatives, the decisionmaker may select one, or more generally a feasible subset, and obtain the associated reward(s). The objective is to maximize the sum of rewards minus total costs invested. We consider this problem under a general model of an alternative as a "Markov Search Process," a type of undiscounted Markov Decision Process on a finite acyclic graph. Even simple cases generalize NP-hard problems such as Pandora's Box with nonobligatory inspection. Despite the apparently adaptive and interactive nature of the problem, we prove optimal prophet inequalities for this problem under a variety of combinatorial constraints. That is, we give approximation algorithms that interact with the alternatives sequentially, where each must be fully explored and either selected or else discarded before the next arrives. In particular, we obtain a computationally efficient $\frac{1}{2}-ε$ prophet inequality for Combinatorial Markov Search subject to any matroid constraint. This result implies incentive-compatible mechanisms with constant Price of Anarchy for serving single-parameter agents when the agents strategically conduct independent, costly search processes to discover their values.

Enrique Nueve, Bo Waggoner, Dhamma Kimpara et al.

In multiclass classification over $n$ outcomes, the outcomes must be embedded into the reals with dimension at least $n-1$ in order to design a consistent surrogate loss that leads to the "correct" classification, regardless of the data distribution. For large $n$, such as in information retrieval and structured prediction tasks, optimizing a surrogate in $n-1$ dimensions is often intractable. We investigate ways to trade off surrogate loss dimension, the number of problem instances, and restricting the region of consistency in the simplex for multiclass classification. Following past work, we examine an intuitive embedding procedure that maps outcomes into the vertices of convex polytopes in a low-dimensional surrogate space. We show that full-dimensional subsets of the simplex exist around each point mass distribution for which consistency holds, but also, with less than $n-1$ dimensions, there exist distributions for which a phenomenon called hallucination occurs, which is when the optimal report under the surrogate loss is an outcome with zero probability. Looking towards application, we derive a result to check if consistency holds under a given polytope embedding and low-noise assumption, providing insight into when to use a particular embedding. We provide examples of embedding $n = 2^{d}$ outcomes into the $d$-dimensional unit cube and $n = d!$ outcomes into the $d$-dimensional permutahedron under low-noise assumptions. Finally, we demonstrate that with multiple problem instances, we can learn the mode with $\frac{n}{2}$ dimensions over the whole simplex.

Enrique Nueve, Bo Waggoner

When agents trade in a Duality-based Cost Function prediction market, they collectively implement the learning algorithm Follow-The-Regularized-Leader. We ask whether other learning algorithms could be used to inspire the design of prediction markets. By decomposing and modifying the Duality-based Cost Function Market Maker's (DCFMM) pricing mechanism, we propose a new prediction market, called the Smooth Quadratic Prediction Market, the incentivizes agents to collectively implement general steepest gradient descent. Relative to the DCFMM, the Smooth Quadratic Prediction Market has a better worst-case monetary loss for AD securities while preserving axiom guarantees such as the existence of instantaneous price, information incorporation, expressiveness, no arbitrage, and a form of incentive compatibility. To motivate the application of the Smooth Quadratic Prediction Market, we independently examine agents' trading behavior under two realistic constraints: bounded budgets and buy-only securities. Finally, we provide an introductory analysis of an approach to facilitate adaptive liquidity using the Smooth Quadratic Prediction Market. Our results suggest future designs where the price update rule is separate from the fee structure, yet guarantees are preserved.

Rafael Frongillo, Bo Waggoner

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 and 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.

Rafael Frongillo, Robert Gomez, Anish Thilagar et al.

Winner-take-all competitions in forecasting and machine-learning suffer from distorted incentives. Witkowski et al. 2018 identified this problem and proposed ELF, a truthful mechanism to select a winner. We show that, from a pool of $n$ forecasters, ELF requires $Θ(n\log n)$ events or test data points to select a near-optimal forecaster with high probability. We then show that standard online learning algorithms select an $ε$-optimal forecaster using only $O(\log(n) / ε^2)$ events, by way of a strong approximate-truthfulness guarantee. This bound matches the best possible even in the nonstrategic setting. We then apply these mechanisms to obtain the first no-regret guarantee for non-myopic strategic experts.

Jessie Finocchiaro, Rafael Frongillo, Bo Waggoner

Given a prediction task, understanding when one can and cannot design a consistent convex surrogate loss, particularly a low-dimensional one, is an important and active area of machine learning research. The prediction task may be given as a target loss, as in classification and structured prediction, or simply as a (conditional) statistic of the data, as in risk measure estimation. These two scenarios typically involve different techniques for designing and analyzing surrogate losses. We unify these settings using tools from property elicitation, and give a general lower bound on prediction dimension. Our lower bound tightens existing results in the case of discrete predictions, showing that previous calibration-based bounds can largely be recovered via property elicitation. For continuous estimation, our lower bound resolves on open problem on estimating measures of risk and uncertainty.

Ariel Avital, Klim Efremenko, Aryeh Kontorovich et al.

We propose a non-parametric variant of binary regression, where the hypothesis is regularized to be a Lipschitz function taking a metric space to [0,1] and the loss is logarithmic. This setting presents novel computational and statistical challenges. On the computational front, we derive a novel efficient optimization algorithm based on interior point methods; an attractive feature is that it is parameter-free (i.e., does not require tuning an update step size). On the statistical front, the unbounded loss function presents a problem for classic generalization bounds, based on covering-number and Rademacher techniques. We get around this challenge via an adaptive truncation approach, and also present a lower bound indicating that the truncation is, in some sense, necessary.

Zhiyuan Liu, Huazheng Wang, Bo Waggoner et al.

We investigate the sparse linear contextual bandit problem where the parameter $θ$ is sparse. To relieve the sampling inefficiency, we utilize the "perturbed adversary" where the context is generated adversarilly but with small random non-adaptive perturbations. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $\mathcal{O}(\sqrt{kT\log d})$ even when $d \gg T$ where $k$ and $d$ are the number of effective and ambient dimension, respectively. Compared to the recent work from Sivakumar et al. (2020), our analysis does not rely on the precondition processing, adaptive perturbation (the adaptive perturbation violates the i.i.d perturbation setting) or truncation on the error set. Moreover, the special structures in our results explicitly characterize how the perturbation affects exploration length, guide the design of perturbation together with the fundamental performance limit of perturbation method. Numerical experiments are provided to complement the theoretical analysis.

Jessie Finocchiaro, Rafael Frongillo, Bo Waggoner

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 $\mathbb{R}^d$, 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. 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. Conversely, we show how to construct a consistent polyhedral surrogate for any given discrete loss. Our framework yields succinct proofs of consistency or inconsistency of various polyhedral surrogates in the literature, and for inconsistent surrogates, it further reveals the discrete losses for which these surrogates are consistent. We show some 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.

Nika Haghtalab, Cameron Musco, Bo Waggoner

In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and density estimation literature, very few losses outside the dominant $log\ loss$ are applied. We aim to understand this fact, taking an axiomatic approach to the design of loss functions for learning distributions. We start by proposing a set of desirable criteria that any good loss function should satisfy. Intuitively, these criteria require that the loss function faithfully evaluates a candidate distribution, both in expectation and when estimated on a few samples. Interestingly, we observe that \emph{no loss function} possesses all of these criteria. However, one can circumvent this issue by introducing a natural restriction on the set of candidate distributions. Specifically, we require that candidates are $calibrated$ with respect to the target distribution, i.e., they may contain less information than the target but otherwise do not significantly distort the truth. We show that, after restricting to this set of distributions, the log loss, along with a large variety of other losses satisfy the desired criteria. These results pave the way for future investigations of distribution learning that look beyond the log loss, choosing a loss function based on application or domain need.

Yahav Bechavod, Katrina Ligett, Aaron Roth et al.

We study an online classification problem with partial feedback in which individuals arrive one at a time from a fixed but unknown distribution, and must be classified as positive or negative. Our algorithm only observes the true label of an individual if they are given a positive classification. This setting captures many classification problems for which fairness is a concern: for example, in criminal recidivism prediction, recidivism is only observed if the inmate is released; in lending applications, loan repayment is only observed if the loan is granted. We require that our algorithms satisfy common statistical fairness constraints (such as equalizing false positive or negative rates -- introduced as "equal opportunity" in Hardt et al. (2016)) at every round, with respect to the underlying distribution. We give upper and lower bounds characterizing the cost of this constraint in terms of the regret rate (and show that it is mild), and give an oracle efficient algorithm that achieves the upper bound.

Rafael Frongillo, Nishant A. Mehta, Tom Morgan et al.

Recent work introduced loss functions which measure the error of a prediction based on multiple simultaneous observations or outcomes. In this paper, we explore the theoretical and practical questions that arise when using such multi-observation losses for regression on data sets of $(x,y)$ pairs. When a loss depends on only one observation, the average empirical loss decomposes by applying the loss to each pair, but for the multi-observation case, empirical loss is not even well-defined, and the possibility of statistical guarantees is unclear without several $(x,y)$ pairs with exactly the same $x$ value. 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 demonstrate practicality of these algorithms in low-dimensional settings, while lower bounds demonstrate intrinsic difficulty in higher dimensions. Finally, we demonstrate the potential benefit of the algorithms over natural baselines that use traditional single-observation losses via both lower bounds and simulations.

Matthew Joseph, Aaron Roth, Jonathan Ullman et al.

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.

Sampath Kannan, Jamie Morgenstern, Aaron Roth et al.

Bandit learning is characterized by the tension between long-term exploration and short-term exploitation. However, as has recently been noted, in settings in which the choices of the learning algorithm correspond to important decisions about individual people (such as criminal recidivism prediction, lending, and sequential drug trials), exploration corresponds to explicitly sacrificing the well-being of one individual for the potential future benefit of others. This raises a fairness concern. In such settings, one might like to run a "greedy" algorithm, which always makes the (myopically) optimal decision for the individuals at hand - but doing this can result in a catastrophic failure to learn. In this paper, we consider the linear contextual bandit problem and revisit the performance of the greedy algorithm. We give a smoothed analysis, showing that even when contexts may be chosen by an adversary, small perturbations of the adversary's choices suffice for the algorithm to achieve "no regret", perhaps (depending on the specifics of the setting) with a constant amount of initial training data. This suggests that "generically" (i.e. in slightly perturbed environments), exploration and exploitation need not be in conflict in the linear setting.

Jinshuo Dong, Aaron Roth, Zachary Schutzman et al.

We study an online linear classification problem, in which the data is generated by strategic agents who manipulate their features in an effort to change the classification outcome. In rounds, the learner deploys a classifier, and an adversarially chosen agent arrives, possibly manipulating her features to optimally respond to the learner. The learner has no knowledge of the agents' utility functions or "real" features, which may vary widely across agents. Instead, the learner is only able to observe their "revealed preferences" --- i.e. the actual manipulated feature vectors they provide. For a broad family of agent cost functions, we give a computationally efficient learning algorithm that is able to obtain diminishing "Stackelberg regret" --- a form of policy regret that guarantees that the learner is obtaining loss nearly as small as that of the best classifier in hindsight, even allowing for the fact that agents will best-respond differently to the optimal classifier.

Sebastian Casalaina-Martin, Rafael Frongillo, Tom Morgan et al.

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.

Katrina Ligett, Seth Neel, Aaron Roth et al.

Traditional approaches to differential privacy assume a fixed privacy requirement $ε$ for a computation, and attempt to maximize the accuracy of the computation subject to the privacy constraint. As differential privacy is increasingly deployed in practical settings, it may often be that there is instead a fixed accuracy requirement for a given computation and the data analyst would like to maximize the privacy of the computation subject to the accuracy constraint. This raises the question of how to find and run a maximally private empirical risk minimizer subject to a given accuracy requirement. We propose a general "noise reduction" framework that can apply to a variety of private empirical risk minimization (ERM) algorithms, using them to "search" the space of privacy levels to find the empirically strongest one that meets the accuracy constraint, incurring only logarithmic overhead in the number of privacy levels searched. The privacy analysis of our algorithm leads naturally to a version of differential privacy where the privacy parameters are dependent on the data, which we term ex-post privacy, and which is related to the recently introduced notion of privacy odometers. We also give an ex-post privacy analysis of the classical AboveThreshold privacy tool, modifying it to allow for queries chosen depending on the database. Finally, we apply our approach to two common objectives, regularized linear and logistic regression, and empirically compare our noise reduction methods to (i) inverting the theoretical utility guarantees of standard private ERM algorithms and (ii) a stronger, empirical baseline based on binary search.

Jacob Abernethy, Yiling Chen, Chien-Ju Ho et al.

We design mechanisms for online procurement of data held by strategic agents for machine learning tasks. The challenge is to use past data to actively price future data and give learning guarantees even when an agent's cost for revealing her data may depend arbitrarily on the data itself. We achieve this goal by showing how to convert a large class of no-regret algorithms into online posted-price and learning mechanisms. Our results in a sense parallel classic sample complexity guarantees, but with the key resource being money rather than quantity of data: With a budget constraint $B$, we give robust risk (predictive error) bounds on the order of $1/\sqrt{B}$. Because we use an active approach, we can often guarantee to do significantly better by leveraging correlations between costs and data. Our algorithms and analysis go through a model of no-regret learning with $T$ arriving pairs (cost, data) and a budget constraint of $B$. Our regret bounds for this model are on the order of $T/\sqrt{B}$ and we give lower bounds on the same order.

Bo Waggoner

The classic problems of testing uniformity of and learning a discrete distribution, given access to independent samples from it, are examined under general $\ell_p$ metrics. The intuitions and results often contrast with the classic $\ell_1$ case. For $p > 1$, we can learn and test with a number of samples that is independent of the support size of the distribution: With an $\ell_p$ tolerance $ε$, $O(\max\{ \sqrt{1/ε^q}, 1/ε^2 \})$ samples suffice for testing uniformity and $O(\max\{ 1/ε^q, 1/ε^2\})$ samples suffice for learning, where $q=p/(p-1)$ is the conjugate of $p$. As this parallels the intuition that $O(\sqrt{n})$ and $O(n)$ samples suffice for the $\ell_1$ case, it seems that $1/ε^q$ acts as an upper bound on the "apparent" support size. For some $\ell_p$ metrics, uniformity testing becomes easier over larger supports: a 6-sided die requires fewer trials to test for fairness than a 2-sided coin, and a card-shuffler requires fewer trials than the die. In fact, this inverse dependence on support size holds if and only if $p > \frac{4}{3}$. The uniformity testing algorithm simply thresholds the number of "collisions" or "coincidences" and has an optimal sample complexity up to constant factors for all $1 \leq p \leq 2$. Another algorithm gives order-optimal sample complexity for $\ell_{\infty}$ uniformity testing. Meanwhile, the most natural learning algorithm is shown to have order-optimal sample complexity for all $\ell_p$ metrics. The author thanks Clément Canonne for discussions and contributions to this work.