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Heteroscedastic Calibration of Uncertainty Estimators in Deep Learning
Bindya Venkatesh, Jayaraman J. Thiagarajan
TL;DR
Deep-learning uncertainty estimates are often poorly calibrated, limiting their usefulness for actionable interpretation. The paper uses the heteroscedastic regression objective to calibrate existing estimators during training, and reports improved calibration and accuracy in benchmark regression experiments.
Problem
Deep-learning uncertainty estimators can be poorly calibrated, while post-hoc recalibration can render even uninformative intervals meaningless for subsequent analysis.
Source: §1 · p. 1Method
Results
The proposed approach achieves significantly improved calibration with MC dropout and quantile-based estimators and produces highly accurate mean estimators.
Source: §1 · p. 1Takeaways & Limitations
The approach calibrates uncertainty estimators during training, dispensing with a separate recalibration step while retaining intervals associated with specific error sources.
Source: §3 · p. 3
Abstract
from arXiv · showhide
The role of uncertainty quantification (UQ) in deep learning has become crucial with growing use of predictive models in high-risk applications. Though a large class of methods exists for measuring deep uncertainties, in practice, the resulting estimates are found to be poorly calibrated, thus making it challenging to translate them into actionable insights. A common workaround is to utilize a separate recalibration step, which adjusts the estimates to compensate for the miscalibration. Instead, we propose to repurpose the heteroscedastic regression objective as a surrogate for calibration and enable any existing uncertainty estimator to be inherently calibrated. In addition to eliminating the need for recalibration, this also regularizes the training process. Using regression experiments, we demonstrate the effectiveness of the proposed heteroscedastic calibration with two popular uncertainty estimators.
1. INTRODUCTION
Deep-learning uncertainty estimates are often poorly calibrated, while post-hoc recalibration can make even uninformative intervals appear calibrated. The paper proposes using the heteroscedastic regression objective to calibrate existing estimators during training.
1. INTRODUCTION
- Uncertainty quantification is important in high-risk deep-learning applications, and calibration evaluates whether prediction intervals contain true targets at the expected rate.The introduction cites healthcare and autonomous driving as critical applications.Source: §1 · p. 1
- MC dropout and quantile-based estimators can be poorly calibrated, prompting separate recalibration steps that adjust their estimates.Random intervals can also be effectively recalibrated, rendering the estimates meaningless for subsequent analysis [16].Source: §1 · p. 1
- Classical heteroscedastic regression estimates variances using a Gaussian likelihood, but those variances are not straightforward to interpret as epistemic or aleatoric uncertainty.The variance estimates arise under a heteroscedastic prior assumption.Source: §1 · p. 1
- The heteroscedastic regression objective is repurposed as a surrogate for calibration, enabling existing uncertainty estimators to produce inherently calibrated intervals.The approach uses the estimator's uncertainty estimates in place of heteroscedastic variances in the Gaussian likelihood.Source: §1 · p. 1
- The proposed training objective is also reported to regularize training and produce highly accurate mean estimators.The paper reports improved calibration with both MC dropout and quantile-based uncertainty estimators.Source: §1 · p. 1
2. BACKGROUND
The background introduces interval calibration and uncertainty estimators, then describes heteroscedastic regression and the estimator types used in the paper. It contrasts interpretable uncertainty sources with direct interval-producing models whose error sources are difficult to interpret.
Heteroscedastic Regression
- Heteroscedastic regression models each sample's prediction as Gaussian, estimates its mean and variance, and optimizes parameters using a Gaussian likelihood objective.This is the objective later repurposed as a calibration surrogate.Source: §2 · p. 2
- A heteroscedastic neural network (HNN) directly produces prediction intervals, but its black-box output makes the measured error sources difficult to interpret.The paper distinguishes this from estimators designed to capture specific uncertainty sources.Source: §2 · p. 2
Calibration
- In regression, prediction intervals are well calibrated when the probability that the true target falls inside the interval matches the true empirical probability.Experiments used desired calibration levels [0.1, 0.3, 0.5, 0.7, 0.9, 0.99].Source: §2 · p. 2
Deep Uncertainties
- Approximate neural-network uncertainty methods include MC dropout, concrete dropout, deep ensembles, and Bayes-by-backprop.The background notes that probabilistic Bayesian models can be computationally expensive and often use simple modeling assumptions.Source: §2 · p. 2
- MC dropout estimates epistemic uncertainty from multiple dropout-enabled forward passes, while quantile regression estimates aleatoric uncertainty using upper and lower quantiles.The inter-quantile range between chosen quantiles is used as an aleatoric uncertainty estimate [7].Source: §2 · pp. 2–3
3. PROPOSED APPROACH
The approach incorporates uncertainty estimators into training and uses the heteroscedastic regression objective as a surrogate for calibration. It develops Dropout-HC and Quantile-HC to calibrate epistemic and aleatoric estimates, respectively.
3. PROPOSED APPROACH
- Heteroscedastic calibration uses an uncertainty estimator within the training objective, avoiding explicit post-hoc recalibration while retaining uncertainty-source attribution.The heteroscedastic regression objective serves as a surrogate for calibration.Source: §3 · p. 3
3.1. Dropout-HC
- Dropout-HC repeats dropout-enabled forward passes M times per sample, estimates the prediction mean and variance, then optimizes the model using those estimates.The resulting calibrated estimator continues to account for model uncertainty.Source: §3.1 · p. 3
3.2. Quantile-HC
- Quantile-HC estimates aleatoric uncertainty from conditional quantiles and jointly optimizes the mean and quantile outputs to produce calibrated intervals.The loss includes terms for estimating the conditional quantiles.Source: §3.2 · pp. 3–4
4. EMPIRICAL RESULTS
Across benchmark regression datasets, the proposed methods are reported to improve both model accuracy and interval calibration over existing baselines. Dropout-HC is also reported to be fairly robust to dropout-rate choice.
4. EMPIRICAL RESULTS
- The evaluation uses eight UCI regression datasets with 400–approximately 20,000 samples and 7–124 dimensions, random 80/20 splits, and RMSE and calibration error metrics.Fully connected networks with five ReLU hidden layers are compared with MC Dropout and HNN baselines.Source: §4 · p. 4
- Both proposed methods significantly outperform baselines in accuracy and calibration error across benchmark regression datasets.Expected-versus-achieved calibration plots show near-perfect calibration on the two datasets displayed.Source: §4 · p. 4
- Dropout-HC is fairly robust to dropout rate p, unlike MC Dropout's large calibration-error variability, and produces heavier-tailed σ distributions at matched p.The paper reports the variability for MC Dropout in Figures 1a and 1c.Source: §4 · p. 4
5. CONCLUSIONS
The paper presents heteroscedastic calibration for uncertainty estimators, including Dropout-HC and Quantile-HC, and reports gains in calibration and accuracy on benchmark regression tasks.
5. CONCLUSIONS
- The proposed strategy adapts the heteroscedastic regression objective for inherent calibration and reports gains in both calibration and accuracy on benchmark regression tasks.The paper develops Dropout-HC and Quantile-HC and uses a simple Gaussian prior on predictions.Source: §5 · p. 5