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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



Mini Search Engine

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We used data structures like Hash Tables, Balanced Trees in order to design a text search engine that gives the frequency of the searched word in a given folder of files. Read more

Clustered Monotone Transforms for Rating Factorization

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We propose Clustered Monotone Transforms for Rating Factorization (CMTRF), a novel approach to perform regression up to unknown monotonic transforms over unknown population segments. For recommendation systems, the technique searches for monotonic transformations of the rating scales resulting in a better fit. This is combined with an underlying matrix factorization regression model that couples the user-wise ratings to exploit shared low dimensional structure. The rating scale transformations can be generated for each user (N-CMTRF), for a cluster of users (CMTRF), or for all the users at once (1-CMTRF), forming the basis of three simple and efficient algorithms proposed, all of which alternate between transformation of the rating scales and matrix factorization regression. Despite the non-convexity, CMTRF is theoretically shown to recover a unique solution under mild conditions. Read more

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, 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


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 and above, the distribution is best described as Gaussian at-least in the early phases of training. Read more

[arXiv] [bib]