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Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal -worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing.
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Wasserstein Distributionally Robust Linear-Quadratic Estimation under Martingale Constraints
We focus on robust estimation of the unobserved state of a discrete-time stochastic system with linear dynamics. A standard analysis of this estimation problem assumes a baseline innovation model; with Gaussian innovations we recover the Kalman filter. However, in many settings, there is insufficient or corrupted data to validate the baseline model. To cope with this problem, we minimize the worst-case mean-squared estimation error of adversarial models chosen within a Wasserstein neighborhood around the baseline. We also constrain the adversarial innovations to form a martingale difference sequence. The martingale constraint relaxes the i.i.d. assumptions which are often imposed on the baseline model. Moreover, we show that the martingale constraints guarantee that the adversarial dynamics remain adapted to the natural time-generated information. Therefore, adding the martingale constraint allows to improve upon over-conservative policies that also protect against unrealistic omniscient adversaries. We establish a strong duality result which we use to develop an efficient subgradient method to compute the distributionally robust estimation policy. If the baseline innovations are Gaussian, we show that the worst-case adversary remains Gaussian. Our numerical experiments indicate that the martingale constraint may also aid in adding a layer of robustness in the choice of the adversarial power.
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- PAR ID:
- 10483198
- Publisher / Repository:
- Proceedings of The 26th International Conference on Artificial Intelligence and Statistics
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 206
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 8629-8644
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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