Source-linked AI summary
Uncertainty Quantification and Deep Ensembles
Rahul Rahaman, Alexandre H. Thiery
TL;DR
The paper addresses calibration of over-parameterized neural networks in the low-data regime, where standard models tend to be over-confident. It studies the interaction of ensembling, mixup augmentation, and temperature scaling, finding that calibration depends on their combination and ordering. The authors advocate pooling predictions before temperature scaling for deep ensembles.
Problem
High-capacity neural networks often produce over-confident probabilistic forecasts, and these calibration problems are amplified when training data are scarce.
Method
The paper empirically studies ensembling, temperature scaling, and mixup augmentation, including pool-then-calibrate procedures for deep ensembles.
Results
Standard ensembling does not generally improve calibration, mixup networks are typically under-confident, and calibration depends on applying temperature scaling after pooling.
Takeaways & Limitations
Reliable uncertainty quantification with deep ensembles requires accounting for interactions among augmentation, prediction averaging, and post-processing calibration.
Takeaways & Limitations
The study is constrained by practical low-data validation settings, where fitting temperature without a validation set does not improve calibration and flexible scaling can overfit.
Abstract
from arXiv · showhide
Deep Learning methods are known to suffer from calibration issues: they typically produce over-confident estimates. These problems are exacerbated in the low data regime. Although the calibration of probabilistic models is well studied, calibrating extremely over-parametrized models in the low-data regime presents unique challenges. We show that deep-ensembles do not necessarily lead to improved calibration properties. In fact, we show that standard ensembling methods, when used in conjunction with modern techniques such as mixup regularization, can lead to less calibrated models. This text examines the interplay between three of the most simple and commonly used approaches to leverage deep learning when data is scarce: data-augmentation, ensembling, and post-processing calibration methods. Although standard ensembling techniques certainly help boost accuracy, we demonstrate that the calibration of deep ensembles relies on subtle trade-offs. We also find that calibration methods such as temperature scaling need to be slightly tweaked when used with deep-ensembles and, crucially, need to be executed after the averaging process. Our simulations indicate that this simple strategy can halve the Expected Calibration Error (ECE) on a range of benchmark classification problems compared to standard deep-ensembles in the low data regime.
1 Introduction
The paper studies uncertainty calibration for highly flexible neural networks in the low-data regime, focusing on how ensembling, temperature scaling, and mixup interact. It challenges the assumption that standard ensembling reliably improves calibration.
- Motivation: Small training sets amplify the over-confidence problems common in high-capacity neural networks.Such calibration errors are problematic when deployed systems must quantify uncertainty.
- Scope: The paper investigates ensembling, temperature scaling, and mixup data augmentation as practical low-data methods.These approaches are examined jointly rather than in isolation.
- Implication: The paper argues that calibration procedures must be adapted to the prediction-averaging process in deep ensembles.Its contributions center on the interaction among augmentation, ensembling, and post-processing calibration.
- Findings: Standard ensembling does not generally produce better-calibrated models despite its widespread use.The paper specifically tests the belief that averaging predictions improves calibration.
- Findings: Mixup-trained networks are typically under-confident, creating subtle interactions between augmentation and ensemble calibration.The induced distributional shift can make standard ensembling worsen calibration issues.
2 Background
The background defines probabilistic predictions, augmentation, model averaging, temperature scaling, and calibration metrics used to study deep ensembles. It also describes practical and methodological constraints in the low-data setting.
- Probabilistic predictions: A neural network produces class probabilities through a softmax mapping, with predicted class and confidence given by the largest probability.The probabilities lie in the class simplex, and confidence is the maximum class probability.
- Data augmentation: Augmentation maps an input-label pair to an altered pair, including standard transformations and mixup.Mixup forms convex combinations of inputs and one-hot labels using a random coefficient and index.
- Model averaging: Model averaging combines predictions from multiple models, while deep ensembles provide a practical approximation to Bayesian averaging.Bayesian averaging weights predictions according to a posterior over neural-network weights.
- Practical constraints: Deep ensembles avoid some intractable Bayesian computations, but sensible priors and augmentation remain difficult to incorporate into the Bayesian framework.The Bayesian posterior integral is described as extremely intractable because of its high-dimensional, multimodal structure.
- Calibration: Temperature scaling transforms probabilistic outputs using a positive temperature and fits that temperature on validation data while keeping model parameters fixed.The paper reports that more flexible vector and matrix scaling can overfit extremely small validation sets in this regime.
- Calibration metrics: ECE measures the discrepancy between confidence and empirical accuracy, while reliability curves distinguish calibrated, under-confident, and over-confident predictions.A calibrated model has accuracy approximately equal to confidence within each confidence bin.
3 Empirical Observations
Experiments show that averaging neural-network predictions systematically reduces confidence, which improves calibration only when individual models are over-confident. Mixup reduces over-confidence and improves accuracy, while temperature scaling before averaging largely removes mixup-related differences.
- Averaging predictions generally produces less confident estimates, benefiting calibration mainly when individual networks are over-confident.When individual models are calibrated or under-confident, averaging is typically detrimental; this pattern also appears in SWAG and MC-Dropout ensembles.
- Calibration mechanism: Entropy is a natural confidence surrogate because it increases under tempering and is concave under prediction averaging.The tractable DC score complements entropy by measuring confidence and vanishing for calibrated classification rules.
- Averaging can reduce calibration even when every individual model is calibrated, especially when their predictions differ.The decrease in confidence is larger for more dissimilar models.
- Distance to the training set: Over-confidence increases with distance from the training set, especially at lower mixup α, and persists regardless of mixup amount.Reliability is evaluated alongside accuracy, NLL, and entropy across distance percentiles.
- Effect of mixup-augmentation: Increasing mixup generally increases entropy, decreases over-confidence, and improves prediction quality through lower NLL and higher accuracy.These effects are reported for deep ensembles trained on small CIFAR10 subsets and across four datasets.
- Temperature Scaling: Individual temperature scaling before averaging almost removes differences caused by mixup augmentation.The experiment uses K = 30 networks and fits each temperature on a validation set of Nvalid = 50.
4 Calibrating Deep Ensembles
The paper compares alternative ways to calibrate deep ensembles, focusing on the ordering of pooling and temperature scaling. Across low-data experiments, pooling first and calibrating the aggregated prediction produces the strongest calibration outcomes, including when mixup is used.
- Calibration methodologies: The study evaluates four ensemble-calibration strategies: no calibration, calibrating each model before aggregation, joint aggregation and calibration, and pooling before calibration.These strategies are combined with average, median, and trimmed pooling rules.
- Calibration methodologies: Pool-then-calibrate fits a temperature after aggregating model predictions by minimizing a proper scoring rule on a validation set.The aggregation and temperature-fitting procedures can be carried out independently.
- Pooling and calibration order: Individually calibrated models can yield an under-confident pooled ensemble, whereas a common temperature can make individual models slightly over-confident and the pooled model nearly calibrated.This demonstrates why calibration order matters for ensemble predictions.
- Experimental comparison: Experiments use small training and validation sets across CIFAR10, Imagenette, Imagewoof, CIFAR100, and Diabetic Retinopathy, with repeated validation-set sampling.Figure 5 uses K = 30 mixup-trained models and repeats experiments 50 times on different validation sets.
- Empirical findings: Mixup combined with pool-then-calibrate provides a clear benefit relative to pool-then-calibrate without mixup, while vanilla ensembling can have worse calibration than single models.The reported pattern appears in low-data experiments and extends to the full-data CIFAR100 setting.
- Empirical findings: The optimal temperature was consistently below one, which the paper associates with under-confident predictions produced by averaging.A temperature below one sharpens the pooled probabilistic forecasts.
5 Discussion
The paper finds that ensembling, mixup, and temperature scaling interact through non-trivial trade-offs, and that calibration depends on applying pooling and temperature scaling in the correct order.
- Standard ensembling alone does not generally produce better-calibrated predictions.Averaging typically reduces confidence, which helps mainly when individual networks are overconfident.
- Mixup augmentation is practically important but involves non-trivial calibration and performance trade-offs.
- Pool-Then-Calibrate first pools individual neural-network predictions and then applies temperature scaling to the pooled result.The approach uses a simple and robust post-processing step after aggregation.
- The order of pooling and temperature scaling is important for obtaining calibrated deep ensembles.
6 Broader Impact
Well-calibrated probabilistic predictions matter because risk management and trusted decisions depend on model outputs. Calibration is also closely tied to public adoption of machine learning, especially in AI-driven medical diagnosis.
- Well-calibrated probabilistic predictions are crucial for risk management and trusted decisions based on probabilistic-model outputs.
- Calibration is important to public adoption of machine-learning methods, particularly in AI-driven medical diagnosis.The passage links calibration to trust in new technologies.
A Additional experiments
Additional experiments examine ensemble size, mixup strength, and numerical performance across low-data setups. They show that performance often saturates around 15 models, high mixup can improve low-data performance while increasing under-confidence, and Pool-Then-Calibrate performs best across reported metrics.
- Size of the ensembles: Around 15 models, performance under most metrics saturates on CIFAR10.The comparison covers groups B–D, while non-calibrated group A is omitted because of significantly worse performance.
- Size of the ensembles: Methods C and D perform similarly on CIFAR10 as ensemble size varies.
- Effect of mixup α: High mixup makes models under-confident with adequate data, while α near 1.0 significantly boosts performance in low-data settings.
- Ablation study: The ablation study uses CIFAR10 with 1000 training examples, 50 validation examples, and 100 different validation sets.
- Detailed numerical results: Pool-Then-Calibrate has the best performance across all reported metrics, while pooled predictions are consistently more under-confident than single models.