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Bias-Reduced Uncertainty Estimation for Deep Neural Classifiers
Yonatan Geifman, Guy Uziel, Ran El-Yaniv
TL;DR
The paper addresses biased uncertainty estimates for highly confident predictions in non-Bayesian deep classifiers. It improves confidence scores by selectively using earlier training snapshots, and extensive experiments show consistent gains over existing methods across evaluated datasets.
Problem
Existing confidence methods can become biased for highly confident instances because later SGD training impairs their confidence estimates.
Method
The paper proposes PES, which selects an appropriate early-stopped snapshot per instance, and AES, which approximates PES by averaging confidence values from evenly spaced snapshots without additional examples.
Results
AES improved baseline methods in 39 of 42 experiments and, with k = 30, reduced baseline E-AURC in every experiment.
Takeaways & Limitations
The resulting confidence scores systematically improve existing estimation techniques across all evaluated datasets, including calibrated uncertainty estimates.
Takeaways & Limitations
PES requires an additional labeled set and expensive training, while current instance-level uncertainty estimation requires several forward passes at inference.
Abstract
from arXiv · showhide
We consider the problem of uncertainty estimation in the context of (non-Bayesian) deep neural classification. In this context, all known methods are based on extracting uncertainty signals from a trained network optimized to solve the classification problem at hand. We demonstrate that such techniques tend to introduce biased estimates for instances whose predictions are supposed to be highly confident. We argue that this deficiency is an artifact of the dynamics of training with SGD-like optimizers, and it has some properties similar to overfitting. Based on this observation, we develop an uncertainty estimation algorithm that selectively estimates the uncertainty of highly confident points, using earlier snapshots of the trained model, before their estimates are jittered (and way before they are ready for actual classification). We present extensive experiments indicating that the proposed algorithm provides uncertainty estimates that are consistently better than all known methods.
1 INTRODUCTION
Non-Bayesian uncertainty methods for deep classifiers are practical but can produce biased confidence estimates, especially for highly confident instances. The paper attributes this problem to SGD training dynamics and proposes using earlier model snapshots to improve confidence estimation.
- Practical uncertainty estimation for deep classifiers commonly extracts signals from standard non-Bayesian networks, while Bayesian approaches face computational hurdles.
- Existing confidence techniques can wrongly estimate uncertainty for highly confident instances because training later focuses on harder points.
- Confidence estimates for easy instances are assessed reliably early in SGD training but become impaired later as optimization concentrates on hard instances.
- The paper focuses on ordinal uncertainty ranking rather than probability calibration, treating these as separate tasks in non-Bayesian estimation.
- The proposed approach uses earlier training snapshots to improve confidence scores, and experiments report consistent gains over existing methods across datasets.
2 PROBLEM SETTING
The paper formalizes uncertainty estimation for supervised multi-class classification using confidence scores derived from a deep network’s softmax outputs. An optimal score should rank instances according to their likelihood of misclassification.
- The setting is supervised multi-class classification over an unknown distribution P(X, Y), with classifier risk defined through an expected loss.
- Deep classifiers use a softmax output vector, and the predicted class is the index of its largest activation.
- Training snapshots f^[i] are recorded after each epoch of mini-batch SGD, providing partially trained models for confidence analysis.
- A confidence function κ(x, i|f) assigns confidence for predicting class i from signals extracted by classifier f and need only induce a partial ordering.
- An optimal confidence score should preserve true loss monotonicity: lower confidence corresponds to a higher probability of misclassification.
3 PERFORMANCE EVALUATION OF CONFIDENCE SCORES BY SELECTIVE CLASSIFICATION
The paper evaluates confidence scores through selective classification, where confidence determines which predictions are retained or rejected. It introduces normalized Excess-AURC as a unitless measure based on the risk-coverage curve.
- Selective classification: Selective classification turns a confidence score into a reject-based classifier whose performance can be assessed through coverage and selective risk.
- Excess-AURC: The proposed normalized metric subtracts the hindsight-optimal AURC from a score’s AURC, enabling meaningful comparisons across problems.
- Risk-coverage evaluation: The risk-coverage curve plots selective risk against coverage, and better confidence scores reject misclassified points earlier, producing lower AURC.
- CIFAR-100 example: On CIFAR-100, softmax confidence produced approximately 0.29 risk at full coverage and approximately 0.06 risk at 0.5 coverage after rejecting the least confident half.
- Normalization: The hindsight-optimal curve’s AURC was 0.04802, approximated as 0.04800 by the integral calculation.
- Excess-AURC: E-AURC is unitless, lies in [0, 1], and equals 0 for an optimal confidence score.
4 RELATED WORK
Related work spans Bayesian-inspired dropout, margin-based confidence, embedding-space KNN, ensembles, and training-snapshot methods. These approaches differ in the signals they use and their computational requirements.
- MC-dropout estimates uncertainty from variance across several dropout-enabled forward passes at test time.
- Softmax values are commonly used as classification-margin signals, with larger values generally associated with higher confidence.
- Embedding-space KNN uses distances as proxies for class-conditional probabilities and extracts uncertainty from non-final network layers.
- Deep ensembles average softmax responses from several trained networks but require substantially large computing resources.
- Prior snapshot ensembles average fully converged models produced under cyclic learning-rate schedules, differing conceptually and quantitatively from this paper’s use of training snapshots.
5 MOTIVATION
Training reveals a confidence-dependent timing effect: highly confident points are learned early, but their confidence estimates later degrade, while low-confidence points continue improving through the final model.
- An independent validation set tracks softmax confidence across intermediate classifiers f[i] during training, enabling evaluation on unseen points.
- The highest-confidence points stabilize near maximal softmax values around epoch 80 and are correctly classified as early as about epoch 25.
- For highly confident points, E-AURC first improves and then degrades, making an intermediate model such as f[130] better than the final model.
- For low-confidence points, confidence estimates improve monotonically during training, with the final model f[T] performing best.
- These dynamics motivate pointwise early stopping, selecting an intermediate snapshot separately for each sample or sample group.
6 SUPERIOR CONFIDENCE SCORE BY EARLY STOPPING
The paper introduces early-stopping methods that improve confidence scores by selecting or averaging intermediate network snapshots, trading additional computation or labeled data against simpler approximation.
- Approach: The supervised method learns an improved confidence score for a trained classifier and scoring function using labeled data, while an approximate method avoids additional training examples.
- Pointwise Early Stopping: PES uses saved training snapshots, a layer-size hyperparameter q, and E-AURC evaluated on each layer to choose models and thresholds.
- Pointwise Early Stopping: PES partitions inputs into confidence layers and selects the best intermediate model and confidence function for each layer.
- Pointwise Early Stopping: At inference, the algorithm searches for the first layer whose confidence score meets its threshold, then adds the layer index to preserve ordering.
- Averaged Early Stopping: AES averages confidence scores from k evenly spaced intermediate models between epochs t and T, using the early learning of easy points to reduce reliance on the final model.
- Averaged Early Stopping: AES is used in most experiments because PES has substantial computational cost, and the approximation works surprisingly well.
7 EXPERIMENTAL RESULTS
Across four image datasets and multiple confidence methods, AES and PES improved uncertainty estimation, with AES outperforming baselines in most experiments and PES reducing softmax E-AURC on every evaluated dataset. Calibration results tracked these improvements, while PES remained computationally expensive.
- AES evaluation: AES was evaluated with four confidence scores across CIFAR-10, CIFAR-100, SVHN, and ImageNet using k values of 10, 30, and 50.The evaluated scores were softmax response, NN-distance, MC-dropout, and Ensemble.
- Metric interpretation: E-AURC values differed across datasets in ways consistent with their reported difficulty levels, supporting its use for comparing learning problems.CIFAR-10 and SVHN had smaller values, while CIFAR-100 and ImageNet had larger values.
- AES results: AES improved the baseline in 39 of 42 experiments, and k = 30 reduced baseline E-AURC in every experiment.AES also improved the ensemble state of the art on all evaluated datasets.
- Calibration: Platt-scaled AES consistently improved NLL and Brier scores alongside raw uncertainty estimates across datasets and underlying uncertainty methods.The authors conclude that E-AURC serves as a reliable proxy for calibrated probabilities.
- Computational cost: PES can require 175,000,000 forward passes for the stated CIFAR-100 MC-dropout setting, limiting its practicality for computationally demanding confidence methods.The cited setting uses n = 7000, T = 250, and d = 100.
- PES results: PES reduced softmax E-AURC on every evaluated dataset, with its best improvement reaching 18% on CIFAR-100.PES was implemented only with softmax response because applying it to other methods required excessive computation.
8 CONCLUDING REMARKS
The paper concludes that SGD training can deform early reliable uncertainty estimates, motivating algorithms that use training snapshots to improve confidence scores. AES is scalable, whereas PES requires extra labeled data and expensive training, and inference still requires several forward passes.
- Motivation: The proposed algorithms are motivated by the observation that reliable early-epoch estimates are later deformed during SGD training.The paper describes this behavior as resembling overfitting in deep neural networks.
- Practical trade-offs: PES requires an additional labeled set and expensive training, while AES is described as simple and scalable.The two methods use snapshot models generated during training.
- Conclusion: The resulting confidence scores systematically improved existing estimation techniques on all evaluated datasets.This is the paper’s broad empirical conclusion.
- Open limitations: Both methods currently require several forward passes through the network for each instance during uncertainty estimation.The authors suggest distillation or a single-model early-stopping approximation as possible ways to reduce inference time.
A.2 ARCHITECTURES AND HYPER PARAMETERS
The experiments use distinct architectures and training configurations for small image datasets and ImageNet, alongside specified implementations for baseline confidence and calibration methods.
- Architectures and training: CIFAR-10, CIFAR-100, and SVHN use a VGG-16-inspired architecture trained for 250 epochs with SGD, momentum 0.9, and batch size 128.The initial learning rate is 0.1, halved every 20 epochs.
- Architectures and training: ImageNet uses ResNet-18 trained for 90 epochs with SGD, momentum 0.9, and batch size 256.The learning rate starts at 0.1 and decays by 0.1 every 30 epochs.
- Baseline implementations: The softmax response baseline uses the relevant class probability as its confidence score.It is defined as κ(x, i|f) = f(x)i.
- Baseline implementations: NN-distance uses k = 500, while its embedding-regularization and adversarial-training extensions are omitted.The omissions preserve the classifier’s performance and avoid adding the extensions to every comparison method.
- Baseline implementations: MC-dropout uses dropout rate p = 0.5 and 100 feed-forward iterations per sample, while the ensemble averages five DNN softmax outputs.These settings define the compared baseline implementations.
- Probability calibration: Platt scaling fits logistic regression from a confidence measure to binary correctness targets using a validation set.The original test set is randomly split 50-50 for calibration training and testing.
B DETAILED RESULTS
Table 4 reports AES E-AURC and percentage improvement across four datasets, baseline methods, and various k values. Standard errors are provided for AES results involving softmax response and NN-distance, but not for other methods due to computational complexity.
- Standard errors are reported for AES with softmax response and NN-distance, while other methods lack standard errors because of computational complexity.
- Table 4 covers CIFAR-10, CIFAR-100, SVHN, and ImageNET, comparing AES across various k values against baseline methods.
- E-AURC values are multiplied by 10^3 for clarity, alongside percentage improvement over each baseline.
C MOTIVATION - EXTENDED EXPERIMENTS
Extended experiments show that training-time overfitting affects uncertainty estimates across MC-dropout and NN-distance, though its severity differs by confidence group and method. These findings support using information from earlier training stages to correct estimates, particularly for easy points.
- Extended experiments show E-AURC degradation for easy points after a stage of training, resembling overfitting.
- MC-dropout exhibits overfitting across training, but the effect is much smaller than for the other evaluated methods.
- NN-distance shows slight overfitting for easy points, while hard instances are affected much more severely.
- The results indicate that the proposed correction strategy is potentially useful for MC-dropout, NN-distance, and the other evaluated method.