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}

Pulkit Gopalani, Ekdeep Singh Lubana, Wei Hu

Recent analysis on the training dynamics of Transformers has unveiled an interesting characteristic: the training loss plateaus for a significant number of training steps, and then suddenly (and sharply) drops to near--optimal values. To understand this phenomenon in depth, we formulate the low-rank matrix completion problem as a masked language modeling (MLM) task, and show that it is possible to train a BERT model to solve this task to low error. Furthermore, the loss curve shows a plateau early in training followed by a sudden drop to near-optimal values, despite no changes in the training procedure or hyper-parameters. To gain interpretability insights into this sudden drop, we examine the model's predictions, attention heads, and hidden states before and after this transition. Concretely, we observe that (a) the model transitions from simply copying the masked input to accurately predicting the masked entries; (b) the attention heads transition to interpretable patterns relevant to the task; and (c) the embeddings and hidden states encode information relevant to the problem. We also analyze the training dynamics of individual model components to understand the sudden drop in loss.

Pulkit Gopalani, Wei Hu

Training Transformers on algorithmic tasks frequently demonstrates an intriguing abrupt learning phenomenon: an extended performance plateau followed by a sudden, sharp improvement. This work investigates the underlying mechanisms for such dynamics, primarily in shallow Transformers. We reveal that during the plateau, the model often develops an interpretable partial solution while simultaneously exhibiting a strong repetition bias in their outputs. This output degeneracy is accompanied by internal representation collapse, where hidden states across different tokens become nearly parallel. We further identify the slow learning of optimal attention maps as a key bottleneck. Hidden progress in attention configuration during the plateau precedes the eventual rapid convergence, and directly intervening on attention significantly alters plateau duration and the severity of repetition bias and representational collapse. We validate that these identified phenomena-repetition bias and representation collapse-are not artifacts of toy setups but also manifest in the early pre-training stage of large language models like Pythia and OLMo.