Sayar Karmakar

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.

Christopher Perez, Sayar Karmakar

COVID-19 has brought about many changes in social dynamics. Stay-at-home orders and disruptions in school teaching can influence bullying behavior in-person and online, both of which leading to negative outcomes in victims. To study cyberbullying specifically, 1 million tweets containing keywords associated with abuse were collected from the beginning of 2019 to the end of 2021 with the Twitter API search endpoint. A natural language processing model pre-trained on a Twitter corpus generated probabilities for the tweets being offensive and hateful. To overcome limitations of sampling, data was also collected using the count endpoint. The fraction of tweets from a given daily sample marked as abusive is multiplied to the number reported by the count endpoint. Once these adjusted counts are assembled, a Bayesian autoregressive Poisson model allows one to study the mean trend and lag functions of the data and how they vary over time. The results reveal strong weekly and yearly seasonality in hateful speech but with slight differences across years that may be attributed to COVID-19.

Sayar Karmakar, Anirbit Mukherjee

A particular direction of recent advance about stochastic deep-learning algorithms has been about uncovering a rather mysterious heavy-tailed nature of the stationary distribution of these algorithms, even when the data distribution is not so. Moreover, the heavy-tail index is known to show interesting dependence on the input dimension of the net, the mini-batch size and the step size of the algorithm. In this short note, we undertake an experimental study of this index for S.G.D. while training a $\relu$ gate (in the realizable and in the binary classification setup) and for a variant of S.G.D. that was proven in Karmakar and Mukherjee (2022) for ReLU realizable data. From our experiments we conjecture that these two algorithms have similar heavy-tail behaviour on any data where the latter can be proven to converge. Secondly, we demonstrate that the heavy-tail index of the late time iterates in this model scenario has strikingly different properties than either what has been proven for linear hypothesis classes or what has been previously demonstrated for large nets.

Soham Bonnerjee, Sayar Karmakar, Subhrajyoty Roy

With the increasing popularity of large language models, concerns over content authenticity have led to the development of myriad watermarking schemes. These schemes can be used to detect a machine-generated text via an appropriate key, while being imperceptible to readers with no such keys. The corresponding detection mechanisms usually take the form of statistical hypothesis testing for the existence of watermarks, spurring extensive research in this direction. However, the finer-grained problem of identifying which segments of a mixed-source text are actually watermarked, is much less explored; the existing approaches either lack scalability or theoretical guarantees robust to paraphrase and post-editing. In this work, we introduce a unique perspective to such watermark segmentation problems through the lens of epidemic change-points. By highlighting the similarities as well as differences of these two problems, we motivate and propose WISER: a novel, computationally efficient, watermark segmentation algorithm. We theoretically validate our algorithm by deriving finite sample error-bounds, and establishing its consistency in detecting multiple watermarked segments in a single text. Complementing these theoretical results, our extensive numerical experiments show that WISER outperforms state-of-the-art baseline methods, both in terms of computational speed as well as accuracy, on various benchmark datasets embedded with diverse watermarking schemes. Our theoretical and empirical findings establish WISER as an effective tool for watermark localization in most settings. It also shows how insights from a classical statistical problem can lead to a theoretically valid and computationally efficient solution of a modern and pertinent problem.

Soham Bonnerjee, Sayar Karmakar, Wei Biao Wu

Federated Learning has gained traction in privacy-sensitive collaborative environments, with local SGD emerging as a key optimization method in decentralized settings. While its convergence properties are well-studied, asymptotic statistical guarantees beyond convergence remain limited. In this paper, we present two generalized Gaussian approximation results for local SGD and explore their implications. First, we prove a Berry-Esseen theorem for the final local SGD iterates, enabling valid multiplier bootstrap procedures. Second, motivated by robustness considerations, we introduce two distinct time-uniform Gaussian approximations for the entire trajectory of local SGD. The time-uniform approximations support Gaussian bootstrap-based tests for detecting adversarial attacks. Extensive simulations are provided to support our theoretical results.

Sanchari Das, Andrew Kim, Sayar Karmakar

Due to the outbreak of COVID-19, users are increasingly turning to online services. An increase in social media usage has also been observed, leading to the suspicion that this has also raised cyberbullying. In this initial work, we explore the possibility of an increase in cyberbullying incidents due to the pandemic and high social media usage. To evaluate this trend, we collected 454,046 cyberbullying-related public tweets posted between January 1st, 2020 -- June 7th, 2020. We summarize the tweets containing multiple keywords into their daily counts. Our analysis showed the existence of at most one statistically significant changepoint for most of these keywords, which were primarily located around the end of March. Almost all these changepoint time-locations can be attributed to COVID-19, which substantiates our initial hypothesis of an increase in cyberbullying through analysis of discussions over Twitter.

Sayar Karmakar, Anirbit Mukherjee

In this work, we demonstrate provable guarantees on the training of a single ReLU gate in hitherto unexplored regimes. We give a simple iterative stochastic algorithm that can train a ReLU gate in the realizable setting in linear time while using significantly milder conditions on the data distribution than previous such results. Leveraging certain additional moment assumptions, we also show a first-of-its-kind approximate recovery of the true label generating parameters under an (online) data-poisoning attack on the true labels, while training a ReLU gate by the same algorithm. Our guarantee is shown to be nearly optimal in the worst case and its accuracy of recovering the true weight degrades gracefully with increasing probability of attack and its magnitude. For both the realizable and the non-realizable cases as outlined above, our analysis allows for mini-batching and computes how the convergence time scales with the mini-batch size. We corroborate our theorems with simulation results which also bring to light a striking similarity in trajectories between our algorithm and the popular S.G.D. algorithm - for which similar guarantees as here are still unknown.

Sayar Karmakar, Anirbit Mukherjee, Theodore Papamarkou

In this work, we study the possibility of defending against data-poisoning attacks while training a shallow neural network in a regression setup. We focus on doing supervised learning for a class of depth-2 finite-width neural networks, which includes single-filter convolutional networks. In this class of networks, we attempt to learn the network weights in the presence of a malicious oracle doing stochastic, bounded and additive adversarial distortions on the true output during training. For the non-gradient stochastic algorithm that we construct, we prove worst-case near-optimal trade-offs among the magnitude of the adversarial attack, the weight approximation accuracy, and the confidence achieved by the proposed algorithm. As our algorithm uses mini-batching, we analyze how the mini-batch size affects convergence. We also show how to utilize the scaling of the outer layer weights to counter output-poisoning attacks depending on the probability of attack. Lastly, we give experimental evidence demonstrating how our algorithm outperforms stochastic gradient descent under different input data distributions, including instances of heavy-tailed distributions.