Pulkit Gopalani

Pulkit Gopalani, Sayar Karmakar, Dibyakanti Kumar et al.

In recent times machine learning methods have made significant advances in becoming a useful tool for analyzing physical systems. A particularly active area in this theme has been "physics-informed machine learning" which focuses on using neural nets for numerically solving differential equations. In this work, we aim to advance the theory of measuring out-of-sample error while training DeepONets - which is among the most versatile ways to solve P.D.E systems in one-shot. Firstly, for a class of DeepONets, we prove a bound on their Rademacher complexity which does not explicitly scale with the width of the nets involved. Secondly, we use this to show how the Huber loss can be chosen so that for these DeepONet classes generalization error bounds can be obtained that have no explicit dependence on the size of the nets. The effective capacity measure for DeepONets that we thus derive is also shown to correlate with the behavior of generalization error in experiments.

Pulkit Gopalani, Anirbit Mukherjee

In this note, we consider appropriately regularized $\ell_2-$empirical risk of depth $2$ nets with any number of gates and show bounds on how the empirical loss evolves for SGD iterates on it -- for arbitrary data and if the activation is adequately smooth and bounded like sigmoid and tanh. This in turn leads to a proof of global convergence of SGD for a special class of initializations. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives. Most critically the amount of regularization required for our analysis is independent of the size of the net.

Pulkit Gopalani, Samyak Jha, Anirbit Mukherjee

In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth $2$ nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.

Chaitanya Devaguptapu, Devansh Agarwal, Gaurav Mittal et al.

Adversarial robustness of deep learning models has gained much traction in the last few years. Various attacks and defenses are proposed to improve the adversarial robustness of modern-day deep learning architectures. While all these approaches help improve the robustness, one promising direction for improving adversarial robustness is unexplored, i.e., the complex topology of the neural network architecture. In this work, we address the following question: Can the complex topology of a neural network give adversarial robustness without any form of adversarial training?. We answer this empirically by experimenting with different hand-crafted and NAS-based architectures. Our findings show that, for small-scale attacks, NAS-based architectures are more robust for small-scale datasets and simple tasks than hand-crafted architectures. However, as the size of the dataset or the complexity of task increases, hand-crafted architectures are more robust than NAS-based architectures. Our work is the first large-scale study to understand adversarial robustness purely from an architectural perspective. Our study shows that random sampling in the search space of DARTS (a popular NAS method) with simple ensembling can improve the robustness to PGD attack by nearly~12\%. We show that NAS, which is popular for achieving SoTA accuracy, can provide adversarial accuracy as a free add-on without any form of adversarial training. Our results show that leveraging the search space of NAS methods with methods like ensembles can be an excellent way to achieve adversarial robustness without any form of adversarial training. We also introduce a metric that can be used to calculate the trade-off between clean accuracy and adversarial robustness. Code and pre-trained models will be made available at \url{https://github.com/tdchaitanya/nas-robustness}