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Conformal prediction beyond exchangeability

Rina Foygel Barber, Emmanuel J. Candes, Aaditya Ramdas, Ryan J. Tibshirani

arXiv:2202.13415v5stat.ME

TL;DR

Conformal prediction depends on exchangeability and symmetric model-fitting algorithms, assumptions that can fail when data drift or recent observations matter more. The paper adds weighted quantiles and randomization to address these departures, reporting reduced coverage loss under nonexchangeability while retaining existing guarantees under exchangeability.

  • Problem

    Conformal prediction relies on exchangeability and symmetric fitting algorithms, while deployed data may drift and motivate treating recent observations as more relevant.

    Source: §1 · p. 2

  • Method

    The paper combines weighted residual quantiles for nonexchangeable data with randomization that permits nonsymmetric model-fitting algorithms.

    Source: §1.2 · p. 5, §3.1 · p. 10, §7 · p. 32

  • Results

    The methods reduce coverage loss under nonexchangeability while retaining existing guarantees under exchangeability; on electricity data, NexCP+LS and NexCP+WLS achieve nearly the desired 90% coverage while CP+LS undercovers.

    Source: §5.2 · p. 25, §7 · p. 32

  • Takeaways & Limitations

    The authors identify application-specific efficiency and robustness as objectives for nonexchangeable conformal prediction.

    Source: §7 · p. 32

    A coverage bound may be uninformative when large, so choosing weights that make the coverage gap likely small requires prior knowledge about the data distribution.

    Source: §4.3 · p. 18

Abstract

from arXiv · show

Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. However, exchangeability is often violated when predictive models are deployed in practice. For example, if the data distribution drifts over time, then the data points are no longer exchangeable; moreover, in such settings, we might want to use a nonsymmetric algorithm that treats recent observations as more relevant. This paper generalizes conformal prediction to deal with both aspects: we employ weighted quantiles to introduce robustness against distribution drift, and design a new randomization technique to allow for algorithms that do not treat data points symmetrically. Our new methods are provably robust, with substantially less loss of coverage when exchangeability is violated due to distribution drift or other challenging features of real data, while also achieving the same coverage guarantees as existing conformal prediction methods if the data points are in fact exchangeable. We demonstrate the practical utility of these new tools with simulations and real-data experiments on electricity and election forecasting.

1 Introduction

The paper extends conformal prediction beyond exchangeable data and symmetric fitting algorithms, using weighted procedures to address distribution drift and other violations while preserving existing guarantees under exchangeability.

1 Introduction

  • Split conformal calibrates prediction intervals using holdout errors and guarantees predictive coverage when observations are exchangeable.The introduction describes this as formalizing the holdout-quantile approach.Source: §1 · p. 2
  • On Australian electricity data, standard conformal coverage fell well below the 90% target over a stretch, while nonexchangeable conformal maintained approximately the target level.The example motivates recovering predictive coverage under distribution drift or other exchangeability violations.Source: §1 · p. 2, Figure 1 · p. 2

1.2 Our contributions

  • The paper generalizes split conformal, full conformal, and jackknife+ to handle nonexchangeable data and nonsymmetric algorithms.With a symmetric algorithm, the procedures recover the original variants.Source: §1.2 · p. 3
  • When data are exchangeable, the methods retain existing coverage guarantees; if exchangeability is mildly violated, coverage guarantees are only slightly lower.With unit weights and a symmetric algorithm, the weighted methods reduce to the usual conformal or jackknife+ methods; weighting does not hurt coverage under exchangeability.Source: §1.2 · pp. 3–4
  • Coverage-gap control requires fixed, non-data-dependent weights that make a weighted sum of total-variation distances small, without assumptions on the joint data distribution.The bound quantifies coverage degradation relative to exchangeability.Source: §1.2 · p. 4
  • The coverage-gap bound is stronger than simply asking whether data are nearly exchangeable: a time series may be close under individual swaps yet far from exchangeable under most permutations.The passage attributes this distinction to possible nonstationarity or dependence over time.Source: §1.2 · p. 4
  • For independent data under drift, assigning smaller weights to points more distributionally distant from the test point can reduce the coverage gap.If dTV(Z_i, Z_{n+1}) decreases with i, the paper gives weights w_i = ρ^(n+1−i), for ρ < 1, as an example.Source: §1.2 · pp. 4–5
  • A stronger bound based on residual distributions can be substantially tighter than one based on full data, particularly when drift affects prediction-irrelevant features.For split conformal, residuals are computed from a pre-fitted model; for full conformal, they use a model fitted to the entire sequence.Source: §1.2 · p. 5

2 Background and related work

This section reviews conformal prediction’s exchangeability and algorithm-symmetry assumptions, its coverage guarantees and efficiency tradeoffs, and related approaches to distribution shift. It distinguishes this paper’s fixed-weight approach from prior methods that require stronger assumptions.

Background and related work

  • Under exchangeable data, full conformal guarantees distribution-free coverage at 1−α when its algorithm treats points symmetrically, and split conformal has the same guarantee.Split conformal uses a holdout set; full conformal includes a hypothesized test point when fitting the model.Source: §2 · pp. 5–7
  • Full conformal can be more precise than split conformal, but requires refitting for each candidate outcome; split conformal loses accuracy because it reserves data for fitting.The precision advantage is not universal, and specialized techniques can reduce full conformal’s computational cost in some cases.Source: §2 · pp. 6–7
  • Unlike Tibshirani et al. [2019]’s covariate-shift method requiring likelihood-ratio weights, this work uses fixed weights for small unknown violations and accommodates nonsymmetric, order-dependent algorithms.It also retains exact coverage under exchangeability, whereas their weighted method does not.Source: §2 · p. 8
  • Prior conformal time-series methods retain exchangeability or other distributional assumptions, while sequential tests detect nonexchangeability rather than guarantee predictive inference on such data [Chernozhukov et al., 2018, Xu and Xie, 2021, Stankeviciute et al., 2021; Vovk, 2021].Source: §2 · p. 9

3 Nonexchangeable conformal prediction

The section extends conformal prediction beyond exchangeable data by using weighted quantiles, then adapts the methods to allow nonsymmetric model-fitting algorithms. For nonsymmetric full conformal, a random tag swap enables the method to account for data order or other tags while preserving predictive coverage.

3.1 Robust inference through weighted quantiles

  • Weighted quantiles let split and full conformal methods emphasize observations believed to come from a distribution closer to the test point.The weights are fixed; for temporal drift, recent observations can receive more weight, while spatial weights can depend on distance to the test location.Source: §3.1 · p. 10

3.2 Enhanced predictions with nonsymmetric algorithms

  • The method allows arbitrary data-dependent fitting algorithms by representing observations as tagged data points, with tags encoding weights, time or location, or observation order.The algorithm can use those tags when fitting, so exchanging two observations while keeping their tags does not have to leave the fitted model unchanged.Source: §3.2 · pp. 11–13
  • For full conformal with a nonsymmetric algorithm, the method draws K independently with probability proportional to each normalized weight and swaps the tags of observation K and the test point before fitting.The random swap is the modification used to maintain predictive coverage; it is applied to the hypothesized test point in full conformal.Source: §3.2 · p. 13
  • Split conformal needs no such modification for a nonsymmetric algorithm because its model is prefitted on separate data and is symmetric in the holdout observations.Its prediction interval remains unchanged.Source: §3.2 · p. 13
  • The tag swap can alter the fitted model and reduce precision, especially in autoregressive settings where the test-point tag has a disproportionate role.The theoretical results do not require stability assumptions, but choosing algorithms that are not too sensitive to the swap is left for future work.Source: §3.2 · pp. 14–15

4 Theory

The theory bounds coverage loss under nonexchangeability using weighted differences between swapped residual-vector distributions, and shows that mild deviations can also limit overcoverage. It further identifies when these guarantees are useful and how they extend beyond data that are nearly exchangeable.

4.1 Lower bounds on coverage

  • Coverage loss for nonexchangeable full conformal is bounded using weighted total-variation differences between swapped residual vectors; the same bound applies to split conformal.The bound depends on residual-vector differences, not the total-variation distance between the swapped raw data vectors.Source: §4.1 · pp. 15–16

4.2 Upper bounds on coverage

  • With distinct candidate test residuals, the method also limits overcoverage; when the test-point weight is small, mild exchangeability violations can cause only mild overcoverage or undercoverage.The direction of the coverage deviation cannot be determined without further assumptions; for example, increasing conditional variance over time can cause undercoverage, while decreasing variance can cause overcoverage.Source: §4.2 · p. 16

4.3 Remarks on the theorems

  • Under exchangeability, fixed weights preserve coverage even with nonsymmetric algorithms, while the bounds also establish robustness of original symmetric conformal methods under mild violations.Weights that are too low reduce effective sample size and can widen intervals; choosing them optimally is left open, and useful guarantees require weights informed by likely distribution differences.Source: §4.3 · pp. 17–18

4.4 Examples

  • Geometrically decaying weights yield small coverage gaps under slight bounded drift and after a changepoint when enough observations follow it.In covariate time series, swapped residual distributions can be close even when swapped raw-data distributions are far apart; for bounded covariance condition number, the residual distance is small when n ≫ p^2.Source: §4.4 · pp. 18–20

4.5 Extensions and explorations

  • The theory allows a multiplicative coverage bound under Huber contamination, but only for symmetric algorithms; the paper otherwise derives additive bounds.Random or data-dependent weights and tags can be accommodated by replacing total-variation terms with suitable conditional versions, while a fuller investigation of data-dependent weights is left for future work.Source: §4.5 · pp. 20–21
  • Coverage-gap bounds can be small even when the data are far from every exchangeable distribution, because they measure swap-specific departures rather than distance to an exchangeable law.An alternating Bernoulli example has a coverage-gap bound of at most 4ϵ without weights, although its total-variation distance to any exchangeable distribution is approximately 1 for large n.Source: §4.5 · p. 21

5 Experiments

Experiments compare original full conformal with weighted nonexchangeable methods, finding better coverage under drift and shorter intervals when using a nonsymmetric algorithm.

5.1 Simulations

  • Across changepoint and drift simulations, NexCP+LS and NexCP+WLS maintained approximately 90% coverage while CP+LS undercovered; NexCP+WLS also produced narrower intervals than CP+LS.In the i.i.d. setting, all three methods had coverage near 90% and similar mean interval widths.Source: Figure 2 · p. 23, §5.1 · pp. 23–24, Table 1 · p. 24

5.2 Electricity data set

  • On original electricity data, CP+LS undercovered while NexCP+LS and NexCP+WLS achieved nearly 90% coverage; on permuted data, all three methods stayed close to 90%.On the original data, NexCP+LS intervals were generally wider than NexCP+WLS intervals; on permuted data, their widths were similar, with higher variability for both weighted methods.Source: Figure 3 · p. 25, §5.2 · pp. 25–26, Table 2 · p. 26

5.3 Election data set

  • The election experiment uses reported Eastern time zone counties for training and other counties for testing, and weights observations using 2016 demographic and socioeconomic information.The weighting approach implicitly assumes that counties with similar 2016 demographics will generate approximately exchangeable 2020 data.Source: §5.3 · pp. 27–28
  • For election data at target coverage 1 − α = 0.9, CP undercovered substantially, particularly with least squares, whereas NexCP, with or without the nonsymmetric algorithm, achieved coverage much closer to 90%.Source: Table 3 · p. 28, §5.3 · p. 28

6 Proofs

The section proves the paper’s theorems by contrasting the exchangeability argument behind classical conformal coverage with proof techniques for weighted quantiles and nonsymmetric algorithms.

6.1 Background: proof of Theorem 1

  • Under exchangeable residuals, coverage fails only if the test point is among the largest residuals, an event bounded in probability by α.The proof uses symmetry to make each residual equally likely to be among the “strange” points, of which there are at most α(n + 1).Source: §6.1 · p. 29
  • The classical argument does not directly extend: unequal weights break quantile symmetry, nonsymmetric fitting can break residual exchangeability, and distribution drift adds further coverage-bound challenges.These obstacles arise even when the data themselves are exchangeable for the first two cases.Source: §6.1 · p. 30

6.2 Proof of Theorem 2

  • Theorem 2’s proof bounds the weighted fraction of “strange” points by α, then links noncoverage to a randomly indexed strange point using the index’s independence from the data.The deterministic weighted-fraction inequality is attributed to [Harrison, 2012, Lemma A.1].Source: §6.2 · pp. 31–32

7 Discussion … B.2 Methods

The paper extends conformal prediction beyond exchangeability with weighted residuals and randomization for nonsymmetric algorithms, while bounding coverage loss without assumptions on the data distribution. The merged sections also extend these methods to general nonconformity scores and develop a nonexchangeable jackknife+ procedure.

7 Discussion

  • Without assumptions on the joint data distribution, existing and proposed conformal methods have coverage guarantees, with coverage gaps bounded by weighted total-variation distances between residual vectors after swaps.The bound uses residual vectors formed by swapping point i with point n + 1.Source: §7 · p. 32
  • The methods aim to recover reasonable coverage when data are globally far from exchangeable but locally deviate mildly, by downweighting distant points and encouraging application-specific weighting and algorithm design.The authors hope to improve efficiency and robustness for specific applications.Source: §7 · p. 32

A Extension to general nonconformity scores

  • The methods extend to non-real-valued responses and general nonconformity scores, and the same theoretical result as Theorem 2 continues to hold.The proof does not fundamentally rely on residual scores.Source: A Extension to general nonconformity… · p. 37

B.1 Background

  • Jackknife+ balances the computational and statistical costs of split and full conformal using leave-one-out models and residuals; CV+ extends the approach to K-fold cross-validation.Leaving each point out of model fitting avoids overfitting that point.Source: §B.1 · p. 38
  • In practice jackknife+ generally achieves coverage near 1 − α, but its worst-case theoretical guarantee is only 1 − 2α.The guarantee improves under a stability assumption on the algorithm; the usual theorem assumes exchangeable data and a symmetric algorithm.Source: §B.1 · p. 38

B Nonexchangeable jackknife+

  • For a symmetric algorithm, nonexchangeable jackknife+ uses prespecified weights, and setting every weight to 1 recovers ordinary jackknife+.Source: §B.2 · p. 39
  • For a nonsymmetric algorithm, the method draws a random index K and fits leave-one-out models that assign point k the test point’s tag.The fitted models do not require knowledge of the test point.Source: §B.2 · p. 39

B.3 Theory

The theory extends jackknife+ to nonexchangeable settings by defining leave-two-out residuals and bounding coverage error through total-variation distances.

B.3 Theory

  • The nonexchangeable jackknife+ construction maps each data sequence to a residual matrix built from leave-two-out models, with zeros on the diagonal.The residual matrix is defined for pairs of distinct indices.Source: §B.3 · p. 40
  • Theorem 5 bounds nonexchangeable jackknife+ coverage error using weighted total-variation distances between residual distributions.The paper also gives a bound using distances between the data distributions, but notes that this more interpretable bound can be substantially looser.Source: §B.3 · p. 40

C Huber-robustness of conformal prediction · D Nonexchangeable cross-conformal · E Additional proofs and calculations

Sections C–D extend nonexchangeable conformal prediction with multiplicative coverage bounds under Huber contamination and an n-fold cross-conformal method that supports arbitrary nonconformity scores.

C Huber-robustness of conformal prediction

  • Section C considers training data with adversarially corrupted points while requiring coverage for a test point drawn from the target distribution.The proposed mixture distance measures Huber-style contamination and, under the stated mixture model, bounds the distance between a training point and the target test point by ϵ.Source: C Huber-robustness of conformal… · p. 41
  • Under Huber contamination, split or full conformal prediction has noncoverage at most α/(1−ϵ), a factor of at most 1/(1−ϵ) above nominal α.The multiplicative coverage gap is proportional to α, which can yield tighter error control at small target error levels when the distribution distances are small.Source: C Huber-robustness of conformal… · p. 42
  • The multiplicative bounds require independent data and a symmetric algorithm, and may be looser than additive bounds when the mixture distance is large.The mixture distance can exceed total variation distance and can equal 1 even when total variation is arbitrarily small.Source: C Huber-robustness of conformal… · p. 42

D Nonexchangeable cross-conformal

  • The method draws a random index and uses leave-one-out scoring functions with the corresponding point removed and tags swapped, allowing arbitrary nonconformity scores.For standard regression scores, its prediction set is contained in the jackknife+ interval.Source: D Nonexchangeable cross-conformal · pp. 42–43
  • For the standard regression score, the n-fold cross-conformal prediction set is contained inside the nonexchangeable jackknife+ prediction interval.The section also notes this containment in the exchangeable setting.Source: D Nonexchangeable cross-conformal · pp. 42–43
  • The n-fold nonexchangeable cross-conformal method with an arbitrary nonconformity score retains the coverage guarantee established for jackknife+ in Theorem 5.The proof is essentially the same as for jackknife+ and is omitted.Source: D Nonexchangeable cross-conformal · p. 43

E.1 Proof of Lemma 1

The proof uses maximal coupling and distributional identities to relate the total variation distance between the original and swapped-index variables to that between coupled variables.

E.1 Proof of Lemma 1

  • Maximal coupling [Ross and Pek¨oz, 2007, Proposition 2.7] provides a joint distribution for the paired variables, and the proof applies the theorem again.The first coupling is introduced as a distribution on a pair of random variables.Source: §E.1 · pp. 43–44
  • Drawing each Z_j independently and then drawing the coupled variables yields Z′ and Z′′ with the same distribution as Z.The resulting identity also gives (Z′′)_i d= Z_i under the swapped-indices notation.Source: §E.1 · pp. 43–44
  • The construction gives dTV(Z, Z_i) = dTV(Z′, (Z′′)_i).This follows from the stated distributional identity for (Z′′)_i and the coupled variables.Source: §E.1 · p. 44

E.2 Proof of Theorem 3 … E.4.1 Background: proof of Theorem 4

These proof sections establish nonexchangeable conformal results through a rank-based coverage argument and a total-variation calculation, then recall the exchangeable jackknife+ proof based on permutation symmetry and strange points.

E.2 Proof of Theorem 3

  • Theorem 3’s proof reduces nonexchangeable split conformal to full conformal and uses a coverage-event characterization to establish the result.The proof then applies an inequality to distinct residual values and bounds terms using an index attaining the maximum.Source: §E.2 · pp. 44–46
  • Theorem 3’s proof verifies its key inequality by selecting a maximizing residual index and showing that only the test-point term can contribute a positive difference.The remaining terms vanish or are bounded by the difference between the test-point weight and the maximizing index’s weight.Source: §E.2 · pp. 45–46

E.3 Calculation for (19)

  • The calculation for (19) reduces the total-variation comparison between residual vectors to one between their normalized directions.It uses the independence of direction and magnitude, then bounds the directional discrepancy through density and matrix calculations.Source: §E.3 · pp. 46–47
  • The calculation for (19) obtains the marginal density of the normalized direction by integrating over matrices orthogonal to that direction.It compares the direction’s density with that of the direction after swapping an entry, using their common normalizing constant and determinant calculations.Source: §E.3 · p. 47

E.4.1 Background: proof of Theorem 4

  • The exchangeable background proof starts from a symmetric fitting algorithm and derives a permutation property for the jackknife+ residual matrix.It relates its entries to the usual leave-one-out model before applying the strange-point argument.Source: §E.4.1 · pp. 49–50
  • The exchangeable jackknife+ proof uses permutation symmetry of its residual matrix to connect coverage failure with the size of the strange-point set.The recalled argument invokes Landau’s tournament theorem for a deterministic bound on that set.Source: §E.4.1 · p. 50

E.4.2 Proof of Theorem 5 · E.4.3 Proof of Lemma 2 · E.5 Proof of Theorem 6

These appendix proofs establish the weighted strange-point bound used in Theorem 5 and prove Theorem 6’s bounds for nonexchangeable conformal prediction and jackknife+ through a contamination-mixture argument.

E.4.2 Proof of Theorem 5

  • For Theorem 5, noncoverage of the prediction at index n + 1 implies that the selected point K is strange, and the weighted fraction of strange points is at most 2α.The proof combines this implication with the weighted strange-point lemma.Source: §E.4.2 · pp. 51–52

E.5 Proof of Theorem 6

  • Lemma 3 bounds an expectation involving a weighted sum of independent Bernoulli variables; its lower bound follows from Jensen’s inequality, while the upper bound is harder.The passage notes special cases used in multiple-testing results, including Storey et al. [2004, Theorem 3], Storey [2002], and Ramdas et al. [2019, Lemma 3].Source: §E.5 · p. 53
  • Theorem 6’s proof represents the data through independent contamination indicators and uses this mixture construction to derive bounds for nonexchangeable conformal prediction and jackknife+.The proof then applies a weighted-Bernoulli lemma to the two methods’ bounds.Source: §E.5 · pp. 53–54
  • The proof verifies the conformal and jackknife+ bounds by controlling total variation distances, with zero distance for observations whose contamination indicators are zero.The jackknife+ verification is described as nearly identical to the conformal calculation.Source: §E.5 · pp. 55–56

E.5.1 Proof of Lemma 3

The proof establishes the lower bound by Jensen’s inequality and proves the upper bound by reducing it to three cases, ending with induction for arbitrary weights and probabilities.

E.5.1 Proof of Lemma 3

  • The upper-bound proof handles arbitrary c ≥ 0 by showing it is equivalent to the lemma’s c = 0 case, augmenting the variables with Bernoulli terms of probability 1 and weight c/m.It chooses an integer m ≥ c and assigns each added term weight c/m.Source: §E.5.1 · pp. 56–57
  • When all weights are 1 and probabilities are equal, the bound reduces to a binomial expectation, with the c > 0 case handled using Jensen’s inequality.For c = 0, the proof invokes the known bound [Storey et al., 2004, Theorem 3].Source: §E.5.1 · p. 57
  • For arbitrary weights with equal probabilities p1=...=pn, conditioning on S=A⊤1 gives A⊤B a Binomial(S,p1) distribution; concavity supports the expectation bound.Source: §E.5.1 · pp. 57–58
  • For arbitrary weights and probabilities, induction on n represents the Bernoulli terms using shared Bernoulli variables, then applies the equal-probability case and Jensen’s inequality.The argument assumes ordered probabilities, treats the zero-maximum case as trivial, and concludes the induction step.Source: §E.5.1 · pp. 58–59

F Simulations for split conformal and jackknife+

The section repeats the full conformal simulations with split conformal prediction and jackknife+. Across these experiments, nonexchangeable methods better maintain coverage under changepoints and distribution drift, while weighted least squares yields narrower intervals.

F Simulations for split conformal and jackknife+

  • The experiments reuse the Section 5.1 simulated data and weight/tag definitions, evaluate each time point after a 100-point burn-in, and average results over 200 independent replications.At each n = 100, . . . , N −1, inference uses observations 1 through n to predict point n + 1.Source: F Simulations for split conformal and… · p. 60
  • The split conformal comparisons are SplitCP+LS, NexSplitCP+LS, and NexSplitCP+WLS; jackknife+ compares Jack+LS, NexJack+LS, and NexJack+WLS.Each method is defined analogously to its full conformal counterpart, replacing full conformal with split conformal or jackknife+.Source: F Simulations for split conformal and… · p. 60
  • For split conformal, odd training-data indices go to the training set and even indices to the holdout set.Source: F Simulations for split conformal and… · p. 60
  • In Settings 2 and 3, the original methods lose coverage substantially, while nonexchangeable split conformal and jackknife+ stay closer to target coverage; weighted least squares narrows intervals.In i.i.d. Setting 1, the three versions of each method have comparable performance.Source: F Simulations for split conformal and… · pp. 60–61, Table 4 · p. 60, Table 5 · p. 60

G Election data set description

The election data set combines county-level demographic and socioeconomic covariates with 2016 and 2020 election vote counts. After matching covariates and vote counts, the merged data covered 3,076 counties split into training and test sets.

G Election data set description

  • Predictors included county-level race, sex, Hispanic-origin, and age proportions, plus median household income and bachelor’s-degree attainment.Demographic and income features used 2019 Census estimates [United States Census Bureau, 2019a] [United States Census Bureau, 2019c]; education used 2015–2019 data [United States Census Bureau, 2019b]. 2019 estimates were used because they were the most recent available.Source: G Election data set description · p. 61
  • The 2016 covariates used the same sources, almost exclusively using figures published by 2016, while election votes came from MIT Election Data and Science Lab [2018] and Leip [2020].The authors note that a few 2016 entries may have been changed in later Census updates.Source: G Election data set description · p. 61
  • Matching covariates and election vote counts yielded 3,111 counties; merging 2016 and 2020 data left 3,076 counties, with 1,119 for training and 1,957 for testing.Source: G Election data set description · p. 61

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