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Existing algorithms for online conformal prediction -- guaranteeing marginal coverage in adversarial settings -- are variants of online gradient descent (OGD), but their analyses of worst-case coverage do not follow from the regret guarantee of OGD. What is the relationship between no-regret learning and online conformal prediction? We observe that although standard regret guarantees imply marginal coverage in i.i.d. settings, this connection fails as soon as we either move to adversarial environments or ask for group conditional coverage. On the other hand, we show a tight connection between threshold calibrated coverage and swap-regret in adversarial settings, which extends to group-conditional (multi-valid) coverage. We also show that algorithms in the follow the perturbed leader family of no regret learning algorithms (which includes online gradient descent) can be used to give group-conditional coverage guarantees in adversarial settings for arbitrary grouping functions. Via this connection we analyze and conduct experiments using a multi-group generalization of the ACI algorithm of Gibbs & Candes [2021] 

We give an efficient reduction through which any machine learning algorithm can be converted into an interactive protocol that can interact with another party (such as a human) to reach agreement on predictions and improve accuracy. The requirements on each party are calibration conditions which are computationally and statistically tractable relaxations of Bayesian rationality --- that are sensible even in prior free settings --- and hence are a substantial generalization of Aumann's classic ``agreement theorem''. In the interactive protocol, the machine learning model first produces a prediction. Then, the human responds to the model's prediction by either conveying agreement, or else providing feedback of some sort. The model then updates its state and provides a new prediction, and the human in turn may update their beliefs. The process continues until the model and the human reach agreement. The first setting we study generalizes past work on Aumann's Agreement Theorem, in which the parties aim to agree on a one-dimensional expectation. At each round, each party simply communicates an estimate of their current prediction for the expectation. In this setting we recover the quantitative convergence theorem of [Aaronson, 2005] (but under our much weaker assumptions). We then move on to the case in which the parties maintain beliefs about a distribution over d outcomes and consider two feedback mechanisms. The first simply corresponds to a vector-valued estimate of the agents' current prediction. The second takes a decision theoretic perspective: if the human needs to take some downstream action from a finite set, and has an arbitrary utility function of their action and the outcome, then we show that the parties can communicate and reach agreement about the correct downstream action to take by simply communicating at each round the action that they believe to be utility maximizing. The number of rounds until agreement remains independent of $$d$$ in this case. We can also generalize our protocols to more than 2 parties, with computational complexity that degrades only linearly with the number of parties. Our protocols are based on simple, efficiently maintainable conditions and result in predictions that are more accurate than any single party's alone. 

In traditional reinforcement learning (RL), the learner aims to solve a single objective optimization problem: find the policy that maximizes expected reward. However, in many real-world settings, it is important to optimize over multiple objectives simultaneously. For example, when we are interested in fairness, states might have feature annotations corresponding to multiple (intersecting) demographic groups to whom reward accrues, and our goal might be to maximize the reward of the group receiving the minimal reward. In this work, we consider a multi-objective optimization problem in which each objective is defined by a state-based reweighting of a single scalar reward function. This generalizes the problem of maximizing the reward of the minimum reward group. We provide oracle-efficient algorithms to solve these multi-objective RL problems even when the number of objectives is exponentially large-for tabular MDPs, as well as for large MDPs when the group functions have additional structure. Finally, we experimentally validate our theoretical results and demonstrate applications on a preferential attachment graph MDP. 

Blasiok et al. [2023] proposed distance to calibration as a natural measure of calibration error that unlike expected calibration error (ECE) is continuous. Recently, Qiao and Zheng [2024] gave a non-constructive argument establishing the existence of an online predictor that can obtain O(√T ) distance to calibration in the adversarial setting, which is known to be impossible for ECE. They leave as an open problem finding an explicit, efficient algorithm. We resolve this problem and give an extremely simple, efficient, deterministic algorithm that obtains distance to calibration error at most 2√T . 

There has been substantial recent concern that pricing algorithms might learn to ``collude.'' Supra-competitive prices can emerge as a Nash equilibrium of repeated pricing games, in which sellers play strategies which threaten to punish their competitors who refuse to support high prices, and these strategies can be automatically learned. In fact, a standard economic intuition is that supra-competitive prices emerge from either the use of threats, or a failure of one party to optimize their payoff. Is this intuition correct? Would preventing threats in algorithmic decision-making prevent supra-competitive prices when sellers are optimizing for their own revenue? No. We show that supra-competitive prices can emerge even when both players are using algorithms which do not encode threats, and which optimize for their own revenue. We study sequential pricing games in which a first mover deploys an algorithm and then a second mover optimizes within the resulting environment. We show that if the first mover deploys any algorithm with a no-regret guarantee, and then the second mover even approximately optimizes within this now static environment, monopoly-like prices arise. The result holds for any no-regret learning algorithm deployed by the first mover and for any pricing policy of the second mover that obtains them profit at least as high as a random pricing would -- and hence the result applies even when the second mover is optimizing only within a space of non-responsive pricing distributions which are incapable of encoding threats. In fact, there exists a set of strategies, neither of which explicitly encode threats that form a Nash equilibrium of the simultaneous pricing game in algorithm space, and lead to near monopoly prices. This suggests that the definition of ``algorithmic collusion'' may need to be expanded, to include strategies without explicitly encoded threats. 

We consider the problem of online learning in the linear contextual bandits setting, but in which there are also strong individual fairness constraints governed by an unknown similarity metric. These constraints demand that we select similar actions or individuals with approximately equal probability, which may be at odds with optimizing reward, thus modeling settings where profit and social policy are in tension. We assume we learn about an unknown Mahalanobis similarity metric from only weak feedback that identifies fairness violations, but does not quantify their extent. This is intended to represent the interventions of a regulator who “knows unfairness when he sees it” but nevertheless cannot enunciate a quantitative fairness metric over individuals. Our main result is an algorithm in the adversarial context setting that has a number of fairness violations that depends only logarithmically on T, while obtaining an optimal O(√T) regret bound to the best fair policy. 

Motivated by settings in which predictive models may be required to be non-discriminatory with respect to certain attributes (such as race), but even collecting the sensitive attribute may be forbidden or restricted, we initiate the study of fair learning under the constraint of differential privacy. Our first algorithm is a private implementation of the equalized odds post-processing approach of (Hardt et al., 2016). This algorithm is appealingly simple, but must be able to use protected group membership explicitly at test time, which can be viewed as a form of “disparate treatment”. Our second algorithm is a differentially private version of the oracle-efficient in-processing approach of (Agarwal et al., 2018) which is more complex but need not have access to protected group membership at test time. We identify new tradeoffs between fairness, accuracy, and privacy that emerge only when requiring all three properties, and show that these tradeoffs can be milder if group membership may be used at test time. We conclude with a brief experimental evaluation.