The Calibration Generalization Gap A. Michael Carrell University of Cambridge

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The Calibration Generalization Gap∗

A. Michael Carrell

University of Cambridge

ac2411@cam.ac.uk

Neil Mallinar

UC San Diego

nmallina@ucsd.edu

James Lucas

NVIDIA

jlucas@cs.toronto.edu

Preetum Nakkiran

Apple

preetum@apple.com

Abstract

Calibration is a fundamental property of a good predictive model: it requires that the model predicts correctly in

proportion to its condence. Modern neural networks, however, provide no strong guarantees on their calibration—

and can be either poorly calibrated or well-calibrated depending on the setting. It is currently unclear which factors

contribute to good calibration (architecture, data augmentation, overparameterization, etc), though various claims

exist in the literature.

We propose a systematic way to study the calibration error: by decomposing it into (1) calibration error on the

train set, and (2) the calibration generalization gap. This mirrors the fundamental decomposition of generalization.

We then investigate each of these terms, and give empirical evidence that (1) DNNs are typically always calibrated on

their train set, and (2) the calibration generalization gap is upper-bounded by the standard generalization gap. Taken

together, this implies that models with small generalization gap (|Test Error - Train Error|) are well-calibrated. This

perspective unies many results in the literature, and suggests that interventions which reduce the generalization

gap (such as adding data, using heavy augmentation, or smaller model size) also improve calibration. We thus

hope our initial study lays the groundwork for a more systematic and comprehensive understanding of the relation

between calibration, generalization, and optimization.

1 Introduction

When machine learning models are deployed in the real world, as components of larger systems, we often want to

know more about their behavior than just overall loss or accuracy. For classication models, for example, it is helpful

to know not only their test error, but also estimates of predictive uncertainty — how condent the model is on various

inputs. Calibrated models can be more useful than uncalibrated ones, because their condences are operationally

meaningful: conditioning on high-condence predictions is “almost as good” as conditioning on high-accuracy

predictions1, but can be done without knowing the ground-truth.

Many machine learning models produce outputs in a probability simplex, and it is natural to ask whether these

outputs are intrinsically calibrated (even without post-hoc modications). For deep neural networks (DNNs), we can

ask, informally:

Are DNNs typically well-calibrated, on standard tasks?

This question has received much attention over the last several decades, but the literature remains muddled: early

works found that networks were reasonably calibrated [Niculescu-Mizil and Caruana, 2005], then Guo et al. [2017a]

found that “modern” networks (in 2017) were poorly calibrated, and most recently Minderer et al. [2021] argued that

current networks (in 2021) are in fact calibrated. The issue is complicated because the notion of “deep neural network”

has itself evolved over time: with dierent architectures, optimizers, and even dierent benchmark datasets on which

calibration is measured.

∗A version of this work appeared at the ICML 2022 Workshop on Distribution-Free Uncertainty Quantication.

1Formally, this is true in expectation, for rst-order moments.

arXiv:2210.01964v2 [cs.LG] 6 Oct 2022

Figure 1: ECE and error of an overparameterized model throughout training. We show reliability diagrams at various

point during training. In the early stage of training (before batch

1000

), both test and train ECEs are small (

<0.05

even though the test error itself is high (

>20%

). After batch

1000

, the generalization gap grows, and the test and

train ECEs also diverge. In the late stage of training, as the model becomes overcondent, the test ECE approaches

test error, while the train ECE goes to zero. Throughout, the dierence in test and train ECE is upper-bounded by the

dierence in test and train error.

A more rened question is thus to ask: when are DNNs calibrated? That is, for what settings of architecture,

optimizer, data distributions, etc. Various prior works have argued that certain individual factors have signicant

impact of calibration— for example, Minderer et al. [2021] claims non-convolutional models tend to be better calibrated,

and Wen et al. [2021] highlights the dierence in calibration when using data augmentations. A priori, a complete

understanding of this question may require jointly understanding all of these factors— but we can hope for better

abstractions which let us study only those aspects of our design choices which aect calibration.

In this work, we take steps to clarify the landscape of calibration in deep neural networks. First, we propose a

simple framework for reasoning about calibration: Just as we can classically decompose the Test Error into the Train

Error and the Generalization Gap, we can similarly decompose the Test Calibration Error (Test ECE) as:

TestECE

|{z }

Calibration on Test Set

≤TrainECE

| {z }

Calibration on Train Set

+|TestECE −TrainECE|

| {z }

Calibration Generalization Gap

(1)

This bounds the test calibration error in terms of an optimization quantity (calibration on the train set) and a general-

ization quantity. We can then study these two terms individually, much like the classical approach of generalization

theory.

With this framework in hand, we then apply it to study the calibration of DNNs. We rst observe that the Train

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摘要：

TheCalibrationGeneralizationGap*A.MichaelCarrellUniversityofCambridgeac2411@cam.ac.ukNeilMallinarUCSanDiegonmallina@ucsd.eduJamesLucasNVIDIAjlucas@cs.toronto.eduPreetumNakkiranApplepreetum@apple.comAbstractCalibrationisafundamentalpropertyofagoodpredictivemodel:itrequiresthatthemodelpredictscorrectl...

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