Raghav Somani

Raghav Somani

Quantitative Analyst at
The D. E. Shaw Group

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Misc

Projects

Sparse Regression and Support Recovery bounds for Orthogonal Matching Pursuit

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We study the problem of sparse regression where the goal is to learn a sparse vector that best optimizes a given objective function. Under the assumption that the objective function satisfies restricted strong convexity (RSC), we analyze Orthogonal Matching Pursuit (OMP) and obtain support recovery result as well as a tight generalization error bound for OMP. Furthermore, we obtain lower bounds for OMP, showing that both our results on support recovery and generalization error are tight up to logarithmic factors. To the best of our knowledge, these support recovery and generalization bounds are the first such matching upper and lower bounds (up to logarithmic factors) for any sparse regression algorithm under the RSC assumption. Read more

Universality Patterns in the Training of Neural Networks

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This work proposes and demonstrates a surprising pattern in the training of neural networks: there is a one to one relation between the values of any pair of losses (such as cross entropy, mean squared error, \(0/1\) error etc.) evaluated for a model arising at (any point of) a training run. This pattern is universal in the sense that this one to one relationship is identical across architectures (such as VGG, Resnet, Densenet etc.), algorithms (SGD and SGD with momentum) and training loss functions (cross entropy and mean squared error). Read more

Scaling laws of optimization algorithms for Deep Learning - the Graphon perspective

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The non-linear training dynamics of two-layer NNs can be modeled as a mean-field interacting particle system, where neurons in the hidden layer act as “particles.” These dynamics often lead to Wasserstein gradient flows, treating the problem as an optimization over probability measures due to the permutation symmetry of neurons. Extending this to multi-layer NNs, which exhibit more complex symmetries as large computational graphs, this work describes the analytical scaling limits of stochastic optimization algorithms as network size grows. By leveraging the theory of exchangeable arrays, graphons, gradient flows on metric spaces, and propagation of chaos, we characterize this scaling limit. We discover a generalized McKean-Vlasov equation on graphons, where propagation of chaos holds, and in the zero-noise limit, this scaling limit becomes a gradient flow on the metric space of graphons. Read more

Publications

Non-Gaussianity of Stochastic Gradient Noise

Abhishek Panigrahi, Raghav Somani, Navin Goyal & Praneeth Netrapalli
Published at: Science meets Engineering of Deep Learning (SEDL) workshop, Neural Information Processing Systems (NeurIPS), 2019

We study the distribution of the Stochastic Gradient Noise during the training and observe that for batch sizes \(256\) and above, the distribution is best described as Gaussian at-least in the early phases of training. Read more

[arXiv] [bib]