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Conformal Risk Control
Anastasios N. Angelopoulos, Stephen Bates, Adam Fisch, Lihua Lei, Tal Schuster
TL;DR
The paper addresses how to provide performance guarantees for pretrained models when error is naturally measured by losses beyond miscoverage. It develops conformal risk control, which selects a conservative post-processing parameter to control expected monotone loss, and reports coverage and application examples. The method is tight up to an O(1/n) factor but requires monotonicity or near-monotonicity for effective control.
Problem
Pretrained models need statistical performance guarantees, including for prediction tasks where the relevant error is not simply miscoverage.
Method
Conformal risk control post-processes model outputs by selecting a conservativeness parameter for bounded losses that decrease as prediction sets grow.
Results
The method generalizes conformal prediction, achieves ≥90% average true-class inclusion when α = 0.1, and is tight up to O(1/n).
Takeaways & Limitations
The framework applies to false negative rate, graph distance, and token-level F1-score across computer vision and natural language processing examples.
Takeaways & Limitations
Risk control does not hold for non-monotone losses without further assumptions, while monotonization is powerful only for near-monotone losses.
Abstract
from arXiv · showhide
We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an $\mathcal{O}(1/n)$ factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.
1 Introduction
The paper extends conformal prediction from miscoverage to expected monotone losses by post-processing a model's outputs, while retaining distribution-free finite-sample risk guarantees. Its procedure applies to losses including false negative rate and F1-score, and is tight up to O(1/n).
- The procedure post-processes a pretrained model's outputs into quantities with statistical performance guarantees for safe deployment.
- Conformal risk control extends conformal prediction to prediction tasks whose natural error is not simply miscoverage.
- For any bounded loss that decreases as prediction sets grow, the method controls expected risk and recovers conformal miscoverage under the indicator loss.
- The framework supports losses such as false negative rate and F1-score, including multilabel prediction sets formed by thresholding classifier scores.
- α = 0.1 yields prediction sets containing ≥90% of true classes on average in the described multilabel setting.
- 1.1 Algorithm and preview of main results: The algorithm chooses a conservativeness parameter from the first n calibration functions and deploys it on the forthcoming test point, with a lower-bound guarantee within 2B/(n+1).
- Limitations: Risk control can fail for non-monotone losses, although monotonizing losses provides a guarantee when the loss is near-monotone.
- 1.2 Related work: Compared with concentration-based approaches, the method is described as simpler, practical to implement, distribution-free, finite-sample, and tight up to O(1/n).
2 Theory
The theory establishes risk control for bounded monotone losses and shows that the procedure is nearly tight. It recovers conformal prediction for miscoverage while exposing limitations for non-monotone losses.
- Risk control: The proposed algorithm controls risk when each loss function is non-increasing, right-continuous, and bounded.This is the core condition behind the conformal risk control guarantee.
- Risk lower bound: The procedure is tight up to an additive factor 2B/(n+1), under i.i.d. losses and an almost-sure absence of discontinuities at any fixed parameter.The lower bound requires nonnegative losses and P(J(Li, λ) > 0) = 0 for any λ.
- Conformal prediction reduction: Conformal prediction is recovered when the loss is the indicator of a miscoverage event, with the conformal threshold selected as a calibration-data quantile.The general risk-control procedure and conformal prediction are algorithmically equivalent under this loss.
- Conformal prediction reduction: For continuous conformal scores, the general lower bound gives miscoverage at least α − 2/(n+1), versus α − 1/(n+1) for the conformal-specific result.The paper notes that the factor-of-two difference is small for moderate n but does not establish whether it can be improved.
- Non-monotone losses: Without monotonicity, conformal risk control can fail arbitrarily badly, although monotonizing the losses restores the guarantee when the risk is near-monotone.The paper also gives an asymptotic procedure for non-monotone losses, but says it is powerful only in the near-monotone case.
3 Examples
The procedure controls diverse monotone risks across computer vision and natural language processing, including false negative rate, hierarchical graph distance, and F1-score. Four experiments report risk near the target level across repeated random data splits.
- Overview: Four tasks use non-binary, monotone losses bounded by 1, demonstrating the procedure across computer vision and natural language processing.The examples target false negative rate, graph distance, and token-level F1-score.
- FNR control in tumor segmentation: 0.0987 mean FNR was obtained for tumor segmentation at α = 0.1 over 1000 trials.The reported standard deviation was 0.0114, using n = 1000 calibration points.
- FNR control in multilabel classification: 0.0996 mean FNR was obtained on MS COCO at α = 0.1 over 1000 trials.The reported standard deviation was 0.0052, using n = 4000 calibration points.
- Control of graph distance in hierarchical image classification: Hierarchical ImageNet controls minimum hierarchical distance loss at α = 0.05 over 1000 random data splits.The experiment uses WordNet with maximum depth D = 14, n = 30000, and 20000 evaluation points.
- F1-score control in open-domain question answering: Natural Questions evaluates F1 risk and average prediction-set size over 1000 independent random data splits.Answers are treated as token bags, with precision and recall combined through their geometric average while ignoring punctuation and articles.
4 Extensions
The paper extends conformal risk control to distribution shifts, including weighted control under train-test changes and covariate-shift settings. These extensions preserve risk control under stated continuity, boundedness, and shift assumptions.
- Risk control under distributional shift: Under distribution shift, the target risk is defined under a test distribution different from the training distribution.The training pairs are sampled from P_train, while P_test is assumed absolutely continuous with respect to P_train for weighted reformulation.
- Risk control under distributional shift: When the shift weight w is known and bounded, weighting the loss preserves the properties needed for conformal risk control.The weighted loss is non-decreasing, bounded, and right-continuous whenever the original loss is.
- Risk control under distributional shift: Under covariate shift with unchanged Y given X, transductive learning can achieve risk control even when w is unbounded.The procedure defines a covariate-specific threshold using the available test covariate.
- Risk control under distributional shift: Proposition 2 establishes the stated distribution-shift risk-control result and generalizes weighted conformal prediction to any monotone risk.The weighted conformal procedure is recovered when the loss is an indicator of prediction-set miscoverage.
Total variation bound
The extensions control risk beyond a single expected loss: arbitrary distribution shifts receive a total-variation degradation bound, while the framework also covers quantile, multiple, adversarial, and U-statistic risks.
- Total variation bound: Arbitrary distribution shifts are handled by a total variation bound describing how standard unweighted conformal risk control degrades.Proposition 3 formalizes this degradation under the conditions of Theorem 1.
- Quantile risk control: Quantile risk control is recast as expected risk control using the indicator loss 1 {L_i(λ) > α}.Proposition 4 states that the resulting quantile guarantee is achieved under Theorem 1.
- Multiple risk control: Multiple monotone risks can be controlled by defining a worst-case risk across a family indexed by γ ∈ Γ.Proposition 5 states that the resulting worst-case risk is controlled.
- Adversarial risk control: Adversarial risks are reduced to conformal risk control by taking the supremum of the perturbed losses over γ ∈ Γ.Proposition 6 states that the resulting choice of λ achieves risk control.
- U-statistic risks: U-risk extends the framework to losses depending on k test points, where the fixed-threshold risk is the expectation of an order-k U-statistic.Under the stated monotonicity and continuity conditions, Proposition 7 establishes the target result.
5 Conclusion
The paper broadens conformal prediction to new applications by providing a more general language for studying validity and by developing mathematical tools for conformal risk control.
- 5 Conclusion: The framework broadens conformal prediction to new applications and supplies technical tools for studying its validity.The paper states that these tools may also have independent technical interest.
A Monotonizing non-monotone risks
The paper extends conformal risk control to non-monotone risks by applying the procedure to a monotonized empirical risk. This yields asymptotic risk control, while finite-sample control remains unresolved for the resulting procedure.
- The proposed algorithm controls non-monotone risks asymptotically by operating on a monotonized version of the empirical risk.The monotonized empirical risk is introduced specifically to enable this extension.
- Under Theorem A.1’s conditions, the analogous procedure achieves asymptotic risk control with an upper bound of α − B/(n + 1).
- For monotone losses, the monotonized threshold ˜λ↑ equals the original threshold ˆλ.
- Finite-sample risk control for ˜λ↑ is currently unknown, although the counterexample in Proposition 1 does not apply to this procedure.
B Proofs
The proofs establish the paper’s risk-control guarantees through bounded jumps, exchangeability, empirical-risk identities, and uniform convergence. They also extend the argument to transformed and worst-case risks.
- Proofs: The Jump Lemma bounds discontinuities in the empirical risk, using bounded loss contributions to control each jump.
- Proofs: The main proof chains empirical-risk inequalities to obtain a lower bound of α − 2B/(n + 1).
- Proofs: Exchangeability and symmetry allow the proofs to relate sample-based thresholds and risks to the held-out loss.
- Proofs: Uniform convergence of the monotonized population risk supports the asymptotic guarantee for the monotonized procedure.
- Proofs: The resulting expected held-out loss satisfies α − 2B/(n + 1) ≤ E[g(Z′)] ≤ α.
- Proofs: Taking pointwise suprema preserves boundedness, monotonicity, and right-continuity, enabling worst-case risk control.