On Computationally Efficient Multi-Class Calibration

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.

Omnipredictors for Regression and the Approximate Rank of Convex Functions

Consider the supervised learning setting where the goal is to learn to predict labels $\mathbf y$ given points $\mathbf x$ from a distribution. An \textit{omnipredictor} for a class $\mathcal L$ of loss functions and a class $\mathcal C$ of hypotheses is a predictor whose predictions incur less expected loss than the best hypothesis in $\mathcal C$ for every loss in $\mathcal L$. Since the work of [GKR+21] that introduced the notion, there has been a large body of work in the setting of binary labels where $\mathbf y \in \{0, 1\}$, but much less is known about the regression setting where $\mathbf y \in [0,1]$ can be continuous. Our main conceptual contribution is the notion of \textit{sufficient statistics} for loss minimization over a family of loss functions: these are a set of statistics about a distribution such that knowing them allows one to take actions that minimize the expected loss for any loss in the family. The notion of sufficient statistics relates directly to the approximate rank of the family of loss functions. Our key technical contribution is a bound of $O(1/\varepsilon^{2/3})$ on the $\epsilon$-approximate rank of convex, Lipschitz functions on the interval $[0,1]$, which we show is tight up to a factor of $\mathrm{polylog} (1/\epsilon)$. This yields improved runtimes for learning omnipredictors for the class of all convex, Lipschitz loss functions under weak learnability assumptions about the class $\mathcal C$. We also give efficient omnipredictors when the loss families have low-degree polynomial approximations, or arise from generalized linear models (GLMs). This translation from sufficient statistics to faster omnipredictors is made possible by lifting the technique of loss outcome indistinguishability introduced by [GKH+23] for Boolean labels to the regression setting.

Agnostically Learning Single-Index Models using Omnipredictors

We give the first result for agnostically learning Single-Index Models (SIMs) with arbitrary monotone and Lipschitz activations. All prior work either held only in the realizable setting or required the activation to be known. Moreover, we only require the marginal to have bounded second moments, whereas all prior work required stronger distributional assumptions (such as anticoncentration or boundedness). Our algorithm is based on recent work by [GHK$^+$23] on omniprediction using predictors satisfying calibrated multiaccuracy. Our analysis is simple and relies on the relationship between Bregman divergences (or matching losses) and $\ell_p$ distances. We also provide new guarantees for standard algorithms like GLMtron and logistic regression in the agnostic setting.

When Does Optimizing a Proper Loss Yield Calibration?

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.

Loss minimization yields multicalibration for large neural networks

Multicalibration is a notion of fairness that aims to provide accurate predictions across a large set of groups. Multicalibration is known to be a different goal than loss minimization, even for simple predictors such as linear functions. In this note, we show that for (almost all) large neural network sizes, optimally minimizing squared error leads to multicalibration. Our results are about representational aspects of neural networks, and not about algorithmic or sample complexity considerations. Previous such results were known only for predictors that were nearly Bayes-optimal and were therefore representation independent. We emphasize that our results do not apply to specific algorithms for optimizing neural networks, such as SGD, and they should not be interpreted as "fairness comes for free from optimizing neural networks".

Characterizing notions of omniprediction via multicalibration

A recent line of work shows that notions of multigroup fairness imply surprisingly strong notions of omniprediction: loss minimization guarantees that apply not just for a specific loss function, but for any loss belonging to a large family of losses. While prior work has derived various notions of omniprediction from multigroup fairness guarantees of varying strength, it was unknown whether the connection goes in both directions. In this work, we answer this question in the affirmative, establishing equivalences between notions of multicalibration and omniprediction. The new definitions that hold the key to this equivalence are new notions of swap omniprediction, which are inspired by swap regret in online learning. We show that these can be characterized exactly by a strengthening of multicalibration that we refer to as swap multicalibration. One can go from standard to swap multicalibration by a simple discretization; moreover all known algorithms for standard multicalibration in fact give swap multicalibration. In the context of omniprediction though, introducing the notion of swapping results in provably stronger notions, which require a predictor to minimize expected loss at least as well as an adaptive adversary who can choose both the loss function and hypothesis based on the value predicted by the predictor. Building on these characterizations, we paint a complete picture of the relationship between the various omniprediction notions in the literature by establishing implications and separations between them. Our work deepens our understanding of the connections between multigroup fairness, loss minimization and outcome indistinguishability and establishes new connections to classic notions in online learning.

A Unifying Theory of Distance from Calibration

We study the fundamental question of how to define and measure the distance from calibration for probabilistic predictors. While the notion of perfect calibration is well-understood, there is no consensus on how to quantify the distance from perfect calibration. Numerous calibration measures have been proposed in the literature, but it is unclear how they compare to each other, and many popular measures such as Expected Calibration Error (ECE) fail to satisfy basic properties like continuity. We present a rigorous framework for analyzing calibration measures, inspired by the literature on property testing. We propose a ground-truth notion of distance from calibration: the $\ell_1$ distance to the nearest perfectly calibrated predictor. We define a consistent calibration measure as one that is a polynomial factor approximation to the this distance. Applying our framework, we identify three calibration measures that are consistent and can be estimated efficiently: smooth calibration, interval calibration, and Laplace kernel calibration. The former two give quadratic approximations to the ground truth distance, which we show is information-theoretically optimal. Our work thus establishes fundamental lower and upper bounds on measuring distance to calibration, and also provides theoretical justification for preferring certain metrics (like Laplace kernel calibration) in practice.

Loss Minimization through the Lens of Outcome Indistinguishability

We present a new perspective on loss minimization and the recent notion of Omniprediction through the lens of Outcome Indistingusihability. For a collection of losses and hypothesis class, omniprediction requires that a predictor provide a loss-minimization guarantee simultaneously for every loss in the collection compared to the best (loss-specific) hypothesis in the class. We present a generic template to learn predictors satisfying a guarantee we call Loss Outcome Indistinguishability. For a set of statistical tests--based on a collection of losses and hypothesis class--a predictor is Loss OI if it is indistinguishable (according to the tests) from Nature's true probabilities over outcomes. By design, Loss OI implies omniprediction in a direct and intuitive manner. We simplify Loss OI further, decomposing it into a calibration condition plus multiaccuracy for a class of functions derived from the loss and hypothesis classes. By careful analysis of this class, we give efficient constructions of omnipredictors for interesting classes of loss functions, including non-convex losses. This decomposition highlights the utility of a new multi-group fairness notion that we call calibrated multiaccuracy, which lies in between multiaccuracy and multicalibration. We show that calibrated multiaccuracy implies Loss OI for the important set of convex losses arising from Generalized Linear Models, without requiring full multicalibration. For such losses, we show an equivalence between our computational notion of Loss OI and a geometric notion of indistinguishability, formulated as Pythagorean theorems in the associated Bregman divergence. We give an efficient algorithm for calibrated multiaccuracy with computational complexity comparable to that of multiaccuracy. In all, calibrated multiaccuracy offers an interesting tradeoff point between efficiency and generality in the omniprediction landscape.

Low-Degree Multicalibration

Introduced as a notion of algorithmic fairness, multicalibration has proved to be a powerful and versatile concept with implications far beyond its original intent. This stringent notion -- that predictions be well-calibrated across a rich class of intersecting subpopulations -- provides its strong guarantees at a cost: the computational and sample complexity of learning multicalibrated predictors are high, and grow exponentially with the number of class labels. In contrast, the relaxed notion of multiaccuracy can be achieved more efficiently, yet many of the most desirable properties of multicalibration cannot be guaranteed assuming multiaccuracy alone. This tension raises a key question: Can we learn predictors with multicalibration-style guarantees at a cost commensurate with multiaccuracy? In this work, we define and initiate the study of Low-Degree Multicalibration. Low-Degree Multicalibration defines a hierarchy of increasingly-powerful multi-group fairness notions that spans multiaccuracy and the original formulation of multicalibration at the extremes. Our main technical contribution demonstrates that key properties of multicalibration, related to fairness and accuracy, actually manifest as low-degree properties. Importantly, we show that low-degree multicalibration can be significantly more efficient than full multicalibration. In the multi-class setting, the sample complexity to achieve low-degree multicalibration improves exponentially (in the number of classes) over full multicalibration. Our work presents compelling evidence that low-degree multicalibration represents a sweet spot, pairing computational and sample efficiency with strong fairness and accuracy guarantees.

KL Divergence Estimation with Multi-group Attribution

Estimating the Kullback-Leibler (KL) divergence between two distributions given samples from them is well-studied in machine learning and information theory. Motivated by considerations of multi-group fairness, we seek KL divergence estimates that accurately reflect the contributions of sub-populations to the overall divergence. We model the sub-populations coming from a rich (possibly infinite) family $\mathcal{C}$ of overlapping subsets of the domain. We propose the notion of multi-group attribution for $\mathcal{C}$, which requires that the estimated divergence conditioned on every sub-population in $\mathcal{C}$ satisfies some natural accuracy and fairness desiderata, such as ensuring that sub-populations where the model predicts significant divergence do diverge significantly in the two distributions. Our main technical contribution is to show that multi-group attribution can be derived from the recently introduced notion of multi-calibration for importance weights [HKRR18, GRSW21]. We provide experimental evidence to support our theoretical results, and show that multi-group attribution provides better KL divergence estimates when conditioned on sub-populations than other popular algorithms.