# Gaussian Loss Smoothing Enables Certified Training with Tight Convex Relaxations

arXiv: https://arxiv.org/abs/2403.07095 · HTML: https://mlsanity.com/paper/2403.07095/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 61 references that are in the map.
- First posted on arXiv 11 Mar 2024.

## Builds on (most cited first, 30 of 37)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Evasion Attacks against Machine Learning at Test Time](https://mlsanity.com/paper/1708.06131/) (2017)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [On Adaptive Attacks to Adversarial Example Defenses](https://mlsanity.com/paper/2002.08347/) (2020)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers](https://mlsanity.com/paper/1906.04584/) (2019)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Bound](https://mlsanity.com/paper/2205.00263/) (2022)
- [First Three Years of the International Verification of Neural Networks Competition (VNN-COMP)](https://mlsanity.com/paper/2301.05815/) (2023)
- [Certified Training: Small Boxes are All You Need](https://mlsanity.com/paper/2210.04871/) (2022)
- [SmoothMix: Training Confidence-calibrated Smoothed Classifiers for Certified Robustness](https://mlsanity.com/paper/2111.09277/) (2021)
- [Boosting Randomized Smoothing with Variance Reduced Classifiers](https://mlsanity.com/paper/2106.06946/) (2021)
- [Fast Certified Robust Training with Short Warmup](https://mlsanity.com/paper/2103.17268/) (2021)
- [Understanding Certified Training with Interval Bound Propagation](https://mlsanity.com/paper/2306.10426/) (2023)
- [Expressive Losses for Verified Robustness via Convex Combinations](https://mlsanity.com/paper/2305.13991/) (2023)

## Built on by (most cited first, 4 of 4)

- [Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certification](https://mlsanity.com/paper/2410.06816/) (2024)
- [Towards Generalized Certified Robustness with Multi-Norm Training](https://mlsanity.com/paper/2410.03000/) (2024)
- [Learning Better Certified Models from Empirically-Robust Teachers](https://mlsanity.com/paper/2602.02626/) (2026)
- [Dual Randomized Smoothing: Beyond Global Noise Variance](https://mlsanity.com/paper/2512.01782/) (2025)

## Abstract (arXiv)

Training neural networks with high certified accuracy against adversarial examples remains an open challenge despite significant efforts. While certification methods can effectively leverage tight convex relaxations for bound computation, in training, these methods, perhaps surprisingly, can perform worse than looser relaxations. Prior work hypothesized that this phenomenon is caused by the discontinuity, non-smoothness, and perturbation sensitivity of the loss surface induced by tighter relaxations. In this work, we theoretically show that applying Gaussian Loss Smoothing (GLS) on the loss surface can alleviate these issues. We confirm this empirically by instantiating GLS with two variants: a zeroth-order optimization algorithm, called PGPE, which allows training with non-differentiable relaxations, and a first-order optimization algorithm, called RGS, which requires gradients of the relaxation but is much more efficient than PGPE. Extensive experiments show that when combined with tight relaxations, these methods surpass state-of-the-art methods when training on the same network architecture for many settings. Our results clearly demonstrate the promise of Gaussian Loss Smoothing for training certifiably robust neural networks and pave a path towards leveraging tighter relaxations for certified training.