Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated.

We study calibration measures in a sequential prediction setup.

Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated.

Calibration requires predictor outputs to be consistent with their Bayesian posteriors. For machine learning predictors that do not distinguish between small perturbations, calibration errors are continuous in predictions, e.g., smooth calibration error (Foster and Hart, 2018), Distance to Calibration (Blasiok et al., 2023a). On the contrary, decision-makers who use predictions make optimal decisions discontinuously in probabilistic space, experiencing loss from miscalibration discontinuously. Calibration errors for decision-making are thus discontinuous, e.g., Expected Calibration Error (Foster and Vohra, 1997), and Calibration Decision Loss (Hu and Wu, 2024). Thus, predictors with a low calibration error for machine learning may suffer a high calibration error for decision-making, i.e., they may not be trustworthy for decision-makers optimizing assuming their predictions are correct. It is natural to ask if post-processing a predictor with a low calibration error for machine learning is without loss to achieve a low calibration error for decision-making. In our paper, we show that post-processing an online predictor with $\epsilon$ distance to calibration achieves $O(\sqrt{\epsilon})$ ECE and CDL, which is asymptotically optimal. The post-processing algorithm adds noise to make predictions differentially private. The optimal bound from low distance to calibration predictors from post-processing is non-optimal compared with existing online calibration algorithms that directly optimize for ECE and CDL.

We initiate the study of the truthfulness of calibration measures in sequential prediction. A calibration measure is said to be truthful if the forecaster (approximately) minimizes the expected penalty by predicting the conditional expectation of the next outcome, given the prior distribution of outcomes. Truthfulness is an important property of calibration measures, ensuring that the forecaster is not incentivized to exploit the system with deliberate poor forecasts. This makes it an essential desideratum for calibration measures, alongside typical requirements, such as soundness and completeness. We conduct a taxonomy of existing calibration measures and their truthfulness. Perhaps surprisingly, we find that all of them are far from being truthful. That is, under existing calibration measures, there are simple distributions on which a polylogarithmic (or even zero) penalty is achievable, while truthful prediction leads to a polynomial penalty. Our main contribution is the introduction of a new calibration measure termed the Subsampled Smooth Calibration Error (SSCE) under which truthful prediction is optimal up to a constant multiplicative factor.

Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class calibration that give strong guarantees of meaningful predictions and can be achieved in time and sample complexities polynomial in $k$? Prior notions of calibration exhibit a tradeoff between computational efficiency and expressivity: they either suffer from having sample complexity exponential in $k$, or needing to solve computationally intractable problems, or give rather weak guarantees. Our main contribution is a notion of calibration that achieves all these desiderata: we formulate a robust notion of projected smooth calibration for multi-class predictions, and give new recalibration algorithms for efficiently calibrating predictors under this definition with complexity polynomial in $k$. Projected smooth calibration gives strong guarantees for all downstream decision makers who want to use the predictor for binary classification problems of the form: does the label belong to a subset $T \subseteq [k]$: e.g. is this an image of an animal? It ensures that the probabilities predicted by summing the probabilities assigned to labels in $T$ are close to some perfectly calibrated binary predictor for that task. We also show that natural strengthenings of our definition are computationally hard to achieve: they run into information theoretic barriers or computational intractability. Underlying both our upper and lower bounds is a tight connection that we prove between multi-class calibration and the well-studied problem of agnostic learning in the (standard) binary prediction setting.

We study a sequential binary prediction setting where the forecaster is evaluated in terms of the calibration distance, which is defined as the $L_1$ distance between the predicted values and the set of predictions that are perfectly calibrated in hindsight. This is analogous to a calibration measure recently proposed by B{\l}asiok, Gopalan, Hu and Nakkiran (STOC 2023) for the offline setting. The calibration distance is a natural and intuitive measure of deviation from perfect calibration, and satisfies a Lipschitz continuity property which does not hold for many popular calibration measures, such as the $L_1$ calibration error and its variants. We prove that there is a forecasting algorithm that achieves an $O(\sqrt{T})$ calibration distance in expectation on an adversarially chosen sequence of $T$ binary outcomes. At the core of this upper bound is a structural result showing that the calibration distance is accurately approximated by the lower calibration distance, which is a continuous relaxation of the former. We then show that an $O(\sqrt{T})$ lower calibration distance can be achieved via a simple minimax argument and a reduction to online learning on a Lipschitz class. On the lower bound side, an $\Omega(T^{1/3})$ calibration distance is shown to be unavoidable, even when the adversary outputs a sequence of independent random bits, and has an additional ability to early stop (i.e., to stop producing random bits and output the same bit in the remaining steps). Interestingly, without this early stopping, the forecaster can achieve a much smaller calibration distance of $\mathrm{polylog}(T)$.

Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated. However, typical machine learning models are trained to approximately minimize loss over restricted families of predictors, that are unlikely to contain the ground truth. Under what circumstances does optimizing proper loss over a restricted family yield calibrated models? What precise calibration guarantees does it give? In this work, we provide a rigorous answer to these questions. We replace the global optimality with a local optimality condition stipulating that the (proper) loss of the predictor cannot be reduced much by post-processing its predictions with a certain family of Lipschitz functions. We show that any predictor with this local optimality satisfies smooth calibration as defined in Kakade-Foster (2008), B{\l}asiok et al. (2023). Local optimality is plausibly satisfied by well-trained DNNs, which suggests an explanation for why they are calibrated from proper loss minimization alone. Finally, we show that the connection between local optimality and calibration error goes both ways: nearly calibrated predictors are also nearly locally optimal.