[Paper Review] Robustly representing uncertainty in deep neural networks through sampling

Patrick McClure, Nikolaus Kriegeskorte|arXiv (Cornell University)|Nov 5, 2016

Adversarial Robustness in Machine Learning14 references16 citations

TL;DR

This paper proposes a sampling-based method to robustly represent uncertainty in deep neural networks using variational inference with dropout and dropconnect. By combining Bernoulli dropout and Gaussian dropconnect, the approach approximates a spike-and-slab variational distribution without increasing parameters, achieving better test accuracy and more reliable uncertainty calibration than either method alone.

ABSTRACT

As deep neural networks (DNNs) are applied to increasingly challenging problems, they will need to be able to represent their own uncertainty. Modeling uncertainty is one of the key features of Bayesian methods. Using Bernoulli dropout with sampling at prediction time has recently been proposed as an efficient and well performing variational inference method for DNNs. However, sampling from other multiplicative noise based variational distributions has not been investigated in depth. We evaluated Bayesian DNNs trained with Bernoulli or Gaussian multiplicative masking of either the units (dropout) or the weights (dropconnect). We tested the calibration of the probabilistic predictions of Bayesian convolutional neural networks (CNNs) on MNIST and CIFAR-10. Sampling at prediction time increased the calibration of the DNNs' probabalistic predictions. Sampling weights, whether Gaussian or Bernoulli, led to more robust representation of uncertainty compared to sampling of units. However, using either Gaussian or Bernoulli dropout led to increased test set classification accuracy. Based on these findings we used both Bernoulli dropout and Gaussian dropconnect concurrently, which we show approximates the use of a spike-and-slab variational distribution without increasing the number of learned parameters. We found that spike-and-slab sampling had higher test set performance than Gaussian dropconnect and more robustly represented its uncertainty compared to Bernoulli dropout.

Motivation & Objective

• To improve uncertainty representation in deep neural networks by evaluating sampling-based variational inference methods.
• To compare the performance and robustness of different variational distributions—Bernoulli and Gaussian dropout, dropconnect, and spike-and-slab approximations—on image classification tasks.
• To develop a parameter-efficient approximation of spike-and-slab variational inference that combines Bernoulli dropout and Gaussian dropconnect.
• To assess how well these methods generalize under distribution shift using noisy test inputs.

Proposed method

• The authors use variational inference to approximate the posterior over network weights, minimizing the KL divergence between the variational distribution and the true posterior.
• They evaluate multiple sampling strategies: Bernoulli and Gaussian multiplicative noise applied to units (dropout) or weights (dropconnect).
• Monte Carlo sampling at prediction time is used to estimate predictive uncertainty and improve calibration.
• A novel method combines Bernoulli dropout and Gaussian dropconnect to approximate a spike-and-slab variational distribution with no increase in learned parameters.
• The approach is evaluated on MNIST and CIFAR-10 using test sets with added Gaussian noise to simulate distribution shift.
• Calibration is assessed via the mean squared error between predicted probabilities and actual label frequencies across noise levels.

Experimental results

Research questions

• RQ1 How does sampling at prediction time affect the calibration of deep neural network predictions under distribution shift?
• RQ2 How do different variational distributions—Bernoulli vs. Gaussian, dropout vs. dropconnect—compare in terms of uncertainty representation and test accuracy?
• RQ3 Can a combined use of Bernoulli dropout and Gaussian dropconnect effectively approximate a spike-and-slab variational distribution without increasing model parameters?
• RQ4 How does the choice of dropout rate and noise variance impact the robustness and performance of Bayesian DNNs under noisy inputs?

Key findings

• Sampling at prediction time consistently improved the calibration of probabilistic predictions across all models, especially under distribution shift.
• Gaussian dropconnect achieved the highest test set accuracy on CIFAR-10 but was less robust to input noise compared to Bernoulli dropout.
• Bernoulli dropout and spike-and-slab dropout showed greater robustness to input noise, maintaining better calibration under high-variance perturbations.
• The proposed combination of Bernoulli dropout and Gaussian dropconnect approximated a spike-and-slab distribution effectively, yielding higher test accuracy than Gaussian dropconnect and better uncertainty calibration than Bernoulli dropout.
• Sampling with higher variance dropout schedules led to greater performance gains, indicating that MC sampling reduces sensitivity to hyperparameter choice.
• On CIFAR-10, MC sampling improved performance more significantly under high-variance dropout, suggesting enhanced regularization and stability.