Understanding Adversarial Training with Energy-based Models

We aim at using Energy-based Model (EBM) framework to better understand adversarial training (AT) in classifiers, and additionally to analyze the intrinsic generative capabilities of robust classifiers. By viewing standard classifiers through an energy lens, we begin by analyzing how the energies of adversarial examples, generated by various attacks, differ from those of the natural samples. The central focus of our work is to understand the critical phenomena of Catastrophic Overfitting (CO) and Robust Overfitting (RO) in AT from an energy perspective. We analyze the impact of existing AT approaches on the energy of samples during training and observe that the behavior of the ``delta energy' -- change in energy between original sample and its adversarial counterpart -- diverges significantly when CO or RO occurs. After a thorough analysis of these energy dynamics and their relationship with overfitting, we propose a novel regularizer, the Delta Energy Regularizer (DER), designed to smoothen the energy landscape during training. We demonstrate that DER is effective in mitigating both CO and RO across multiple benchmarks. We further show that robust classifiers, when being used as generative models, have limits in handling trade-off between image quality and variability. We propose an improved technique based on a local class-wise principal component analysis (PCA) and energy-based guidance for better class-specific initialization and adaptive stopping, enhancing sample diversity and generation quality. Considering that we do not explicitly train for generative modeling, we achieve a competitive Inception Score (IS) and Fréchet inception distance (FID) compared to hybrid discriminative-generative models.

@article{mirza2025_2505.22486,
  title={ Understanding Adversarial Training with Energy-based Models },
  author={ Mujtaba Hussain Mirza and Maria Rosaria Briglia and Filippo Bartolucci and Senad Beadini and Giuseppe Lisanti and Iacopo Masi },
  journal={arXiv preprint arXiv:2505.22486},
  year={ 2025 }
}