Title: An Information Theoretic Perspective on Conformal Prediction
Venue: NeurIPS 2024
Accepted: yes
Keywords: conformal prediction

information theory

uncertainty quantification
Abstract: Conformal Prediction (CP) is a distribution-free uncertainty estimation framework that constructs prediction sets guaranteed to contain the true answer with a user-specified probability. Intuitively, the size of the prediction set encodes a general notion of uncertainty, with larger sets associated with higher degrees of uncertainty. In this work, we leverage information theory to connect conformal prediction to other notions of uncertainty. More precisely, we prove three different ways to upper bound the intrinsic uncertainty, as described by the conditional entropy of the target variable given the inputs, by combining CP with information theoretical inequalities. Moreover, we demonstrate two direct and useful applications of such connection between conformal prediction and information theory: (i) more principled and effective conformal training objectives that generalize previous approaches and enable end-to-end training of machine learning models from scratch, and (ii) a natural mechanism to incorporate side information into conformal prediction. We empirically validate both applications in centralized and federated learning settings, showing our theoretical results translate to lower inefficiency (average prediction set size) for popular CP methods.

An Information Theoretic Perspective on Conformal Prediction

Abstract

Conformal Prediction (CP) is a distribution-free uncertainty estimation framework that constructs prediction sets guaranteed to contain the true answer with a userspecified probability. Intuitively, the size of the prediction set encodes a general notion of uncertainty, with larger sets associated with higher degrees of uncertainty. In this work, we leverage information theory to connect conformal prediction to other notions of uncertainty. More precisely, we prove three different ways to upper bound the intrinsic uncertainty, as described by the conditional entropy of the target variable given the inputs, by combining CP with information theoretical inequalities. Moreover, we demonstrate two direct and useful applications of such connection between conformal prediction and information theory: (i) more principled and effective conformal training objectives that generalize previous approaches and enable end-to-end training of machine learning models from scratch, and (ii) a natural mechanism to incorporate side information into conformal prediction. We empirically validate both applications in centralized and federated learning settings, showing our theoretical results translate to lower inefficiency (average prediction set size) for popular CP methods.

1 Introduction

Machine learning (ML) models have rapidly grown in popularity and reach, having now found use in many safety-critical domains like healthcare [1] and autonomous driving [28]. In these areas, predictions must be accompanied by reliable measures of uncertainty to ensure safe decision-making. However, most ML models are designed and trained to produce only point estimates, which capture only crude notions of uncertainty with no statistical guarantees. Conformal prediction (CP) [64], in particular its split variant (SCP) [47], has recently gained in popularity as a principled and scalable solution to equip any , potentially black-box, model with proper uncertainty estimates in the form of prediction sets; in loose terms, larger sets are associated with higher degrees of uncertainty.

In this work, we take a closer look at conformal prediction through the lens of information theory (IT), establishing a connection between conformal prediction and the underlying intrinsic uncertainty of the data-generating process, as captured by the conditional entropy H ( Y | X ) = -E P XY [log P Y | X ] of the target variable Y given the inputs X . We prove conformal prediction can be used to bound H ( Y | X ) from above in three different ways: one derived from the data processing inequality, which we dub DPI bound , and two coming from a variation of Fano's inequality [19], a model agnostic one, which we call (simple) Fano bound , and another informed by the predictive model itself, to which we refer as model-based Fano bound . To the best of our knowledge, these bounds represent the first bridge connecting information theory and conformal prediction, which we hope will bring new tools to both fields. We already present two such tools in this paper: (i) we show our upper bounds serve as

principled training objectives to learn classifiers that are more amenable to SCP, and (ii) we advance a systematic way to incorporate side information into the construction of prediction sets. In a number of classification tasks, we empirically validate that both these applications of our theoretical results lead to better predictive efficiency, i.e., narrower and, consequently, more informative prediction sets.

The rest of the paper is organized as follows. In Section 2, we first introduce the necessary background to guide the reader through our main theoretical results. We introduce our three new upper bounds to the intrinsic uncertainty in Section 3, and their applications to conformal training and side information in Sections 4 and 5, respectively. Thereafter, we explore the related work in CP and IT in Section 6, present and analyze our experimental results in Section 7, and finally conclude in Section 8.

2 Background

In this section, we present the needed background on conformal prediction and list decoding [18, 22, 68], an area of information theory that, as we show, is closely related to CP and especially useful in deriving our main results. We start with the necessary notation. As usual, we denote random variables in uppercase letters and their realization in lowercase, e.g., X = x . We reserve calligraphic letters, e.g. X , for sets and use P X , Q X , . . . to denote probability measures on the space X . To simplify the notation, we use P, Q, . . . when the underlying space is clear. For example, given a probability measure Q XY , the probability of the event { ( X,Y ) : Y ∈ C ( X ) } is denoted as Q ( Y ∈ C ( X )) .

2.1 Conformal Prediction

Conformal prediction (CP) is a theoretically grounded framework that provides prediction sets with finite-sample guarantees under minimal distribution-free assumptions. Concretely, given a set of n data points ( X i , Y i ) ∈ X × Y , i = 1 , . . . , n drawn from some (unknown) joint distribution P XY , CP allows us to construct sets C ( X ) ∈ Y , such that for a new data point from the same distribution ( X test , Y test ) we have the following guarantee for a target error rate α ∈ (0 , 1)

where the probability is over the randomness in the sample { ( X i , Y i ) } n i =1 ∪{ ( X test , Y test ) } . To make this more tangible, the reader can picture C ( X ) as a subset of the possible labels in a classification problem, or as a confidence interval around the point estimate of a regressor in a regression setting.

In this work, we focus on a variant called split conformal prediction (SCP) [47] that gained popularity in the ML community, since it can leverage any pre-trained model f : X → Y in the construction of prediction sets. In this setting, the aforesaid n data points constitute a calibration data set D cal , which must be disjoint from the training data set used to fit the predictive model f . This separation between training and calibration data is what gives the name split to the method.

The first step in SCP is to define a nonconformity score function s f : X × Y → R , which is itself a function of model f and captures the magnitude of the prediction error at a given data point; the higher the score s f ( x, y ) , the higher the disagreement between input x and prediction y . At calibration time, we evaluate the score function at every ( X i , Y i ) ∈ D cal to get a collection of scores { S i = s f ( X i , Y i ) } n i =1 , and at test time, we construct prediction set C ( X test ) as

where Quantile (1 -α ; { S i } n i =1 ) is the level 1 -α quantile of the empirical distribution defined by { S i } n i =1 . The central result in conformal prediction, which we restate below for completeness, proves that prediction sets thus constructed achieve marginal valid coverage , i.e., satisfy (1).

Theorem 2.1 ([31, 64]) . If { ( X i , Y i ) } n i are i.i.d. (or only exchangeable), then for a new i.i.d. draw ( X test , Y test ) , and for any α ∈ (0 , 1) and for any score function s such that { S i } n i =1 are almost surely distinct, then C ( X test ) as defined above satisfies

See Appendix B for the proof and a more thorough introduction to CP. It is worth noting that valid coverage is not sufficient; the uninformative set predictor that always outputs C ( X test ) = Y trivially satisfies (1). We would also like our prediction sets to be as narrow as possible, and that is why CP methods are often compared in terms of their (empirical) inefficiency , i.e., the average prediction set size 1 / | D test | ∑ x ∈D test |C ( x ) | for some test data set D test . This is, in fact, not the only type of inefficiency criterion, but we use it as our main performance metric since it is the most common [65].

We depict the split conformal prediction procedure in Figure 1, where we include two extra variables that will be useful later in the text: the model prediction ˆ Y = f ( X ) , and the event of valid coverage E = { Y ∈ C ( X ) } , i.e., the event of the prediction set containing the correct class.

2.2 Conformal Prediction as List Decoding

In a nutshell, we can see conformal prediction as defining a map from an input x to a set of candidates in the target set Y . It turns out that the conformal prediction framework is equivalent to (variable-size) list decoding, an error-recovery

X

Y

ˆ

Y

C

(

X

)

D

cal

E

model going back to the works of Elias [18] and Wozencraft [68] in communication theory-we review some of these results in Appendix C.2. In particular, consider the mapping from the true label y to the input x as a noisy communication channel p ( x | y ) . The goal of an error-correcting code is then to decode the input x and recover the one true label y . List decoding generalizes this idea, allowing the decoder to return a set of outcomes (a list) instead of a pointwise prediction. If the correct solution is not part of the set output by the decoder, an error is declared. Although conformal prediction and list decoding were developed for different purposes, namely uncertainty quantification and error correction, it is easy to see that, if we allow for variable-size lists, the list decoding problem for the channel p ( x | y ) as described above is equivalent to the conformal prediction problem.

To our knowledge, this link between conformal prediction and information theory (and list decoding in particular) has gone unnoticed in the literature, and in this paper we leverage it in two directions. First, we apply information-theoretic inequalities for list decoding to upper bound the conditional entropy H ( Y | X ) of the data-generating process. This leads to new objectives for conformal training (see Section 4) and new bounds on the inefficiency of a given model (see Appendices E and G.3). Second, the information-theoretic interpretation of CP gives us an effective and theoretically grounded way of incorporating side-information into CP to improve predictive efficiency (see Section 5).

3 Information Theory Applied to Conformal Prediction

In this section, we develop our main results, which link information theory and conformal prediction. Concretely, we provide three novel upper bounds on the conditional entropy H ( Y | X ) : one coming from the data processing inequality and two derived in a similar way to Fano's inequality. We defer the proofs to Appendix D for the sake of conciseness, but in broad strokes, our results come from relating the bounds on the error probability provided by these information-theoretic inequalities (typically in the context of error-correcting codes) to the guarantees provided by CP in Theorem 2.1.

3.1 Data Processing Inequality for Conformal Prediction

We start by using the classical data processing inequality (DPI) in the context of conformal prediction. Specifically, we focus on the DPI for f -divergences, which we discuss thoroughly in Appendix C.1. In brief, for a convex function f with f (1) = 0 , the f -divergence between two probability measures P and Q is defined as (see [58])

In particular, with f ( x ) = x log x we recover the familiar notion of KL-divergence [27]. The DPI for f -divergences states that for any two probability measures P X and Q X defined on a space X , and any map W Y | X , which maps ( P X , Q X ) to ( P Y , Q Y ) , we have

We can apply the DPI for f -divergence above in the context of conformal prediction by considering the probability of the event of valid coverage { Y ∈ C ( X ) } under two different probability measures P and Q . Taking P as the data-generating distribution P := P X P Y | X and constructing Q := P X Q Y | X for an arbitrary Q Y | X (e.g., a machine learning model), we get the following proposition.

Proposition 3.1 (DPI Bound) . Consider any conformal prediction method satisfying the upper and lower bounds of Theorem 2.1 for α ∈ (0 , 0 . 5) . For any arbitrary conditional distribution Q Y | X , the true conditional distribution P Y | X and the input measure P X , define the following two measures Q := P X Q Y | X and P := P X P Y | X . Then, we have

Note that the entropy term H ( Y | X ) is computed using the measure P . We relegate the proof to Appendix D.1. In the bound in (3), the term Q ( Y ∈ C ( X )) appears inside a log, so an empirical estimate ˆ Q ( Y ∈ C ( X )) would result in a lower bound and would be biased. We can provide an upper confidence bound on this estimate using the empirical Bernstein inequality [41] and use that instead. Based on the empirical Bernstein inequality, with probability 1 -δ , we have

with V n ( Z ) the empirical variance of Z = ( Z 1 , . . . , Z n ) , Z i = Q ( y i ∈ C ( x i )) . Using these bounds, we get the following inequality with probability 1 -δ :

H ( Y | X ) ≤ h b ( α ) + (1 -α ) log ˜ Q ( Y ∈ C ( X )) + α n log ˜ Q ( Y / ∈ C ( X )) -E P [ log Q Y | X ] . (4) This upper bound is one of our main results, which we dub the DPI bound . Note that for the last expectation, we can use the empirical estimate, as it is an unbiased approximation.

3.2 Model-Based Fano's Inequality and Variations

Next, we present an inequality which is a variation of Fano's inequality [19], a classical result that, among other things, is used to prove Shannon's classical theorem on channel capacity. See Appendix C.4 for more details. In our context, we can use Fano's inequality to relate the conditional entropy H ( Y | X ) to the probability of error, i.e., P ( Y / ∈ C ( X )) . From that insight, we obtain Proposition 3.2 by modifying the classical proof of Fano's inequality, which can be found in [14], and applying the conformal guarantees from Theorem 2.1 to bound the probability of error.

Proposition 3.2 (Model-Based Fano Bound) . Consider any conformal prediction method satisfying the upper and lower bounds of Theorem 2.1 for α ∈ (0 , 0 . 5) . Then, for the true distribution P , and for any probability distribution Q , we have

Note that we have one term conditioned on the event of valid coverage, { Y ∈ C ( X ) } , and another conditioned on { Y / ∈ C ( X ) } . We provide the proof in Appendix D.2. A good choice for Q is the predictive model itself, and that is why we refer to the bound above as Model-Based (MB) Fano bound . Another natural choice for Q is the uniform distribution, which gives us the following result. Corollary 3.1 (Simple Fano Bound) . Consider any conformal prediction method satisfying the upper and lower bounds of Theorem 2.1 for α ∈ (0 , 0 . 5) . Then, for the true distribution P we have

The proof follows directly from Proposition 3.2 by replacing Q with the uniform distribution. We refer to the bound in (6) as (simple) Fano bound , since it is model agnostic and can be approximated directly with only empirical estimates of the prediction set size. This last bound explicitly relates the central notion of uncertainty in conformal prediction, the prediction set size, to an informationtheoretic concept of uncertainty in H ( Y | X ) . This reinterpretation of conformal prediction as a form of list decoding introduces various information-theoretic tools, potentially useful for various applications. In Appendix E we derive some new inequalities for conformal prediction, in particular offering new lower bounds on the inefficiency of the conformal prediction. In the next section, we discuss how these inequalities can be used as conformal training schemes.

4 Conformal Training

Although split conformal prediction is applicable to any pretrained ML model as a post-processing step, the overall performance of any CP method (commonly measured by its inefficiency) is highly dependent on the underlying model itself. Therefore, previous works have proposed to take CP into account already during model training and directly optimize for low predictive inefficiency [8, 13, 61]. We use the term conformal training to refer to such approaches in general (see Appendix F for an overview of the topic). In particular, we focus on ConfTr [61] since it generalizes and outperforms [8]. In ConfTr [61] each training batch B is split into calibration B cal and test B test halves to simulate the SCP process (see Section 2) for each gradient update of model f and minimize the following size loss

where C f ( x ) is constructed using the statistics of the nonconformity scores computed on B cal . We use the notation C f ( x ) to emphasize the dependence of the prediction set on model f . Still, SCP involves step functions and Stutz et al. [61] introduce two relaxations to recover a differentiable objective: (i) the computation of quantiles is relaxed via differentiable sorting operators [9, 17, 50]; (ii) the thresholding operation in the construction of prediction sets in (2) is replaced by smooth assignments via the logistic sigmoid. The latter relaxation gives 'soft' prediction sets ˆ C f ( x ) , which contain each of the labels y ∈ Y with a certain probability. See Algorithm 1 for a depiction of conformal training.

Our upper bounds on H ( Y | X ) , namely DPI, MB Fano and simple Fano, presented in the previous section can be made differentiable in the same way, and thus can also serve as proper loss functions for conformal training. The motivation for doing so is twofold. First, the conditional entropy H ( Y | X ) captures the underlying uncertainty of the task, or equivalently, the uncertainty under the true labelling distribution P Y | X . Thus, by minimizing these upper bounds, we can hope to push the model f closer to the true distribution, which is known to achieve minimal inefficiency [65]. Interestingly, the cross-entropy loss also bounds H ( Y | X ) from above and thus can be motivated as a conformal training objective from the same angle. In that regard, the DPI bound from Proposition 3.1 is particularly advantageous as it is provably tighter than the cross-entropy (see Appendix D.1).

Second, we can connect the simple Fano bound from Corollary 3.1 to the size loss (7) from [61]. In Appendix F.1, we show that via Jensen's inequality and log( | Y |-| C ( X ) | ) ≤ log | Y | the bound in (6) can be further upper bounded as

Since λ α and (1 -α n ) are constants, they do not affect optimization, and minimizing the right hand side in (8) is equivalent to minimizing the size loss in (7). Therefore, we ground ConfTr as minimizing an upper bound to the true conditional entropy that is looser than the simple Fano bound and likely also looser than the model-based Fano bound for an appropriate choice for Q.

Algorithm 1: Conformal training algorithm.

5 Side Information

With the information theoretical interpretation of conformal prediction, we can translate various intuitions from information theory, for example, about different types of channels or network information theory, to conformal prediction. In this section, we consider the notion of side information.

Let X be the input covariates, Y be the target variable, Z be some side information about the task, and let Q Y | X be the model which we use to perform conformal prediction. As we can relate CP with Q Y | X to an upper bound on the conditional entropy H ( Y | X ) , we would like to do the same for the case of the conditional entropy when side information is available, i.e., H ( Y | X,Z ) . Since we know that the conditional entropy directly affects the expected set size, i.e., the inefficiency of CP, and given that H ( Y | X ) ≥ H ( Y | X,Z ) we can expect that with the additional side information the inefficiency of CP will decrease. We can take side information into account by defining conformity scores as a function of Q Y | X,Z instead of Q Y | X . A simple way to do that would be via the Bayes rule

where Q Z | X,Y is an auxiliary model that predicts the side information given the input and target variables. Such a model could be learned separately from the main model Q Y | X given access to a dataset D side = { ( x i , y i , z i ) } . Therefore, we can now calibrate with CP by taking into account the side information and then, at test time, given access to the input and side information, we can use the appropriate probabilities Q Y | X,Z to construct the prediction sets. Intuitively, the prediction sets from such a procedure should be smaller than the prediction sets obtained from using Q Y | X directly. It should be noted that, in the case of side information not being available, we can marginalize Q Y,Z | X over Z , which, by construction, falls back to the original model Q Y | X . If the availability pattern of Z is consistent between the calibration and test sets, the conformal prediction guarantees still hold by defining the model as

With this addition to the split conformal prediction tool set, if new (side) information is made available at test time, we can properly incorporate it into the CP pipeline without having to retrain the main classifier. One needs only train an auxiliary classifier Q Z | X,Y , which might be much simpler than Q Y | X (in our experiments, Q Z | X,Y is given by a single linear layer) and require much less data. One notable example of side information arises naturally in the distributed setting, which we discuss next.

5.1 The Distributed Learning Setting

Consider the case where we have a dataset that is distributed among a set of m devices and want to run conformal training to get a global model Q Y | X trained on all the data. Further, assume it is hard to gather the data at a central location (e.g., due to privacy reasons), and thus we have to work with the data staying locally on each device. An example of this would be federated learning (or FL) [42]. In this case, if Z ∈ { 1 , . . . , m } identifies the device, the entropy H ( Y | X ) can be expressed as

which decomposes into a weighted average of local entropy functions H ( Y | X,Z = z ) . We can now use any of our proposed bounds for each of the conditional entropies H ( Y | X,Z = z ) by calibrating with CP independently on each device, ending up with

where H ub ( Y | X,Z = z ) corresponds to an upper bound of the conditional entropy H ( Y | X,Z = z ) . Furthermore, for the mutual information term we have that

where the first inequality is due to Z being discrete and having non-negative entropy and the second is due to Gibbs inequality with Q Z | X being an auxiliary model trained to predict the user id Z = z

given input X = x . A similar upper bound has been considered before in a federated setting [36]. With this upper bound we get

This gives us an upper bound on the entropy of the entire population that decomposes into a sum of local functions, one for each client, only requiring local information. Thus, we can easily carry out conformal training for Q Y | X by minimizing this upper bound in the federated setting with, e.g., federated averaging [42]. At test time, we can take the device ID z as side information. To this end, we can either train a model Q Z | X,Y in parallel and use (9) with the global model Q Y | X at test time to get Q Y | X,Z , or we can obtain Q Y | X,Z by fine-tuning the global model Q Y | X with local data.

6 Related Work

Conformal prediction, a powerful framework for uncertainty quantification developed by Vovk and collaborators [60, 64], has recently witnessed a wide adoption in many fields, e.g., healthcare [3, 38, 39, 48] and finance [6, 67]. The marriage of conformal prediction and machine learning has been especially fruitful. Since the seminal work by Vovk et al. [64], many extensions and applications have been proposed, covering topics such as survival analysis [10], treatment effect evaluation [32], classification [4, 21] and regression [54] settings, risk control [5, 7], and covariate shift [62].

To our knowledge, our work represents the first attempt to bridge conformal prediction and information theory. Among other things, this allows us to build on the conformal training ideas of Bellotti [8] and Stutz et al. [61], deriving principled learning objectives that generalize their approaches, dispense with some of their hyperparameters and result in more efficient prediction sets. Further, we empirically show that our conformal training objectives provide a strong enough learning signal to train complex architectures from scratch, with strong results on ResNet-34 and ResNet-50 [23] fitted on CIFAR10 and CIFAR100, respectively. In contrast, the previous state-of-the-art approach, ConfTr, struggles in those settings (see experiments in Section 7) and required pretrained models for consistent results [61]. Further, our information-theoretic interpretation of CP provides a new simple and effective mechanism to leverage side information in split conformal prediction. We are unaware of any other approaches to treat side information within the conformal prediction framework in the literature.

On the information theory side, the notion of f -divergence and related inequalities have appeared in many different works. The use of f -divergence goes back to works of Ali and Silvey, Csiszár, and Morimoto in the 60s, as in, for instance, [2, 15, 44]. A key f -divergence inequality is the data processing inequality-see [57, 58] for an extensive survey. It provides a unified way of obtaining many classical and new results, including Fano's inequality [19]. The tightness of the data processing inequalities is discussed in terms of Bregman's divergence in [12, 34] and in terms of χ 2 -divergence in [57]. List decoding, which is closely connected to CP, was introduced in the context of communication design by Elias [18] and Wozencraft [68]. A generalization of Fano's inequality to list decoding was given in [16] in the context of multi-user information theory, see also the general Fano inequality for list decoding presented in [53]. Variable-size list decoding was discussed in [57] using ideas first introduced in [52] and [35]. A selection of relevant inequalities for list decoding can be found in [57].

7 Experiments

In this section, we empirically study two applications of our theoretical results, namely conformal prediction with side information and conformal training with our upper bounds on the conditional entropy as optimization objectives. We focus our experiments on classification tasks since this is the most common setting in previous works in conformal training [8, 13, 61].

7.1 Conformal Training

We test the effectiveness of our upper bounds as objectives for conformal training in five data sets: MNIST [29], Fashion-MNIST [69], EMNIST [11], CIFAR10 and CIFAR100 [25]. In addition to our three upper bounds, we also evaluate the cross-entropy loss (CE, also another upper bound to the entropy), and the two main variants proposed in [61], namely ConfTr, which minimizes (7) and ConfTrclass that optimizes an additional classification loss term (see Appendix F). We follow a similar

optimization procedure and experimental setup to that of [61], but with the key differences that we learn the classifiers from scratch in all cases (without the need of pretrained CIFAR models), and that we use the larger 'by class' split of EMNIST. For each data set, we use the default train and test splits but transfer 10% of the training data to the test data set. We train the classifiers only on the remaining 90% of the training data and, at test time, run SCP with 10 different calibration/test splits by randomly splitting the enlarged test data set. See Appendix G for a complete description of the experimental setup, with extra results and details on model architectures and hyperparameter search.

In Table 1, we report the empirical inefficiency on test data considering two SCP methods, threshold CP with probabilities (or THR) [56] and APS [55]-see Appendix G.1.1 for results with RAPS [4]. In all cases, our upper bounds proved effective loss functions to train efficient classifiers end-to-end and from scratch. For the simpler data sets (MNIST, Fashion-MNIST and EMNIST), all conformal training methods achieved similar results, but both ConfTr methods proved less consistent. This is noticeable in the oftentimes sharp difference in performance between THR and APS, since even after fine-tuning the hyperparameters (see Appendix G) some of the models failed to converge properly. For the remaining and more challenging data sets, both ConfTr variants lagged behind, probably because they do not provide a strong enough signal to train ResNets from scratch (on CIFAR data sets, Stutz et al. [61] only used ConfTr to fine tune pretrained models). A similar observation applies to the simple Fano bound (6), where the relaxed prediction set size is the only learning signal.

In all experiments, we run conformal training with a target coverage rate of 99%, i.e., α = 0 . 01 . It is then important to assess whether the performance of the resulting models deteriorates at different coverage rates, 'overfitting' to the value of α used for training. In Table 2, we see how inefficiency varies with α at test time for models trained via conformal training with α = 0 . 01 . In particular, we can contrast their performance against that of models trained via the CE loss, which is agnostic to the desired coverage rate. In all cases, our model-based Fano and DPI bound performs best with the THR and APS methods, respectively, proving conformal training is worthwhile even if the desired coverage rate might vary at test time. Still, as noticed in [61], there is a drop in performance in comparison to the CE loss for higher values of α at test time. This could be due to some degree of overfitting, but it could also be attributed to the conformal prediction problem becoming easier for lower coverage rates, thus reducing the gap between our bounds and the CE loss.

7.2 Side Information

As a first experiment, we consider datasets for which a natural grouping of the labels exists and use the group assignment as side information Z . In CIFAR100, there is a disjoint partition of the 100 classes into 20 superclasses, so we define z as the superclass to which example ( x, y ) belongs. In EMNIST, z indicates whether the example is a digit, uppercase or lowercase letter. We train an auxiliary model R Z | X,Y and, at test time, assume access to side information z to recompute class probabilities Q Y | X,Z = z from the original classifier Q Y | X as in equation (9).

We report results for two different scenarios in Table 3. The first is the standard SCP setting, where we assess the inefficiency of THR and APS methods, with side information Z observed for 10, 30 and 100% of the instances. We redefine the classifier f as in (10) to account for when Z is missing, but otherwise, the SCP process remains unchanged. The second scenario is Mondrian or group-balanced CP [64], where one splits D cal into groups and runs CP for each of them individually. In this setting, we group the calibration data points according to Z and base the score function on Q Y | X,Z . In all cases, taking the side information into account reduced the inefficiency considerably.

7.3 Federated Learning (FL)

A practically relevant application of side information arises in FL, where we can take the device ID as side information Z . In the federated setting, we train two extra heads on top of the main classifier, one computing Q Z | X so that we can optimize the proper upper bound in (11), and another computing Q Z | X,Y (while detaching gradients to the main classifier so as to not affect the upper bound optimization) that we use to integrate side information into CP using (9). Besides being a practically relevant application of side information to CP, FL also serves as a more challenging test bed for our conformal training methods, which has not been explored in previous work. We ran federated averaging [42] with CE, ConfTr, ConfTrclass, and our upper bounds as local loss functions. In this setting, we consider CIFAR10, CIFAR100, and EMNIST with 100, 500, and 1K devices, resp. We assign data points to devices imposing a distribution-based label imbalance [33], i.e., we sample a marginal label distribution for each device from a Dirichlet distribution Dir (1 . 0) . See Appendix G for results with Dir (0 . 5) and Dir (0 . 1) . As hyperparameter search in FL is notably challenging and costly [66], we keep the same hyperparameters found for the centralized case in Section 7.1.

After convergence, we ran SCP with the final global model assuming calibration and test data sets at the server, or equivalently that the clients share their scores with the server. This reflects the best inefficiency results we can hope for with the global model, as in practice we might need to resort to privacy-preserving methods that are likely to hurt performance. See Appendix G for a discussion and extra results on other possible settings. We report inefficiency results for the global model with THR in Table 4 (see Table 5 in the appendix for APS results), where we observe similar trends to the centralized experiments in Table 1. ConfTr methods still perform well on EMNIST but struggle on CIFAR data (with the notable exception on CIFAR100, where ConfTr excelled) while our methods delivered consistent results across all data sets. This probably reflects the sensitivity of both ConfTr objectives to hyperparameters, which makes them hard to use in practice, especially in FL where hyperparameter optimization is difficult. Conversely, our bounds seem more robust to such variations, as the hyperparameters found in the centralized setting seem to translate well to the federated case.

One marked difference between Tables 1 and 4 is that the simple Fano bound (6), which lagged behind the DPI bound and its model-based counterpart in the centralized setting, achieved the best results on the federated setting. We hypothesize this could be due to overfitting of the local optimization procedures to the individual data distribution of each device, which hurts the convergence of the global model. This effect is exacerbated on CIFAR100, where we have 500 devices, each of which with very few data points. The simple Fano bound is less vulnerable to such overfitting since it relies on the main classifier to a much lesser degree. Finally, in almost all cases, our bounds outperformed the CE loss, reassuring the potential of conformal training. Moreover, the inclusion of side information reduced inefficiency in all settings, and markedly so in a few instances. This confirms the effectiveness of our side information approach in a complex and practically relevant scenario, like federated learning.

8 Conclusion

In this work, we established a link between notions of uncertainty coming from conformal prediction and information theory (or more precisely variable-size list decoding). We proved that one can use split conformal prediction methods to upper bound the conditional entropy of the target variable given the inputs, and that these upper bounds form principled objectives for conformal training. We empirically validated our approach to conformal training, with strong results in both centralized and federated settings. Furthermore, the information-theoretic perspective also offers a simple yet rigorous approach to incorporate side information into conformal prediction, which we experimentally show leads to better predictive efficiency. To the best of our knowledge, this is the first attempt at connecting information theory and conformal prediction. Given the limited communication between these two research communities thus far, we expect our work to incite a fruitful exchange of not only ideas but also theory and algorithms between these two research domains. In this paper, we concentrated our exposition and experiments in classification tasks, but we see an extension of our methods to the regression setting, as a particularly promising avenue for future work.

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A Broader Impact, Limitations and Other Remarks

Broader Impact. This work explores the connection between conformal prediction and information theory, with the end goal of advancing the state of the art of uncertainty quantification in machine learning. With that in mind, we believe its potential societal consequences are chiefly positive, since our work might contribute to a larger adoption of uncertainty estimates, in a number of safety-critical applications. Notwithstanding, conformal prediction, like any other uncertainty estimation method, should be applied with care and a proper understanding of the provided guarantees so as not to create an illusion of safety when there is none. That is why our work aims to develop new methods and algorithms from first principles, with the hope of providing new techniques that we can study and understand at a deeper level so as to mitigate, or at least foresee, some of their failure modes.

Limitations. Our work is not without limitations. In practice, estimating the conditional entropy H ( Y | X ) is a difficult problem. That is part of the reason upper bounds might be useful, but that also means it is hard to evaluate how tight our upper bounds are. This also limits potential applications of our upper bounds like, for instance, estimating the expected prediction set size. Our experiments in Appendix G.3 rely on quantization to get a reasonable lower bound on the conditional entropy, and thus lower bound the expected prediction set size. Finally, in terms of our experimental results, we unfortunately have not been able to single out which of our new upper bounds performs best. The general trend we observe is that the DPI and model-based Fano bounds perform better on complex classification tasks, while simple Fano seems to excel in relatively simple tasks or where there is a high risk of overfitting, like in the federated setting.

Regression Setting. In this paper, we focus our experiments on the classification setting, similarly to previous works on conformal training [8, 13, 61]. However, in principle, our bounds pose no assumptions on the underlying prediction problem and we see the application of our results to the regression setting as a promising avenue for future work. In practice, only the simple Fano bound in (6) would not be directly applicable to the regression setting, but that is mainly because it assumes a uniform distribution over the output space and we get the, potentially infinite, |Y| term. The other two bounds we propose, DPI in (4) and model-based Fano in (5), do not include the |Y| and | C ( X ) | terms and can be applied to regression problems as is.

Computational Cost. The computational cost of using our bounds for conformal training is the same as that of previously proposed conformal training [61]. In comparison to regular training, i.e. minimizing the cross-entropy loss, the additional cost is given by the (differentiable) sorting operation of the scores, which in our case was performed with diffsort [50] with bitonic networks, which has complexity O ( b 2 log b 2 ) for b the batch size. Since the batch size is typically small, the additional computation cost is only marginal in practice.

B Background on Conformal Prediction

In this section, we outline a brief introduction to conformal prediction, providing the reader with the necessary background to follow our main results. Readers already familiar with conformal prediction can safely skip this section. We start by reviewing the definitions of quantiles and exchangeability, which are central to the main results in conformal prediction.

B.1 Exchangeability, Ranks and Quantiles

The main assumption in conformal prediction is that the data points used for calibration and testing are exchangeable. Next, we define the concept of exchangeability and discuss how it leads to the main results in conformal prediction via properties of ranks of exchangeable random variables. Our exposition is markedly brief, and we refer the reader to [26] for a more thorough discussion on exchangeability and its importance in conformal prediction.

Formally, the concept of exchangeability can be defined as follows.

Definition B.1 (Exchangeable Random Variables) . Random variables X 1 , . . . , X n are said to be exchangeable if for any permutation π : { 1 , . . . , n } → { 1 , . . . , n } , the sequences ( X 1 , . . . , X n ) and ( X π (1) , . . . , X π ( n ) ) have the same joint probability distribution.

Note that exchangeability is a weaker assumption than the i.i.d. (independent and identically distributed) assumption commonly relied upon in machine learning. More precisely, exchangeable random variables must be identically distributed but not necessarily independent [26]. Naturally, i.i.d. random variables are also exchangeable.

One relevant consequence of exchangeability that is central to conformal prediction is that the ranks of exchangeable random variables are uniformly distributed. We define ranks and this property more formally in Definition B.2 and Lemma B.4, respectively.

Definition B.2 (Rank) . For a set of n elements X = { x 1 , . . . , x n } , the rank of any one element x i in X is defined as

for ξ ≥ 0 and U = { U 1 , . . . , U n } a set of i.i.d. random variables uniformly distributed in [ -1 , -1] .

Remark B.3 . The addition of i.i.d. uniform noise serves as a tie-breaking mechanism. Since { U 1 , . . . , U n } are almost surely distinct, { x i + ξU i } n i =1 are also distinct with probability one. This is necessary to render the rank independent of the distribution of X i , which is key to ensure the distribution-free quality of conformal prediction.

Lemma B.4. If ( X 1 , . . . , X n ) are exchangeable random variables, then

In words, Lemma B.4 tell us that the ranking of exchangeable random variables is uniformly distributed among all possible permutations π : { 1 , . . . , n } → { 1 , . . . , n } . That means the probability of observing any one ranking is equal to 1 / n ! and, importantly, independent of the distribution of X . The corollary below follows directly from Lemma B.4.

Corollary B.1. If ( X 1 , . . . , X n ) are exchangeable random variables, then

for t ∈ [0 , n ] and ⌊ t ⌋ the smallest integer smaller or equal to t . Moreover, we can define a valid p-value as P := rank ( X i ; { X 1 , . . . , X n } ) /n since

Proof.

where the first equality follows because rank ( . ) returns an integer, and the third equality follows directly from Lemma B.4: each permutation of ( rank ( X i ; { X 1 , . . . , X n } )) n i =1 has the same probability 1 / n ! , and there are ( n -1)! configurations where rank ( X i ; { X 1 , . . . , X n } ) = i since X i is fixed at rank i , and we can permute the other ( n -1) variables.

As we shall see, Corollary B.1 is central to the main result in conformal prediction, but before proving that result, we should first define the concept of quantiles.

Definition B.5 (Quantile) . For Z a random variable with probability distribution F , the level β quantile of distribution F is defined as

Similarly, for a sample { z i } n i =1 and δ z i a point mass concentrated at z i , the quantile of the empirical distribution is given by

B.2 Conformal Prediction

Armed with the definitions of quantiles and exchangeability, we are prepared to study conformal prediction, a distribution-free uncertainty quantification framework with the following goal: given a set of data points { ( X i , Y i ) } n i =1 sampled from some distribution P on X × Y , to construct a set predictor C : X → 2 Y such that for a new data point ( X test , Y test ) and target error rate α ∈ (0 , 1) , we have the guarantee

where the probability is over the randomness of { ( X i , Y i ) } n i =1 ∪ { ( X test , Y test ) } . Since the probability is taken over both X test and Y test , this means that on average across all possible values in X , the correct label Y test is included in the constructed set C ( X test ) with probability at least 1 -α. This property is known as marginal coverage . This is in contrast to the stronger conditional coverage , which requires

That is, the guarantee holds for each X test individually, with X test fixed and the probability taken over the randomness of Y test only. However, distribution-free conditional coverage is known to admit no non-trivial solution (i.e., besides C ( X test ) = Y ) [30, 63]. Therefore, in this paper, whenever we refer to the lower and upper bounds provided by conformal prediction, we mean those provided by Theorem 2.1, which guarantees only marginal coverage.

There are different ways to achieve marginal coverage, but for simplicity we will focus on split conformal prediction or SCP [47], since it is easier to grasp than other variants of conformal prediction and is also the main object of study in this paper. We start by assuming a calibration dataset D cal which consists of n i.i.d. samples ( X i , Y i ) drawn from an unknown distribution over X ×Y . We also assume access to a model f : X → ˆ Y , where the output space ˆ Y can be different from Y . Prediction sets satisfying the guarantee above can be constructed via the following three steps.

Define a non-conformity score function s : X × Y → R , which assigns high scores to unusual pairs ( x, y ) . The score function is typically a function of the model f itself.

Compute S i = s ( X i , Y i ) for all ( X i , Y i ) ∈ D cal and compute Quantile (1 -α ; { S i } n i =1 ∪ {∞} ) , the empirical 1 -α quantile of the scores in D cal .

Construct prediction sets C ( X test ) = { y : s ( X test , y ) ≤ Quantile (1 -α ; { S i } n i =1 ∪{∞} ) }

Theorem 2.1. (same result as in the main text (Lei et al. [31], Vovk et al. [64]) Let { ( X i , Y i ) } n i be i.i.d. (or only exchangeable) random variables, and { S i } n i =1 be the corresponding set of scores S i = s ( X i , Y i ) given to each pair ( X i , Y i ) by some score function s : X × Y → R . Then for a new i.i.d. draw ( X test , Y test ) and any target error rate α ∈ (0 , 1) , the prediction set constructed as

satisfies the marginal coverage property

Moreover, if { S i } n i =1 are almost surely distinct, this probability is upper bounded by 1 -α + 1 / n +1 .

Proof. For simplicity, we assume that the set of scores { S i } n i =1 are distinct (or have been made distinct by a suitable random tie-breaking rule). Denote S test = s ( X test , Y test ) and observe that since s ( · ) is applied element-wise to each pair ( X i , Y i ) it preserves exchangeability, and thus random variables { S i } n i =1 ∪{ S test } are also exchangeable. Next, we show that the following events are all equivalent

(i) follows from the construction of the prediction set itself C ( X test ) itself.

(ii) can be easily verified as follows. If S test ≤ Quantile (1 -α ; { S i } n i =1 ∪{∞} ) , then shifting all values S i ≥ S test to arbitrary values larger than S test will not change the validity of the inequality, since the 1 -α quantile remains unchanged. In particular, the inequality holds when replacing { S test } with {∞} and vice-versa.

(iii) follows from the fact that if S test ≤ Quantile (1 -α ; { S i } n i =1 ∪ { S test } ) , then S test is among the ⌈ ( n +1)(1 -α ) ⌉ smallest values of the set { S i } n i =1 ∪ { S test } .

Finally, this is where the crucial exchangeability assumption comes into play, allowing us to apply Corollary B.1 to get

From there, it is easy to verify that the right hand side is at least 1 -α and at most 1 -α + 1 / n +1 .

C Background on List Decoding and Basic Information Theoretic Results

In this section, we enunciate some classical results from information theory that are instrumental in deriving our main theoretical results, as we show in the detailed proofs in Appendix D. We also provide a brief introduction to list decoding and demonstrate conformal prediction can be framed as a list decoding problem.

C.1 Data processing inequalities for f-divergence

We start by presenting the data processing inequality (DPI) for f -divergence, which we define below. Definition C.1 ( f -Divergence) . Consider two probability measures P and Q and assume that the measure P is absolutely continuous with respect to Q , i.e., P ≪ Q . For a convex function f : (0 , ∞ ) → R with f (1) = 0 , the f -divergence is defined as:

where dP dQ is a Radon-Nikodym derivative. In particular, using f ( x ) = x log x we recover the familiar notion of KL-divergence

We can similarly define conditional f -divergence.

Definition C.2 (Conditional f -Divergence) . Consider two probability measures P and Q such that P := P X P Y | X and Q := P X Q Y | X and that the measure P is absolutely continuous with respect to Q . For a convex function f : (0 , ∞ ) → R with f (1) = 0 , the conditional f -divergence is defined as:

Theorem C.3 is the classical data processing inequality (DPI), which we restate below for the sake of completeness. The classical versions of DPI, stated in terms of mutual information, can be found in classical information theoretic text books like [14], while the generalization of DPI to f -divergences can be found in other works with a comprehensive survey in [51, 58].

Theorem C.3 (Data Processing Inequality for f -divergence) . Consider a conditional distribution W Y | X . Suppose that P Y and Q Y are two distributions obtained by marginalization of P X W Y | X and Q X W Y | X over X . For any f -divergence, we have

The proof can be found in standard textbooks in information theory see, for example, Chapter 7, Section 7.2 in [51] for a derivation of the DPI enunciated in the same form as above. Next, we consider the application of the DPI to conformal prediction.

Theorem C.4 (Data Processing Inequality for Conformal Prediction) . For any set function C : X → 2 Y , any f -divergence, and all distribution pairs P, Q on ( X,Y ) , we have:

where d f ( p || q ) is the binary f -divergence, namely d f ( p || q ) = qf ( p/q ) + (1 -q ) f (1 -p/ 1 -q ) .

Proof. Consider the random variable E = 1 ( Y ∈ C ( x )) denoting the event of valid coverage, and the conditional distribution P E | X,Y = E D cal [ 1 ( Y ∈ C ( x )) | X,Y ] . Note that P E | X,Y maps distributions P X,Y and Q X,Y to P E and Q E , resp. From here, the result follows directly from Theorem C.3.

The construction in Theorem C.4 can be further improved as follows.

Theorem C.5 (Conditional Data Processing Inequality for Conformal Prediction) . For any set function C : X → 2 Y , any f -divergence, and all conditional distribution pairs P Y | X , Q Y | X , and P X , we have:

Proof. Consider the conditional distribution P E | X = x,Y = E D cal [ 1 ( Y ∈ C ( x )) | X = x, Y ] . We have that

and from the monotonicity property of f-divergences [51] we have that

By taking the expectation with respect to P X , we conclude the proof.

C.2 List Decoding

List decoding [18, 68] is a notion coming from coding theory, a large branch of engineering and mathematics that arises from the application of information theory to the design of reliable communication systems and robust information processing and storage. In particular, we are interested in channel coding, a field concentrated on the design of so-called error-correcting codes to enable reliable communication over inaccurate or noisy communication channels.

The general setup studied in channel coding, including list decoding, can be summarized as follows. A message y ∈ Y is encoded and transmitted over a noisy channel, and a message x ∈ X is received. The noisy channel is governed by probability density p ( x | y ) that describes the probability of observing output x ∈ X given input y ∈ Y . The receiver attempts to decode x , that is, to guess the originally transmitted message y from the received one, x . At this point, parallels to machine learning should already have become clear. If the receiver provides a single guess of the transmitted message y , we are in a unique-decoding scenario, which is akin to a point prediction in machine learning. Conversely, if the receiver is allowed to guess a set (or a list in the list decoding formalism) of the most likely messages, we have list decoding, which closely resembles conformal prediction. Note that in many settings, a list decoding algorithm is constrained to output a list of fixed size. While simple conformal prediction methods for regression settings show the same limitation, the parallel between the two domains is more pertinent when we consider variable-size list decoding [57].

More formally, a list-decoding algorithm can be defined by a set predictor L : X → 2 Y , with maximum output set size | L ( X ) | ≤ M. Naturally, the goal is to design the function L so as to maximize the probability of Y ∈ L ( X ) . Similarly, for a given input-label pair ( X,Y ) the goal of conformal prediction is to provide a set that contains Y with a certain pre-determined probability. It is this connection that motivates our bounds on the conditional entropy H ( Y | X ) . However, the nonconformity score in conformal prediction is typically a function of the output of a given model

f : X → Y . For instance, ˆ Y = f ( X ) could be the logits output by the model, i.e., a vector with the dimension equal to number of classes. If we consider ˆ Y the noisy observation of the ground-truth Y , then the problem of determining a set containing Y is again the list decoding problem. Therefore, we can also consider the communication channel p (ˆ y | y ; x ) directly, which justifies applying the same upper bounds to H ( Y | ˆ Y ) . Note that in this last case, the input data X can also be taken as side information, although it is rarely used directly in building the conformal prediction set.

This reinterpretation of conformal prediction as list decoding allows us to apply some of the standard results from the list decoding literature to conformal prediction, as we show in the next section.

C.3 List decoding: Information Theoretic Inequalities

Fano's inequality for variable size list decoding. The following generalization of Fano's inequality is given in [[53], Appendix 3.E].

Theorem C.6. Consider a scenario where a decoder upon observing ˆ Y provides a nonempty list L ( ˆ Y ) that contains another random variable Y ∈ Y with |Y| = M . Define P e := P ( Y / ∈ L ( ˆ Y )) . We have:

Optimal list decoding and conformal prediction. It can be shown that the optimal list decoder consists of selecting L (ˆ y ) elements of Y with highest conditional probability p ( y | ˆ y ) . That is, consider the sorted posteriors under the true distribution p ( y 1 | ˆ y ) ≥ p ( y 2 | ˆ y ) ≥ · · · ≥ p ( y M | ˆ y ) and choose the first { y 1 , . . . , y | L | } for some list size | L | . However, this rule is for fixed-size list decoding and does not determine how to select the coverage set size to guarantee a given coverage. We can modify this rule to obtain a variable-size list decoding with the required coverage. Assuming again the sorted posteriors p ( y 1 | ˆ y ) ≥ p ( y 2 | ˆ y ) ≥ · · · ≥ p ( y M | ˆ y ) , we can select the set as follows:

It is easy to see that the above set is the smallest set given each y and the confidence level 1 -α (see, for example, [43]). The same result holds in conformal prediction [55, 65].

C.4 Fano and Data Processing Inequalities for Conformal Prediction

First, from Fano's inequality for list decoding, Theorem C.6, we get the next result 'out-of-the-box'.

Proposition C.7. Suppose that |Y| = M . Any conformal prediction method with the prediction set C ( x ) and confidence level 1 -α , α ∈ (0 , 0 . 5) , satisfies the following inequality:

where h b ( · ) is the binary entropy function, [ x ] + := max { x, 0 } and H ( Y | X ) is computed using the true distribution P XY . When the conformal prediction is merely based on the model output ˆ Y = f ( X ) , the inequality can be modified to:

The proposition follows easily from Theorem C.6 by using the condition P ( Y ∈ C ( x )) ≥ 1 -α . Note that in Theorem C.6, we assume non-empty lists, whereas in Proposition C.7 we allow empty prediction sets but apply the maximum operator [ x ] + := max { x, 0 } . This is justified because the last term of Fano's inequality relates to the probability of correct assignments Y ∈ C ( X ) , which never happens for empty sets. See Proposition E.1 for the proof, where the same result appears.

The bounds that we present in the main paper, model-based and simple Fano bounds, are actually derived through a slightly different root by leveraging the lower and upper bounds in the finite-sample guarantee of conformal prediction (Theorem 2.1). We derive these other two bounds in Appendix D.2.

C.5 Related Work on Information Theoretic Inequalities

The use of f -divergences goes back to works of Ali and Silvey, Csiszár, and Morimoto in the 60s-see for instance [2, 15, 44]. A key f -divergence inequality is the data processing inequality, which was used in information theory to establish various upper bounds on the achievability of coding schemes for different tasks-see [57, 58] for an extensive survey. Moreover, the data processing inequality for f -divergences provides a unified way of obtaining many classical and new results, for example Fano's inequality [19]. The tightness of data processing inequalities is discussed in terms of Bregman's divergence in [12, 34] and in terms of χ 2 -divergence in [57].

When it comes to list decoding, there are a number of relevant inequalities in the literature. List decoding was introduced in the context of communication design by Elias [18] and Wozencraft [68]. See also [22] for a more recent overview of list decoding. For fixed list size, the information theoretic bounds on list decoding were obtained in [20] using error exponent analysis. A generalization of Fano's inequality to list decoding was given in [16] in the context of multi-user information theory, see also the general Fano inequality for list decoding presented in [53]. For fixed list size, stronger inequalities, some based on Arimoto-Rényi conditional entropy, were presented in [59]. Variable size list decoding was discussed in [57] using the notion of E γ resolvability first introduced in [52] related to the dependence testing bound. It was used again in the context of channel resolvability in [35], where some relevant inequalities have been obtained and discussed. A selection of the most relevant inequalities for list decoding can be found in [57].

D Proofs of Main Theoretical Results

In this section, we provide the proofs of our main results, namely the DPI bound in Proposition 3.1, the model-based Fano bound in Proposition 3.2, and the simple Fano bound in Corollary 3.1. For notational convenience, we use the shorthand α n = α -1 / n +1 in most of the steps of the derivations.

D.1 DPI Bound

We start with the DPI bound which we restate and proof below using the data processing inequalities discussed in Appendix C. Note that, when clear from the context, we remove explicit dependence on the calibration set D cal from the derivations. It is implicitly assumed that the probability of the event Y ∈ C ( x ) is computed by marginalizing over D cal .

Proposition 3.1. Consider any conformal prediction method with the prediction set C ( x ) with the following finite sample guarantee:

for any α ∈ (0 , 0 . 5) . For any arbitrary conditional distribution Q Y | X , the true conditional distribution P Y | X and the input measure P X , define the following two measures Q := P X Q Y | X and P := P X P Y | X . We have for any α ∈ (0 , 0 . 5)

Proof. Consider an arbitrary distribution Q Y | X . Then we can use P XY , and P X × Q Y | X in the data processing inequality for KL-divergence (Theorem C.4) to get:

Now note that we can decompose D KL ( P X P Y | X || P X Q Y | X ) in terms of the conditional entropy H ( Y | X ) and the cross-entropy -E P XY [log Q Y | X ] :

With the decomposition above, we can rearrange the terms in (12) to get the following upper bound on H ( Y | X )

We can then apply the upper and lower bounds from conformal prediction, i.e. P ( Y ∈ C ( x )) ≥ 1 -α and P ( Y / ∈ C ( x )) ≥ α n , to upper bound d KL ( P ( Y ∈ C ( x )) || Q ( Y ∈ C ( x ))) as follows.

where h b ( · ) is the binary entropy function, that is, h b ( α ) = -α log( α ) -(1 -α ) log(1 -α ) . The equality in the second line follows from the definition of the binary KL divergence, and we get the inequality simply by upper bounding each of the terms individually. In particular, note that log Q ( Y ∈ C ( x )) and log Q ( Y / ∈ C ( x )) are both negative, and h b ( · ) is decreasing in [0 . 5 , 1 . 0] and symmetric about 0 . 5 , such that for typical values of α < 0 . 5

Finally, we can replace the upper bound above into (13) to conclude the proof.

Remark D.1 . The DPI bound always provides a tighter upper bound on the conditional entropy H ( Y | X ) than the cross-entropy, which is easily verified in (13) since the KL term is always nonnegative. This serves as an important motivation to optimize the DPI bound instead of the crossentropy in conformal training.

Remark D.2 . The derivation of the DPI bound places no assumptions on the conditional distribution Q Y | X . However, in practice, the underlying model f often already provides such a distribution, and since it is typically trained to approximate P Y | X well, it makes sense to take Q Y | X as the distribution defined by the model itself. That is how we evaluate the DPI in all of our experiments. Finally, we can again use H ( Y | ˆ Y ) instead of H ( Y | X ) if the conformal method uses merely ˆ Y .

Remark D.3 . Typically, we have α ∈ (0 . 0 , 0 . 5) , and we use this fact in the proof to bound the binary entropy h b ( P ( Y ∈ C ( x ))) . The same could have been done for α ∈ (0 . 5 , 1 . 0) , but since 1 -α lands in the increasing part of the binary entropy function between 0 and 0 . 5 , we have to resort to the lower bound from conformal prediction to get

Remark D.4 . One of the appeals of the DPI bound is that the terms can be computed in a data-driven way using samples. While the cross-entropy can be estimated in an unbiased way with samples from the true distribution P XY , we must be careful when estimating Q ( Y ∈ C ( x )) . The main challenge is that Q ( Y ∈ C ( x )) appears inside a log , and thus an empirical estimate ˆ Q ( Y ∈ C ( x )) would yield a lower bound of the negative KL divergence and would be biased. We can get an upper confidence bound on this estimate via the empirical Bernstein inequality [41], which we restate below.

Theorem D.5 (Empirical Bernstein Inequality [41]) . Let Z, Z 1 , . . . , Z n be i.i.d. random variables with values in [0, 1] and let δ > 0 . Then with probability at least 1 -δ in the i.i.d. vector Z = ( Z 1 , . . . , Z n ) we have that

where V n ( Z ) is the empirical variance over the ( Z 1 , . . . , Z n ) samples.

By using this bound, we have with probability 1 -δ :

where Z i = Q ( y i ∈ C ( x i )) for x i sampled from P X and a prediction set C ( x i ) obtained with a calibration dataset sampled from P D cal . This yields the following inequality with probability 1 -δ :

We can use this inequality to evaluate the bounds.

D.2 Model-Based Fano Bound

In this section, prove the model-based Fano bound, which we restate in Proposition 3.2 below.

Proposition 3.2. Consider any conformal prediction method with the prediction set C ( x ) , and any distribution Q , with the following finite sample guarantee:

for α ∈ (0 , 0 . 5) . Then, for the true distribution P , and for any probability distribution Q , we have

Proof. For notational convenience, define E = 1 ( Y ∈ C ( x )) , which by our assumption on the conformal prediction method means that 1 -α ≤ P ( E = 1) ≤ 1 -α n . The starting point is similar to the well-known proof of Fano's inequality:

where the last step follows because knowing X,Y and D cal , we know if Y ∈ C ( x ) , and therefore H ( E | Y, X, D cal ) = 0 . Furthermore, given the structure of the graphical model of the conformal prediction process (c.f. Figure 1), we have that H ( Y | X ) = H ( Y | X, D cal ) , We now find an upper bound on H ( E,Y | X, D cal ) :

where the first inequality follows from the fact that H ( E | X, D cal ) ≤ H ( E ) , and the last one comes from the same argument for a < 0 . 5 in (14). We can continue as follows

The first equality comes from the definition of the conditional entropy, whereas in the second equality we replace the true distribution P Y | X with an arbitrary conditional distribution Q Y | X plus the KL divergence between the two distributions. The last inequality then follows simply from the fact that the KL divergence is non-negative. Finally, we can leverage the finite-sample guarantees from conformal prediction, namely P ( E = 0) ≤ α and P ( E = 1) ≤ 1 -α n , to upper bound each term, which yields the proof.

Remark D.6 . When the conformal prediction is merely based on the model output ˆ Y = f ( X ) , the model-based Fano bound can be modified to

D.3 Simple Fano

In the derivation of the model-based Fano bound above, we placed no assumptions on the distribution Q Y | X . One simple choice that we consider in this section is the uniform distribution Q Y | X = 1 / |Y| , akin to the classical Fano's inequality [19] and the list decoding result in Proposition C.7.

Corollary 3.1. Consider any conformal prediction method with the prediction set C ( x ) , and any distribution Q , with the following finite sample guarantee:

For α ∈ (0 , 0 . 5) we have the following inequality:

Proof. Note that if we make an error ( E = 0 ) the correct class will not be inside C ( X ) and since Q Y | X is uniform the probability will be spread equally among the remaining |Y|-|C ( X ) | labels, and we have Q Y | X, C ( x ) ,E =0 = 1 |Y|-|C ( X ) | . Through the same logic, we get that Q Y | X, C ( x ) ,E =1 = 1 |C ( X ) | , and plugging both into (17) we get the simple Fano bound

Remark D.7 . Once again, when the conformal prediction prediction is merely based on the model output ˆ Y = f ( X ) , the inequality can be modified to

E Further Theoretical Results and Proofs on the Prediction Set Size

E.1 Fano's inequality for maximal prediction set size

If we leverage the upper bound on the prediction set size, we can find a lower bound on the maximum coverage set size.

Proposition E.1. Suppose that |Y| = M . Consider any conformal prediction method that constructs the prediction set C ( X ) with the following finite-sample guarantee:

for α ∈ (0 , 0 . 5) . Then, we have the following inequality:

We can similarly replace H ( Y | X ) with H ( Y | ˆ Y ) .

Proof. We use the conditional data processing inequality with f ( x ) = x log x , P = P D cal XY , and Q = P D cal X × U M where U M is the uniform distribution over Y . We fix the input to X = x , and the calibration set D cal . The conditional f -divergence, conditioned on D cal and X , is given by:

where the last step follows from the independence of the calibration set D cal and ( X,Y ) . This means that

On the other hand, we have

Now, we can plug both (18) and (19) into the data processing inequality of Theorem C.5. Rearranging the terms and getting the expectation from both sides w.r.t. D cal and X would yield:

Note that the log M terms in the third line cancel each other out, and that the second inequality comes from resolving the expectations and log( M -|C ( X ) | ) ≤ log M. Further, in the last inequality, we used the concavity of binary entropy as in (14). Finally, using respectively the inequalities P ( Y ∈ C ( x ) |D cal , X ) ≤ 1 and log |C ( x ) | ≤ sup D cal sup x ∈X log |C ( x ) | , we get two inequalities:

The first one is the Fano's inequality for list decoding from Proposition C.7 while the second one yields the theorem. Note that if C ( x ) is an empty set, the probability P ( Y ∈ C ( x ) |D cal , X ) is zero, and the term P ( Y ∈ C ( x ) |D cal , X ) log |C ( x ) | disappears. Therefore, if we use the inequality P ( Y ∈ C ( x ) | sup D cal , X ) ≤ 1 , we need to introduce the term [log |C ( x ) | ] + to keep the expectation well defined.

E.2 Fano's inequality for lower bound on prediction set size

In a similar manner, we can also obtain lower bounds for the set size. More specifically, we have that

where u ( y C ( x ) ) and u ( y ¯ C ( x ) ) denote uniform distributions over the labels in the confidence set. By considering the standard conformal prediction bounds on the error probabilities, we have that

which leads to

Note that when E = 1 , we know that Y ∈ C ( x ) , and therefore |C ( x ) | > 0 and [log |C ( x ) | ] + = log |C ( x ) | . Using this, we can find an upper bound on E E =1 (log( |C ( x ) | )) as follows:

This leads to:

All the above terms can be approximated from samples. We summarize this in the following proposition.

Proposition E.2. For any conformal prediction scheme with the coverage guarantee of 1 -α , and any distribution q ( · ) , we have:

where ˆ Q 0 Y | X = q ( y | x ) I [ y / ∈ C ( x )] and ˆ Q 1 Y | X = q ( y | x ) I [ y ∈ C ( x )] . When the conformal prediction is merely based on the model output ˆ Y = f ( X ) , the inequality can be modified to

Remark E.3 . If we use the uniform distribution in the above bound, we get a bound similar to what is obtained from Fano's inequality given in Proposition C.7, but with an additional factor of 1 -α 1 -α + 1 n +1 . Since the factor is smaller than one, the current bound with the choice of uniform distribution is looser than Fano's bound, although the gap vanishes for large n .

F Conformal Training

Split conformal prediction (SCP) [47] has quickly become a popular framework for uncertainty quantification, largely thanks to its computational efficiency. One only needs access to a separate calibration data set to derive prediction sets with valid marginal coverage from any pretrained model. Given that training new machine learning models is becoming ever more time-consuming and expensive with new, larger architectures, this ability to apply conformal prediction to existing models is invaluable in a number of applications. Yet, it is reasonable to expect that the performance of the final set predictor could be improved if the conformal prediction process were to be accounted for during training of the model as well, steering the model towards better predictive efficiency, as it were. That is the motivation behind the line of work that we broadly refer to as conformal training [8, 13, 61]. In a nutshell, conformal training introduces a differentiable, and hence approximate, conformal prediction step during training so that one can directly optimize for desired properties of the set predictor, most notably its predictive efficiency. In what follows we give an overview of conformal training, focusing on classification tasks.

The idea of conformal training has been proposed concomitantly in [8, 13]. Here we follow the approach in [8], where the key idea is to relax the prediction set defined by the model f and define 'soft' prediction sets ˆ C f ( x ) , which contain each of the labels y ∈ Y with a certain probability. That is, if C f ( x, y ) ∈ { 0 , 1 } is the hard assignment of label y to C ( x ) , a corresponding soft version of this assignment can be defined as

where s f ( x, y ) is the non-conformity score function defined by model f evaluated at ( x, y ) , ˆ q is a thresholding value, σ is the logistic sigmoid function, and T is a temperature parameter controlling the smoothness of the soft assignment. Then we can define ˆ C f ( x ) as the vector collection of all the soft assignments ˆ C f ( x, y ) for all labels y ∈ Y . Similarly the size of ˆ C f ( x ) can be naturally defined as

Bellotti [8] then proposes the following loss functions which are computed for each training batch B

where α is the desired coverage rate and g is a user-defined function of the prediction set size, e.g. the log function. Intuitively, L size encourages small (efficient) prediction sets, whereas L coverage penalizes deviations from the target coverage of 1 -α . Naturally, there is a trade-off between these two objectives, inefficiency and coverage, so both loss terms are optimized together with a hyperparameter λ governing the influence of each term:

Importantly, Bellotti [8] argues the choice of the threshold ˆ q in (24) is immaterial since the model can learn to shift its outputs (in logit space) accordingly to match the constraints in L coverage ( f ) . We can then directly optimize (25) via stochastic gradient descent methods during training since it is fully differentiable with respect to the model parameters.

Stutz et al. [61] build on the work of Bellotti [8] by noticing that the calibration step is an important component in conformal prediction that should also be accounted for during training. To that end, they propose to split each training batch B in two: the B cal half used for calibration, and the B test used for testing. Now, instead of using an arbitrary threshold ˆ q , we compute it using the quantile of B cal , or concretely

With this modification, we no longer need to enforce valid coverage via L coverage ( f ) and can optimize for low inefficiency directly by minimizing L size ( f ) on B test . In that case, however, we only get a learning signal from B test , since the quantile operation applied to B cal is non-differentiable. Stutz et al. [61] bypass that limitation via differentiable sorting operators [9, 17, 50], in particular via

a version of differentiable sorting networks. In our experiments, we considered both fast sort [9] and the monotonic differentiable sorting networks of [50] but finally chose the latter since they proved more stable and provided richer gradient signals.

This version of their approach, which we refer to as ConfTr, only optimizes the size loss, but Stutz et al. [61] also proposed another variant which includes a classification loss term as follows

̸

where δ is the indicator function, and L is a user-defined square matrix of size |Y| 2 with L y, ˆ y capturing some similarity notion between y and ˆ y . In our experiments, as well as most experiments in the original paper [61], no prior information about the classification problem is assumed, in which case L is taken to be the identity matrix of size |Y| . Therefore, we have two variants of conformal training as proposed in [61]: ConfTr that optimizes only L size , and ConfTrclass that jointly optimizes L size and L class , both of which are included in our experiments.

Our own approach to conformal training follows the same recipe from [61], i.e., we also simulate a split conformal prediction step during training by splitting each training batch into two and using differentiable sorting operators (see Algorithm 1). The key difference is in how we define the training objectives, which we derive from first principles and standard information theory inequalities. Not only do our upper bounds, DPI (4), MB Fano (5) and Fano (6), outperform the ConfTr objectives in many cases, but they also do away with a few hyperparameters. Namely, the function g in L size , and hyperparameters controlling the relative importance of L size and L class .

F.1 Deriving Conformal Training from Fano's bound

Through Proposition 3.2, we can connect Fano's bound for list decoding to the size loss from [61], proposed for conformal training. Assuming a uniform distribution for Q we can show that

Note that in the first line we have the simple Fano bound, whereas in the last one we have the ConfTr objective, namely log E [ |C ( x ) | ] , multiplied by 1 -α n plus a constant that depends only on α. Therefore, we ground ConfTr as minimizing a looser upper bound to the true conditional entropy of the data than the simple Fano bound we provide in Corollary 3.1. Moreover, the simple Fano bound can be further improved with an appropriate choice of Q , for instance as given by the model f , in the model-based Fano bound of Proposition 3.2.

G Experiments

In this section, we present further experimental results for conformal training in the centralized and federated setting. We start by defining the splits and architectures used for each data set, which are listed in Table 6. In most aspects, we follow the experimental design of [61]. All experiments were conducted on commercially available NVIDIA GPUs using our own implementation in Python 3 and Pytorch [49], which can be found at github.com/Qualcomm-AI-research/info_cp . All data sets were retrieved directly from torchvision [40].

Regarding the architectures, we also closely follow the experimental setup in [61]. For MNIST we have a simple linear model, whereas for Fashion-MNIST and EMNIST we use 2-layer MLPs with 64 and 128 hidden units for first and second layers, respectively. For the CIFAR data sets, we use the default ResNet implementations from torchvision [40], but changing the first convolution to have a kernel size of 3 and unitary stride and padding. We use Pytorch's default weight initialization strategy for all architectures. For all datasets, we use a regular SGD optimizer with momentum 0 . 9 and Nesterov gradients, accompanied by a step scheduler multiplying the initial learning rate by 0 . 1 after 2 / 5 , 3 / 5 and 4 / 5 of the total number of epochs. We only use data augmentations on the CIFAR datasets, and differently from [61], we only apply random flipping and cropping for both CIFAR10 and CIFAR100.

G.1 Centralized Setting

We followed the experimental procedure of [61], and for each dataset and each method, we ran a grid search over the following hyperparameters using ray tune [45]:

Batch size with possible values in { 100 , 500 , 1000 } .

Learning rate with possible values in { 0 . 05 , 0 . 01 , 0 . 005 } .

Temperature used in relaxing the construction of prediction sets at training time. We considered temperature values in { 0 . 01 , 0 . 1 , 0 . 5 , 1 . 0 } .

Steepness of the differentiable sorting algorithm (monotonic sorting networks with Cauchy distribution [50]), which regulates the smoothness of the sorting operator; the higher the steepness value, the closer the differentiable sorting operator is to standard sorting. We considered steepness values in { 1 , 10 , 100 } .

In Tables 7, 8, 9, 10 and 11, we report the best hyperparameters found for each dataset and method as well as the average prediction set size for threshold CP [56] computed in the probability domain (THR) and APS [55], as well as the test accuracy. Importantly, similarly to [61], in all cases we only train the models to optimize threshold CP with log-probabilities. We confirm the observation in [61] that other methods, and notably APS, are unstable during training, probably because it forces us to operate in the probability domain, as opposed to the more optimization-friendly logits or logprobabilities. Nevertheless, we still select hyperparameters according to the best performance with respect to each CP method, and that is why we have different optimal hyperparameters for THR and APS for each data set and each conformal training objective. We note ConfTr and ConfTrclass require extra hyperparameters like the target size and weights attributed to each loss term (see Appendix F). For those hyperparameters, we use the best values for each data set as reported in [61].

As described in the main paper, we use the default train and test splits of each data set but transfer 10% of the training data to the test data set. We train the classifiers only on the remaining 90% of the

training data and, at test time, run SCP with 10 different calibration/test splits by randomly splitting the enlarged test data set. All results reported in the paper are given by the average ( ± one standard deviation) computed across these 10 random splits. Crucially, to avoid overfitting to the test data, the grid search was done solely on the 90% of the training data not used for testing.

G.1.1 Results with Regularized Adaptive Prediction Sets (RAPS)

Wereport additional results with a conformal method known as RAPS (regularized adaptive prediction sets) [4]. In a nutshell, RAPS works exactly like APS but adds a penalization term λ reg to the score of each label that would make the prediction set larger than k reg . In Table 12, we have results with RAPS hyperparameters set as k reg =1 and λ reg =0 . 01 , where we can see a pattern of performance across conformal training objectives that is similar to what we previously observed for THR and APS. As expected, the regularization introduced by RAPS results in prediction sets that are smaller than those obtained via APS in almost all cases. Again, similarly to [61], in all of our experiments we perform conformal training by thresholding on log-probabilities, which we observed to work best. The results we report for thresholding on probabilities (THR), APS and now RAPS show that this translates well to other score functions as well. Nonetheless, the improvement we get via RAPS still does not offset the gains from conformal training. In most cases, RAPS applied to a model trained via CE produces less efficient prediction sets than APS applied to a model trained to optimize one of our bounds.

Wealso study the impact of the two hyperparameter in RAPS, namely λ reg and k reg in the inefficiency of each of the conformal training methods. We concentrate our analysis on CIFAR100 and report results with varying k reg in Table 13 and varying λ reg in Table 14. Interestingly, RAPS was much more sensitive to variations in k reg than in λ reg. Still, we observe models trained via our model-based Fano and DPI bounds produce more efficient prediction sets than the baselines in all cases.

G.2 Federated Setting

In the federated setting, we run conformal training exactly in the same fashion, but including the additional Q Z | X term in (11) to get the proper distributed bound that can be optimized locally and independently in each device. We optimize the conformal training objective with SGD for one epoch in each device, and then communicate the resulting 'personalized' model to the server, which aggregates the model parameters of each device via federated averaging [42]. After aggregation, the

global model thus computed is communicated to the client and the process restarts. We do 5K such communication rounds for EMNIST, and 10K for CIFAR10 and CIFAR100. As described in the main text, we divide the data among 100, 500, and 1K clients for CIFAR10, CIFAR100 and EMNIST, respectively. We assign data points to devices imposing a distribution-based label imbalance [33], where we sample a marginal label distribution for each device from a Dirichlet distribution. We use Dir (1 . 0) for all experiments, but also study the effect of Dir (0 . 5) and Dir (0 . 1) on CIFAR10, as shown in Tables 15 and 16. The remainder of the experimental setup is similar to that used for the centralized setting, with the same architectures, data augmentation strategies, and optimal hyperparameters reported in Tables 7, 8, 9, 10. In Tables 15 and 16, as well as 17 and 18, we report inefficiency results for two different settings, global (GLO) and personalized (PER).

GLO We run SCP with the final global model assuming calibration and test data sets at the server, or equivalently that the clients share their scores with the server. This reflects the best inefficiency results we can hope for with the global model, as in practice we might need to resort to privacypreserving methods that are likely to hurt performance. Notably, we considered the quantile of quantiles approach of Humbert et al. [24] as well as the simpler alternative proposed in [37], but in

both cases we got varying degrees of coverage due to the data heterogeneity among devices introduced in the distribution-based label imbalance setup. Addressing these shortcomings is a promising avenue for future research for conformal prediction in the federated setting.

PER After learning the global model, we fine-tune it on the local training data of each device to obtain a personalized model. We then run SCP individually for each device with local calibration and test data sets and report the average prediction sets across all clients. Importantly, since each client has access to only a small number of data points, we do not always achieve valid coverage in the personalized setting. More precisely, all personalized models on CIFAR10 and EMNIST achieved marginal coverage of around 97%, while for CIFAR100 that value dropped to 90%. Nonetheless, all methods get similar coverages so the results remain comparable. Interestingly, even with personalized models, which implicitly already estimate Q Y | X,Z , updating the personalized model as in (9) with the global head Q Z | X,Y , still results in non-negligible improvements in performance, as shown in Tables 15 and 16, as well as 17 and 18.

G.3 Evaluating the lower bounds on the set size

In this section, we evaluate our two lower bounds on the expected [log | C ( X ) | ] + . The first one can be obtained by rearranging the simple Fano bound (c.f. Proposition C.7) whereas the second one can be obtained from the model-based Fano entropy upper bounds (c.f. Theorem E.2). The main challenge in evaluating these bounds is in that we require the ground truth entropy H ( Y | X ) or H ( Y | ˆ Y ) which in general are not available. To proceed, we adopt the versions of the bounds that depend on H ( Y | ˆ Y ) . When ˆ Y is discrete, we can get a tractable lower bound to H ( Y | ˆ Y ) via a maximum likelihood estimate of the entropy [46]. More specifically, we have that

where | ˆ Y| is the cardinality of ˆ Y .

(a) CIFAR 10

(b) CIFAR 100

4.5

Fano lower bound

Fano lower bound

1.5

Model-based Fano lower bound

4.0

Model-based Fano lower bound

Empirical estimate

Empirical estimate

3.5

(+[（(x)0]

1.0

(+[(x)|60])

3.0

2.5

0.5

2.0

1.5

F0°0

1.0

0.5

0.5

0.000

0.0250.0500.0750.100

0.1250.150

0.175

0.200

0.000

0.0250.0500.0750.100

0.125 0.1500.175

0.200

α

α

Based on this, we evaluate our set size bounds on ResNet models trained with CE on CIFAR 10 and CIFAR 100. As the logits ˆ Y used for CP are not discrete, we perform K-means clustering on them and construct a vector quantized ˆ Y vq by assigning to each logit vector its closest cluster centroid. We then use ˆ Y vq to perform CP with thresholding and also use ˆ Y vq to obtain a lower bound on H ( Y | ˆ Y vq ) via a maximum likelihood estimate for H ( Y, ˆ Y vq ) . For this maximum likelihood estimate we also perform the Miller-Madow bias correction [46].

For CIFAR 10 we cluster the logits into 32 clusters whereas for CIFAR 100 we use 256 clusters. The centroids are learned on a calibration set of 5k logits. For the model-based Fano bound, as it needs to compute terms that depend on the model probabilities and on whether the label was correctly covered by CP or not, we further split the calibration set into two equal-sized chunks; the first is used to find the quantile for thresholding and the second one is used to evaluate the terms of the bound. The obtained lower bounds on [log | C ( X ) | ] + for various α 's can be seen at Figure 2. We also include an 'empirical estimate' which is obtained by computing the quantile with the quantized calibration logits and then measuring the average [log | C ( X ) | ] + on the test set via thresholding on the quantized test logits. We see that model-based Fano provides relatively tight estimates for small values of alpha.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper provides a series of upper bounds for the conditional entropy of the labels given the attributes. The right-hand side of the bounds depends on the coverage probability of the corresponding Conformal Prediction sets. The authors propose to use the bound as an alternative objective function for conformal training.\n\nSTRENGTHS:\n- The link between Conformal Prediction (CP) and List Decoding is inspiring, even if it is partly unclear how it is used in the bounds.  \n- Using CP distribution-free and finite sample properties to compute empirical bounds is interesting and may have an impact beyond the CP community.\n- I like the idea of using the bounds for training CP algorithms.\n\nWEAKNESSES:\n- The authors may explain more intuitively the role of the arbitrary distribution $Q$. Later in the paper $Q$ seems to be linked to the underlying point-predictor. An explicit construction (in the Introduction) would help motivate the proposed approach.\n- The practical consequences of the authors' findings could be clarified better.\n- A few practical examples would be helpful. For example, the authors may add a concrete way to define $Q$ at the end of Section 3.1.\n- The technical contribution seems a relatively straightforward consequence of the inequality $H(Y|X) < H(w_Y(Y, X)|w_X(Y, X))$.\n- The challenging part of optimizing a CP algorithm is smoothing the sorting operation. The proposed training method does not seem to solve or address that problem. The authors should justify better the use of conditional entropy instead of ${\\rm E}(|{\\cal C}|)$. In particular, how an upper bound could be more efficient than the original objective function?\n\nQUESTIONS:\n- Is the prediction set in Equation 3 calibrated using $P$?  \n- What do you mean by *to learn machine learning models that are more amenable to SCP*?\n- Where is the conditional entropy defined? Is the right-hand side of Equation 3 guaranteed to be positive? \n- Is $Q$ supposed to be an approximation of $P$ learned from data?\n- Is $W$ required to be a stochastic map? What happens if $W$ is deterministic?\n- Would it be possible to add a baseline trained in a standard way, i.e. without conformal training?\n\nLIMITATIONS:\nI appreciate the authors addressed their work's limitations in a dedicated section. Perhaps, the paper misses a theoretical and intuitive justification for using the bounds for conformal training.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper observes a connection between the efficiency of CP (tightness of its prediction sets) and information theory. It uses this to derive 3 bounds on a CP's efficiency depending on the entropy of the label given a data record H(Y|X). It then uses these bounds as the optimization objective for the underlying model (nonconformity scorer). The bounds are empirically shown to be valid (i.e., reasonable) training objectives.\n\nSTRENGTHS:\n- This paper is _truly_ well written: it explains extremely well prior literature as well as the concepts being presented. The appendix is also well-curated.\n- The idea behind this paper (to connect information theory and CP) is very interesting, and I'm sure it'll lead to more research.\n- The theoretical analysis is non-trivial and insightful.\n\nWEAKNESSES:\n- Bounds based on Fano's inequality may be loose.\n- Empirically, the method does not outperform SOTA; however, I do think there's more value in this work than just improving CP numerically.\n\nQUESTIONS:\nThank you for your submission!\n\nI'm afraid I could not understand how the connection to list decoding was used when deriving your results. I can see this is being discussed in the appendix, but perhaps it'd be best to mention in the main body which ideas are borrowed from that theory, and how they're being used here. (Can you please clarify, in your rebuttal?)\n\nI did not quite understand why federated learning was introduced in this paper; I didn't feel like it was necessary, and it almost feels like an afterthought. Perhaps it'd be best to motivate it better in the introduction.\n\n\"To the best of our knowledge, Bellotti [8] were the first to advance the idea of conformal training.\"\nNote that, concurrently to Bellotti, this idea was also discovered by Colombo and Vovk in \"Training conformal predictors\".\n\n\nWhile your work is the first one to deal with the interesting intersection between information theory and CP efficiency, please note that other works have looked at:\n1. connecting CP efficiency to the goodness of the underlying algorithm; e.g.:\n- (Dhillon et al.) On the Expected Size of Conformal Prediction Sets\n- (Cherubin) How do the performance of a Conformal Predictor and its underlying algorithm relate?\n\nand also\n\n2. how to best characterize the efficiency of a CP: (Vovk et al.) Criteria of efficiency for conformal prediction.\n\nLIMITATIONS:\nThese were appropriately described by the paper.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper aims to relate the uncertainty measured by conformal prediction by the uncertainty measured by the conditional entropy $H(Y|X)$. Towards this end, they make the following contributions:\n1. They observe that the miscoverage quantity from conformal prediction is the same as the decoding-error quantity from list decoding in information theory.\n2. They exploit this observation to upper bound the conditional entropy $H(Y|X)$ in terms of quantities from conformal prediction.\n3. They use these bounds as objectives for conformal training in order to maximize efficiency i.e keep the prediction sets as small as possible. \n4. They empirically verify that their modifications to conformal training using their upper bounds improve efficiency.\n\nSTRENGTHS:\n**Originality** The problem of relating the uncertainty measured in information theory to that from conformal prediction is very interesting avenue of exploration.\n\nWEAKNESSES:\n**Clarity and Quality** \n\nIn general, it is confusing to interpret what the results are really saying. In particular, I have the following confusion about the theoretical result that remains after reading the paper:\n1. $H(Y|X)$ only measures aleatoric uncertainty whereas the conformal interval captures both aleatoric and epistemic uncertainty. So then, it is not surprising that the conditional entropy would be upper bounded by something in terms of the conformal size $|C(X)|$. Is this the correct interpretation for the Propositions, or is there something else going on here?\n\nFor the empirical result:\n\n2. In Section 4, the authors state that minimizing the RHS of the Fano bound is equivalent to the ConfTr objective from [7]. However, in the tables, the rows corresponding to ConfTr and Fano admit different efficiency - what is the cause of this?\n\n3. What is the intuition for why minimizing the RHS of the bounds is better than minimizing $|C(X)|$ directly?\n\nQUESTIONS:\n1. Could you answer W1 above, and in general connect the theoretical results to epistemic/aleatoric uncertainty?\n\nI am happy to change my assessment/confidence if the authors are able to alleviate my concerns through discussion.\n\nLIMITATIONS:\nYes\n\n\n---------\n\nEDIT POST REBUTTAL:\n\nAfter discussing with the authors, I have increased my score to 6. The authors have done a great job in the discussion of contextualizing the results in terms of epistemic&aleatoric uncertainty. My hope is that the authors in the final version will:\n1. Modify the exposition of the paper to discuss the role of these categories of uncertainty.\n2. In the discussion section, be upfront about the looseness of the bounds when epistemic uncertainty is high.\n3. In the discussion, mention the challenges to quantifying the epistemic uncertainty theoretically.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-13"}

comments: SUMMARY:
This work develops the upper bound of conditional entropy H(Y|X) of data generation process by information theoretic inequality. New objective for conformal training and upper bound for the inefficiency (size of prediction set) for a trained model. Including side information to improve the efficiency of CP is also introduced by the proposed upper bound.

STRENGTHS:
(1) Detailed background information about conformal prediction helps readers build basic understanding easily.
(2) Extensive theoretical works are presented, demonstrating the soundness of proposed upper bound of the inefficiency of CP for a given model.
(3) Five datasets and three baselines are included in the experiments. Standard deviation of experimental results are presented as well.

WEAKNESSES:
(1) Lack of clear notation clarification. For example, the subscripts of Q in Eq. (5) is not explained, making it hard to understand. Also, even if line 136 explains the meaning of H(Y|X), I suggest to write it explicitly for more clearness.
(2) Lack of full algorithm representation. The work only states that including the upper bound as an additional loss term in line 186, but there is no enough explanation on how the proposed algorithm really works.
(3) Proof is not self-contained. For instance, the paper leads readers to Appendix D.1 for the proof of DPI bound. And in Appendix D.1, line 806 further lead readers to Theorem C.4, which is not proved in detail.

QUESTIONS:
The reason of limiting alpha in (0,0.5) is not explained in Proposition 3.1, as Theorem 2.1 works for any alpha belongs to (0,1).

Setting alpha=0.01, as stated in the caption of Table.1, is a very limited experiment setup.

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-17