Abstract
Entropy-regularized optimization is a promising framework for improving robustness, stability and generalization of deep learning models in high-dimensional uncertainty settings. Standard optimization techniques for deep NNs are often based on empirical risk minimization, thereby resulting in overfitting, miscalibrated probabilities and sensitivity to adversarial perturbations. On the other hand, entropy-based regularization develops a principled way of controlling the distributional behaviour of model predictions by promoting smoothness in probability landscapes and discouraging overconfidence. The paper conducts an exploration of general entropy-regularized optimization frameworks, treating the entropy not simply as a measure of uncertainty, but rather an active controllable variable that determines the geometry of the loss surface and drives learning dynamics toward stable minima.This paper shines light on the different theoretical underpinning of various forms of entropy from information theory, statistical physics, and information geometry perspectives and connects them via a shared optimization perspective. We demonstrate that coupling entropy constraints to objective functions via Fullback–Lealer (KL) divergence, variation principles, and optimal transport formulations can lead models to learn more distributionally stable representations making them robust against distributional shifts and adversarial noise. In addition, we study entropy regularization with stochastic optimization methods, i.e. Langevin dynamics where implicit noise injection approximates the behavior of entropy maximization to encourage exploration when optimizing the model during training.We also explain the significance of entropy-based methods over recent deep learning architectures, which include convolutional neural networks and transformer-based models, in aiding predictive calibration and uncertainty quantification. Model trained with entropy-regularization are robust even in presence of adversarial noise or input noise, experimental insight show that these models are doing better than traditional regularisation. Although these benefits provide crucial turn-offs and are sensitive to hyper-parameter tuning. In synthesis, this work lays out a framework connecting the theoretical with the practical; it establishes entropy-regularized optimization as a unifying paradigm for making deep learning robust. It provides fresh directions for building practical learning approaches that work well on accuracy, reliability and robustness in the deployment phase.
Keywords
Entropy Regularization, Adversarial Robustness, Robust Optimization, Information Geometry, Distributional Stability, Deep Neural Networks, Optimal Transport, Uncertainty Calibration (UQ), Stochastic Optimization(Summary)
References
[1] García Trillos, N., Kaplan, Z., & Sanz-Alonso, D. (2019). Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning. Entropy, 21(5), 511.
[2] Cayci, S., He, N., & Srikant, R. (2024). Convergence of Entropy-Regularized Natural Policy Gradient with Linear Function Approximation. SIAM Journal on Optimization.
[3] Jagatap, G., & Joshi, A. (2022). Adversarially Robust Learning via Entropic Regularization. Frontiers in Artificial Intelligence.
[4] Neu, G., Jonsson, A., & Gómez, V. (2017). A Unified View of Entropy-Regularized Markov Decision Processes. NeurIPS Workshop.
[5] Huang, Y.-J., Wang, Z., & Zhou, Z. (2025). Convergence of Policy Iteration for Entropy-Regularized Stochastic Control Problems. SIAM Journal on Control and Optimization.
[6] Liu, D., Vu, M. T., Chatterjee, S., & Rasmussen, L. K. (2019). Entropy-Regularized Optimal Transport Generative Models. ICASSP.
[7] Ito, K., & Kashima, K. (2020). Maximum Entropy Optimal Density Control of Discrete-Time Systems.
[8] Cen, S., Cheng, C., Chen, Y., Wei, Y., & Chi, Y. (2020). Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization. arXiv.
[9] Arriojas, A., Adamczyk, J., Tiomkin, S., & Kulkarni, R. V. (2021). Entropy Regularized Reinforcement Learning Using Large Deviation Theory. arXiv.
[10] Della Vecchia, R., Shilova, A., Preux, P., & Akrour, R. (2022). Entropy Regularized Reinforcement Learning with Cascading Networks. arXiv.
[11] Grandvalet, Y., & Bengio, Y. (2005). Semi-supervised Learning by Entropy Minimization.
[12] Pereyra, G., Tucker, G., Chorowski, J., Kaiser, Ł., & Hinton, G. (2017). Regularizing Neural Networks by Penalizing Confident Output Distributions.
[13] Mnih, V. et al. (2016). Asynchronous Methods for Deep Reinforcement Learning.
[14] Haarnoja, T., Zhou, A., Abbeel, P., & Levine, S. (2018). Soft Actor-Critic: Off-Policy Maximum Entropy Deep RL.
[15] Nachum, O., Norouzi, M., Xu, K., & Schuurmans, D. (2017). Bridging the Gap Between Value and Policy-Based RL.
[16] Amos, B., Xu, L., & Kolter, J. Z. (2017). Input Convex Neural Networks.
[17] Villani, C. (2009). Optimal Transport: Old and New.
[18] Cuturi, M. (2013). Sinkhorn Distances: Lightspeed Computation of Optimal Transport.
[19] Alemi, A. A., Fischer, I., Dillon, J. V., & Murphy, K. (2017). Deep Variational Information Bottleneck.
[20] Achille, A., & Soatto, S. (2018). Information Dropout: Learning Optimal Representations Through Noisy Computation.