Density-Softmax: Scalable and Calibrated Uncertainty Estimation under Distribution Shifts

Prevalent deterministic deep-learning models suffer from significant over-confidence under distribution shifts. Probabilistic approaches can reduce this problem but struggle with computational efficiency. In this paper, we propose Density-Softmax, a fast and lightweight deterministic method to improve calibrated uncertainty estimation via a combination of density function with the softmax layer. By using the latent representation's likelihood value, our approach produces more uncertain predictions when test samples are distant from the training samples. Theoretically, we show that Density-Softmax can produce high-quality uncertainty estimation with neural networks, as it is the solution of minimax uncertainty risk and is distance-aware, thus reducing the over-confidence of the standard softmax. Empirically, our method enjoys similar computational efficiency as a single forward pass deterministic with standard softmax on the shifted toy, vision, and language datasets across modern deep-learning architectures. Notably, Density-Softmax uses 4 times fewer parameters than Deep Ensembles and 6 times lower latency than Rank-1 Bayesian Neural Network, while obtaining competitive predictive performance and lower calibration errors under distribution shifts.