# Strengthening the Internal Adversarial Robustness in Lifted Neural Networks

arXiv: https://arxiv.org/abs/2503.07818 · HTML: https://mlsanity.com/paper/2503.07818/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 10 Mar 2025.

## Builds on (most cited first, 16 of 16)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Beyond Backprop: Online Alternating Minimization with Auxiliary Variables](https://mlsanity.com/paper/1806.09077/) (2018)
- [Lifted Proximal Operator Machines](https://mlsanity.com/paper/1811.01501/) (2018)
- [Contrastive Learning for Lifted Networks](https://mlsanity.com/paper/1905.02507/) (2019)
- [Bilevel Programs Meet Deep Learning: A Unifying View on Inference Learning Methods](https://mlsanity.com/paper/2105.07231/) (2021)
- [Two Tales of Single-Phase Contrastive Hebbian Learning](https://mlsanity.com/paper/2402.08573/) (2024)

## Abstract (arXiv)

Lifted neural networks (i.e. neural architectures explicitly optimizing over respective network potentials to determine the neural activities) can be combined with a type of adversarial training to gain robustness for internal as well as input layers, in addition to improved generalization performance. In this work we first investigate how adversarial robustness in this framework can be further strengthened by solely modifying the training loss. In a second step we fix some remaining limitations and arrive at a novel training loss for lifted neural networks, that combines targeted and untargeted adversarial perturbations.