Certified Robust Neural Networks: Generalization and Corruption Resistance

Adversarial training aims to reduce the problematic susceptibility of modern neural networks to small data perturbations. Surprisingly, overfitting is a major concern in adversarial training of neural networks despite being mostly absent in standard training. We provide here theoretical evidence for this peculiar ``robust overfitting'' phenomenon. Subsequently, we advance a novel loss function which we show both theoretically as well as empirically to enjoy a certified level of robustness against data evasion and poisoning attacks while ensuring guaranteed generalization. We indicate through careful numerical experiments that our resulting holistic robust (HR) training procedure yields SOTA performance in terms of adversarial error loss. Finally, we indicate that HR training can be interpreted as a direct extension of adversarial training and comes with a negligible additional computational burden.

Adaptive Learning on Time Series: Method and Financial Applications

We formally introduce a time series statistical learning method, called Adaptive Learning, capable of handling model selection, out-of-sample forecasting and interpretation in a noisy environment. Through simulation studies we demonstrate that the method can outperform traditional model selection techniques such as AIC and BIC in the presence of regime-switching, as well as facilitating window size determination when the Data Generating Process is time-varying. Empirically, we use the method to forecast S&P 500 returns across multiple forecast horizons, employing information from the VIX Curve and the Yield Curve. We find that Adaptive Learning models are generally on par with, if not better than, the best of the parametric models a posteriori, evaluated in terms of MSE, while also outperforming under cross validation. We present a financial application of the learning results and an interpretation of the learning regime during the 2020 market crash. These studies can be extended in both a statistical direction and in terms of financial applications.