Cross-conformal e-prediction

01/16/2020 ∙ by Vladimir Vovk, et al. ∙ 0

This note discusses a simple modification of cross-conformal prediction inspired by recent work on e-values. The precursor of conformal prediction developed in the 1990s by Gammerman, Vapnik, and Vovk was also based on e-values and is called conformal e-prediction in this note. Replacing e-values by p-values led to conformal prediction, which has important advantages over conformal e-prediction without obvious disadvantages. The situation with cross-conformal prediction is, however, different: whereas for cross-conformal prediction validity is only an empirical fact (and can be broken with excessive randomization), this note draws the reader's attention to the obvious fact that cross-conformal e-prediction enjoys a guaranteed property of validity.

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1 Introduction

Conformal prediction is based on the notion of a p-value. At this time p-values are widely discussed (see, e.g., [1]

), and several alternatives to p-values have been proposed. Perhaps the most popular alternative is Bayes factors, which, when stripped of their Bayesian context, are referred to as e-values in

[10]. In fact, e-values were used (under the name of i-values) when discussing precursors of conformal prediction in the 1990s. One early description is [3]. In this note, it will be convenient to distinguish between conformal e-prediction (using e-values) and conformal p-prediction (standard conformal prediction using p-values).

Conformal e-prediction was superseded by conformal (p-)prediction mainly for two reasons:

  1. Conformal predictions can be packaged as prediction sets [9, Section 2.2]

    , in which case the property of validity of conformal predictors is very easy to state: we just say that the probability of error is at most

    for a pre-specified significance level [9, Proposition 2.3].

  2. In the on-line mode of prediction, smoothed conformal predictors make errors independently, and so small probabilities of errors manifest themselves, with high probability, as a low frequency of errors [9, Corollary 2.5].

These are important advantages of conformal p-prediction, which do not appear to be counterbalanced by any clear disadvantages.

Whereas the notion of a conformal e-predictor does not appear particularly useful, using e-values in place of p-values in cross-conformal prediction [8] has a clear advantage. Cross-conformal predictors are not provably valid [8, Appendix], and this sometimes even shows in experimental results [6]. The limits of violations of validity are given by Rüschendorf’s result (see, e.g., [11]): when merging p-values coming from different folds by taking arithmetic mean (this is essentially what cross-conformal predictors do), the resulting arithmetic mean has to be multiplied by 2 in order to guarantee validity. In the recent method of jackknife+, introduced in [2] and closely related to cross-conformal prediction, there is a similar factor of 2 [2, Theorem 1], which cannot be removed in general [2, Theorem 2].

The situation with e-values is different: the arithmetic mean of e-values is always an e-value. This is an obvious fact, but it is shown in [10] that arithmetic mean is the only useful merging rule. Therefore, the version of cross-conformal prediction based on e-values, which we call cross-conformal e-prediction in this note, is always valid.

2 Conformal e-predictors

Suppose we are given a training set consisting of labelled objects and our goal is to predict the label of a new object . In this note we consider predictors of the following type: for each potential label for we would like to have a number reflecting the plausibility of being the true label of . An example is conformal transducers [9, Section 2.5], which, in the terminology of this note, may be called conformal p-predictors. The output

of a conformal p-predictor is the full conformal prediction for the label of ; e.g., it determines the prediction set at each significance level. We will sometimes write , where , instead of .

We will use the notation for the object space and for the label space (both assumed non-empty). These are measurable spaces from which the objects and labels, respectively, are drawn. Full observations are drawn from . For any non-empty set , will be the set of all non-empty finite sequences of elements of .

A conformal e-predictor is a function that maps any finite sequence , , to a finite sequence of nonnegative numbers with average at most 1,

(1)

that satisfies the following property of equivariance: for any , any permutation of , any , and any ,

The conformal e-predictor proposed in [4] is

where SV

is the set of indices of support vectors:

if and only if is a support vector for the SVM constructed from as training set. When given a training set and a new object , this conformal e-predictor goes through all potential labels for and for each constructs an SVM and outputs . It makes it computationally inefficient.

The following obvious proposition asserts the validity of conformal e-predictors.

Proposition 1.

For any , if are IID,

3 Split conformal e-predictors

Let us fix a measurable space (a summary space). A -valued split conformity measure is a measurable function . Intuitively, encodes how well conforms to . A normalizing transformation is an equivariant measurable function that maps every non-empty finite sequence of elements of to a finite sequence of the same length of nonnegative numbers whose average is at most 1 (i.e., satisfying (1)).

To apply split conformal e-prediction to a training set , we split it into two parts, the training set proper and the calibration set . For a new object and a potential label for it, we set

(2)

where is defined using the following steps:

For many choices of and , the split conformal e-predictor (2) will be computationally efficient; this is the case when:

  1. Processing the training set proper only once, we can find an easily computable rule transforming into .

  2. The normalizing transformation is easily computable.

An example of an easily computable normalizing transformation is

where the summary space is supposed to be .

Proposition 1, our statement of validity, continues to hold for split conformal e-predictors.

4 Cross-conformal e-predictors

A -valued split conformity measure is a -valued cross-conformity measure if does not depend on the order of its first arguments. Given such an and a normalizing transformation , the corresponding cross-conformal e-predictor (CCEP) is defined as follows. The training sequence is randomly split into non-empty multisets (folds) , , of equal (or as equal as possible) sizes , where is a parameter of the algorithm, is a partition of the index set , and consists of all , . For each and each potential label of the new object , find the output of the split conformal e-predictor on the new object and its postulated label with as training set proper and as calibration set, where is the complement to the fold . The corresponding CCEP is defined by

(A slight modification, still provably valid, of this definition is where the arithmetic mean is replaced by the weighted mean with the weights proportional to the sizes of the folds.)

Proposition 1 still holds for cross-conformal e-predictors; this trivially follows from the arithmetic mean of e-values being an e-value.

Remark 2.

To compare the outputs of cross-conformal p-predictors (CCPP) and CCEP, we can use the rough transformation discussed in [10]: a p-value of roughly corresponds to an e-value of . Under this transformation, the arithmetic average of e-values corresponds to the harmonic average of p-values, and the harmonic average is always less than or equal to the arithmetic average [5, Theorem 16]. This suggests that CCEP produce better results than CCPP do. In the opposite direction, the arithmetic average of p-values corresponds to the harmonic average of e-values, which again suggests that CCEP produce a better result than CCPP do.

5 Validity in the time domain

Advantage 1 of conformal p-prediction given on p. 1 is that, when discussing validity, we can talk about probabilities instead of p-values. It mostly disappears when we move to conformal e-prediction: validity has to be defined in terms of e-values (however, it has been argued [7] that e-values are more intuitive than p-values). In this section we discuss advantage 2, which requires the online mode of prediction. We will see that it still holds, albeit in a weakened form.

The notion of validity asserted in Proposition 1 (applied to CCEP) is stated in the “space domain”: the output of the CCEP on the true label is an e-value, i.e., its mean value at a fixed time over the probability space does not exceed 1. Now we will complement the validity in the space domain by the validity in the time domain assuming that the CCEP is bounded.

In the online prediction protocol, we observe an object , apply the CCEP to compute the values for all possible labels , observe the true label , observe another object , apply the CCEP to compute the values for all possible labels , observe the true label , etc.

Remark 3.

Intuitively, using CCEP in the online mode defeats the purpose of cross-conformal prediction: once we process , we would like to apply the rules that we have found (see item 1 on p. 1) to a large number of new objects rather than just one. But more realistic settings are also more complicated.

Let the values for the true labels be . It follows from, e.g., [9, Lemma 3.15] that, if the observations are IID,

We can see that the long-term time average of the values for the true labels is bounded above by 1. In this sense they are time-wise e-values.

Acknowledgments

This research was partially supported by Astra Zeneca and Stena Line.

References

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