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Posts

Note on the Kadison-Singer Problem and its Solution

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The Kadison-Singer problem arose from the work on quantum mechanics done by Paul Dirac in the 1930s. The problem is equivalent to fundamental problems in areas like Operator theory, Hilbert and Banach space theory, Frame theory, Harmonic Analysis, Discrepancy theory, Graph theory, Signal Processing and theoretical Computer Science. The Kadison-Singer problem had been long standing and defied the efforts of most Mathematicians until it was recently solved by Adam Wade Marcus, Daniel Alan Spielman and Nikhil Srivastava in 2013. Read more

A note on Conformal Symplectic and Relativistic Optimization

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This note on a spotlight paper at NeurIPS 2020, has been made while I had been reading the literature on the principle connections between continuous and discrete optimization. The motivation is to understand and create accelerated discrete large scale optimization algorithms from first principles via considering the geometry of phase spaces and numerical integration, specifically symplectic integration. Recent works successfully have been able to throw sufficient light on the two and therefore has attracted my attention. Read more

Geometry of Relativistic Spacetime Physics

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This article introduces and describes the mathematical structures and frameworks needed to understand the modern fundamental theory of Relativistic Spacetime Physics. The self-referential and self-contained nature of Mathematics provides enough power to prescribe a rigorous language needed to formulate the building components of the standard Einstein’s General Theory of Relativity like Spacetime, Matter, and Gravity, along with their behaviors and interactions. In these notes, we will introduce and understand these abstract components, starting with defining the arena of smooth manifolds and then adding the necessary and suffcient differential geometric structures needed to build the primers to the General Theory of Relativity. Read more

Dual spaces and the Fenchel conjugate

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Dual spaces lie at the core of linear algebra and allows us to formally reason about the concept of duality in mathematics. Duality shows up naturally and elegantly in measure theory, functional analysis, and mathematical optimization. In this post, I have tried to learn and explore the nature of duality via Dual spaces, its interpretation in general linear algebra, all of which was motivated by the so called convex conjugate, or the Fenchel conjugate in mathematical optimization. Read more

A survey on Strongly Rayleigh measures and their mixing time analysis

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Strongly Rayleigh measures are natural generalizations of measures that satisfy the notion of negative dependence. The class of Strongly Rayleigh measures provides the most useful characterization of Negative Dependence by grounding it in the theory of multivariate stable polynomials. This post attempts to throw some light on the origin of Strongly Rayleigh measures and Determinantal Point Processes and highlights the fast mixing time analysis of the natural MCMC chain in the support of a Strongly Rayleigh measure as shown by Anari, Gharan and Rezaei 2016. Read more

Analysis of Newton’s Method

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In optimization, Netwon’s method is used to find the roots of the derivative of a twice differentiable function given the oracle access to its gradient and hessian. By having super-liner memory in the dimension of the ambient space, Newton’s method can take the advantage of the second order curvature and optimize the objective function at a quadratically convergent rate. Here I consider the case when the objective function is smooth and strongly convex. Read more

Deriving the Fokker-Planck equation

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In the theory of dynamic systems, Fokker-Planck equation is used to describe the time evolution of the probability density function. It is a partial differential equation that describes how the density of a stochastic process changes as a function of time under the influence of a potential field. Some common application of it are in the study of Brownian motion, Ornstein–Uhlenbeck process, and in statistical physics. The motivation behind understanding the derivation is to study Levy flight processes that has caught my recent attention. Read more

Nesterov’s Acceleration

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This post contains an error vector analysis of the Nesterov’s accelerated gradient descent method and some insightful implications that can be derived from it. Read more

A survey on Large Scale Optimization

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This post contains a summary and survey of the theoretical understandings of Large Scale Optimization by referring some talks, papers, and lectures that I have come across in the recent. Read more

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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Understanding scaling limits of discrete Euclidean optimization algorithms on large unlabeled graphs. This problem is motivated by the problem of optimizing permutation invariant risk functions of (single layer and deep) Neural Networks (NNs). Theoretical aspects stem from the original theory of gradient flows on the Wasserstein space, which have been used to understand scaling limits of (stochstic) gradient descent ((S)GD) processes in the case of single hidden layer neural networks. There are also other related questions that are specific to the qualitative nature of the stochasticity in the SGD process, and the role of depth in NNs. 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]