These metrics admit differentiable sample estimates, making it easy to incorporate a calibration objective into empirical risk minimization.

When facing uncertainty, decision-makers want predictions they can trust. A machine learning provider can convey confidence to decision-makers by guaranteeing their predictions are distribution calibrated--- amongst the inputs that receive a predicted vector of class probabilities q, the actual distribution over classes is given by q. For multi-class prediction problems, however, directly optimizing predictions under distribution calibration tends to be infeasible, requiring sample complexity that grows exponentially in the number of classes C. In this work, we introduce a new notion---decision calibration---that requires the predicted distribution and true distribution over classes to be ``indistinguishable'' to downstream decision-makers. This perspective gives a new characterization of distribution calibration: a predictor is distribution calibrated if and only if it is decision calibrated with respect to all decision-makers. Our main result shows that under a mild restriction, unlike distribution calibration, decision calibration is actually feasible. We design a recalibration algorithm that provably achieves decision calibration efficiently, provided that the decision-makers have a bounded number of actions (e.g., polynomial in C). We validate our recalibration algorithm empirically: compared to existing methods, decision calibration improves decision-making on skin lesion and ImageNet classification with modern neural network predictors.

Calibration has emerged as a foundational goal in ``trustworthy machine learning'', in part because of its strong decision theoretic semantics. Independent of the underlying distribution, and independent of the decision maker's utility function, calibration promises that amongst all policies mapping predictions to actions, the uniformly best policy is the one that ``trusts the predictions'' and acts as if they were correct. But this is true only of \emph{fully calibrated} forecasts, which are tractable to guarantee only for very low dimensional prediction problems. For higher dimensional prediction problems (e.g. when outcomes are multiclass), weaker forms of calibration have been studied that lack these decision theoretic properties. In this paper we study how a conservative decision maker should map predictions endowed with these weaker (``partial'') calibration guarantees to actions, in a way that is robust in a minimax sense: i.e. to maximize their expected utility in the worst case over distributions consistent with the calibration guarantees. We characterize their minimax optimal decision rule via a duality argument, and show that surprisingly, ``trusting the predictions and acting accordingly'' is recovered in this minimax sense by \emph{decision calibration} (and any strictly stronger notion of calibration), a substantially weaker and more tractable condition than full calibration. For calibration guarantees that fall short of decision calibration, the minimax optimal decision rule is still efficiently computable, and we provide an empirical evaluation of a natural one that applies to any regression model solved to optimize squared error.

This can be written succinctly as Y, b Y | ( X) (18) A.2 Calibration in Classification ECE used for break ties. For each model and dataset, the best performing model is then re-run with 50 random seeds to gather information about standard errors and statistical significance. Kernel Bandwidth We select the RBF kernel bandwidth for training on each dataset using the aforementioned hyperparameter optimization. For each county, we track the weather sequence of each year into a few summary statistics for each month (average/maximum/minimum temperatures, precipitation, cooling/heating degree days). All other hyperparameters are held constant, including the number of training steps.

These metrics admit differentiable sample estimates, making it easy to incorporate a calibration objective into empirical risk minimization.

When facing uncertainty, decision-makers want predictions they can trust.

Fueled by discussions around "trustworthiness" and algorithmic fairness, calibration of predictive systems has regained scholars attention. The vanilla definition and understanding of calibration is, simply put, on all days on which the rain probability has been predicted to be p, the actual frequency of rain days was p. However, the increased attention has led to an immense variety of new notions of "calibration." Some of the notions are incomparable, serve different purposes, or imply each other. In this work, we provide two accounts which motivate calibration: self-realization of forecasted properties and precise estimation of incurred losses of the decision makers relying on forecasts. We substantiate the former via the reflection principle and the latter by actuarial fairness. For both accounts we formulate prototypical definitions via properties $Γ$ of outcome distributions, e.g., the mean or median. The prototypical definition for self-realization, which we call $Γ$-calibration, is equivalent to a certain type of swap regret under certain conditions. These implications are strongly connected to the omniprediction learning paradigm. The prototypical definition for precise loss estimation is a modification of decision calibration adopted from Zhao et al. [73]. For binary outcome sets both prototypical definitions coincide under appropriate choices of reference properties. For higher-dimensional outcome sets, both prototypical definitions can be subsumed by a natural extension of the binary definition, called distribution calibration with respect to a property. We conclude by commenting on the role of groupings in both accounts of calibration often used to obtain multicalibration. In sum, this work provides a semantic map of calibration in order to navigate a fragmented terrain of notions and definitions.

When model predictions inform downstream decision making, a natural question is under what conditions can the decision-makers simply respond to the predictions as if they were the true outcomes. Calibration suffices to guarantee that simple best-response to predictions is optimal. However, calibration for high-dimensional prediction outcome spaces requires exponential computational and statistical complexity. The recent relaxation known as decision calibration ensures the optimality of the simple best-response rule while requiring only polynomial sample complexity in the dimension of outcomes. However, known results on calibration and decision calibration crucially rely on linear loss functions for establishing best-response optimality. A natural approach to handle nonlinear losses is to map outcomes $y$ into a feature space $\phi(y)$ of dimension $m$, then approximate losses with linear functions of $\phi(y)$. Unfortunately, even simple classes of nonlinear functions can demand exponentially large or infinite feature dimensions $m$. A key open problem is whether it is possible to achieve decision calibration with sample complexity independent of~$m$. We begin with a negative result: even verifying decision calibration under standard deterministic best response inherently requires sample complexity polynomial in~$m$. Motivated by this lower bound, we investigate a smooth version of decision calibration in which decision-makers follow a smooth best-response. This smooth relaxation enables dimension-free decision calibration algorithms. We introduce algorithms that, given $\mathrm{poly}(|A|,1/\epsilon)$ samples and any initial predictor~$p$, can efficiently post-process it to satisfy decision calibration without worsening accuracy. Our algorithms apply broadly to function classes that can be well-approximated by bounded-norm functions in (possibly infinite-dimensional) separable RKHS.