Risk Control — Getting Started

Terminology

In theoretical parts of the documentation:

• alpha is equivalent to 1 - confidence_level — it can be seen as a risk level.
• calibrate and calibration are equivalent to conformalize and conformalization.

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Overview

Three methods of risk control have been implemented in MAPIE: RCPS (Risk-Controlling Prediction Sets) ¹, CRC (Conformal Risk Control) ², and LTT (Learn Then Test) ³.

MAPIE supports risk control for binary classification and multi-label classification (including image segmentation).

Risk Control Method	Type of Control	Assumption	Non-monotonic Risks	Binary Classification	Multi-label Classification
RCPS	Probability	i.i.d.
CRC	Expectation	Exchangeable
LTT	Probability	i.i.d.

For multi-label classification: CRC and RCPS are used for recall control, while LTT is used for precision control.

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1. What is Risk Control?

Consider a binary classification model that separates data into two classes using a threshold on predicted probabilities. Suppose we want to find a threshold that guarantees a certain precision level.

A naive approach: evaluate how precision varies with different thresholds on a validation dataset.

Naive approach: no guarantees on unseen data.

The Problem

While the chosen threshold works on validation data, it offers no guarantee on new, unseen data.

Risk control adjusts a model parameter \(\lambda\) so that a given risk stays below a desired level with high probability on unseen data.

Risk control: statistically guaranteed thresholds.

Mathematical Formulation

• \(\alpha\): target level below which we want the risk to remain
• \(\delta\): confidence level associated with the risk control

The three methods provide different guarantees:

• CRC: Requires exchangeable data → \(\mathbb{E}(R) \leq \alpha\)
• RCPS and LTT: Require i.i.d. data → \(\mathbb{P}(R \leq \alpha) \geq 1 - \delta\)

Comparison of expectation vs. probability guarantees.

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2. Theoretical Description

2.1 Risk-Controlling Prediction Sets (RCPS)

General Settings

• \(\mathcal{T}_{\hat{\lambda}}: X \to Y'\) — a set-valued function indexed by \(\lambda\) with nesting:

\[ \lambda_1 < \lambda_2 \Rightarrow \mathcal{T}_{\lambda_1}(x) \subset \mathcal{T}_{\lambda_2}(x) \]

• \(L: Y \times Y' \to \mathbb{R}^+\) — a loss function with:

\[ S_1 \subset S_2 \Rightarrow L(y, S_1) \geq L(y, S_2) \]

The goal is to compute an Upper Confidence Bound \(\hat{R}^+(\lambda)\) and find:

\[ \hat{\lambda} = \inf\{\lambda \in \Lambda: \hat{R}^+(\lambda') < \alpha, \;\forall \lambda' \geq \lambda\} \]

Guarantee

\(\mathbb{P}(R(\mathcal{T}_{\hat{\lambda}}) \leq \alpha) \geq 1 - \delta\)

Bounds

The empirical risk: \(\hat{R}(\lambda) = \frac{1}{n}\sum_{i=1}^n L(Y_i, T_{\lambda}(X_i))\)

Hoeffding Bound:

\[ \hat{R}_{\text{Hoeffding}}^+(\lambda) = \hat{R}(\lambda) + \sqrt{\frac{1}{2n}\log\frac{1}{\delta}} \]

Bernstein Bound:

\[ \hat{R}_{\text{Bernstein}}^+(\lambda) = \hat{R}(\lambda) + \hat{\sigma}(\lambda)\sqrt{\frac{2\log(2/\delta)}{n}} + \frac{7\log(2/\delta)}{3(n-1)} \]

Waudby-Smith–Ramdas (recommended for bounded losses):

\[ \hat{R}_{\text{WSR}}^+(\lambda) = \inf \left\{ R \geq 0 : \max_{i=1,\ldots,n} K_i(R, \lambda) > \frac{1}{\delta}\right\} \]

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2.2 Conformal Risk Control (CRC)

Controls any monotone and bounded loss:

\[ \mathbb{E}\left[L_{n+1}(\hat{\lambda})\right] \leq \alpha \]

To find \(\hat{\lambda}\):

\[ \hat{\lambda} = \inf \left\{ \lambda: \frac{n}{n+1}\hat{R}_n(\lambda) + \frac{B}{n+1} \leq \alpha \right\} \]

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2.3 Learn Then Test (LTT)

Controls any loss (including non-monotonic) through multiple hypothesis testing:

For each \(\lambda_j\) in a discrete set \(\Lambda = \{\lambda_1, \ldots, \lambda_n\}\):

1. Estimate the risk on calibration data.
2. Associate hypothesis \(\mathcal{H}_j: R(\lambda_j) > \alpha\).
3. Compute p-value using Hoeffding-Bentkus.
4. Apply FWER control (e.g., Bonferroni correction).

Return \(\hat{\Lambda} = \mathcal{A}(\{p_j\})\) — the set of \(\lambda\) values that control the risk.

Guarantee

\(\mathbb{P}(R(\mathcal{T}_{\lambda}) \leq \alpha) \geq 1 - \delta\) for all \(\lambda \in \hat{\Lambda}\).

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References

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1. Bates, S., Angelopoulos, A., Lei, L., Malik, J., & Jordan, M. "Distribution-free, risk-controlling prediction sets." CoRR, 2021. ↩

2. Angelopoulos, A. N., Bates, S., Fisch, A., Lei, L., & Schuster, T. "Conformal Risk Control." 2022. ↩

3. Angelopoulos, A. N., Bates, S., Candès, E. J., Jordan, M. I., & Lei, L. "Learn then test: Calibrating predictive algorithms to achieve risk control." 2021. ↩