Let , where and such that . The goal is to find . Using the Sherman–Morrison–Woodbury formula, we have:
Abuse ΘϜ Notation
In this blog post, we introduce OMD and state an upperbound for its regret. Using this upperbound, we analyze the regret of the algorithm we proposed in our previous blog.
In this post, we consider the Online Linear Optimization problem where the decision set is confined to the vertices of a Hypercube. Naively applying Hedge to this problem achieves optimal regret in but requires maintaining an exponential number of variables. We show that it is in fact possible to apply Hedge by using only linear number of variables.
This blog is primarily about an easier way of understanding the Lovasz extension of submodular functions.
This is a continuation of my previous blog post on Hedge. Here we consider Bandit feedback instead of full information.
Assume you are a grad student taking two courses this semester. One of which is Machine Learning Theory, which is your favorite course. You don’t like the other course so much, so you never study for it. You still have to do well in this course’s exam, so you decide to use your ML theory knowledge to get through it.
There are two acceptable ways to write the digit ‘7’. It can be written with or without a line in the middle. Given an unlabelled dataset of 7s written in both ways, the task is to find which ones have a dash and which ones don’t.
Consider the following problem. There is an unknown discrete probability distribution on the alphabet . Let this distribution be . The mode of this distribution is and let . You have access to a noisy oracle. The goal is to query this oracle for rounds such that the following regret is minimized.
Recall the Babylonian method for computing square roots, which you might have learnt in highschool. I propose a higher dimensional variant of that, which I use to solve a few Non-Convex problems, specifically computing Matrix Square root, Positive Semidefinite matrix completion and Euclidean Distance Matrix completion.
Let be a real matrix. It can be interpreted as points, . The Euclidean Distance Matrix(EDM) is . In the EDM completion problem, we are given some entries of which are randomly sampled according to a probability . The goal is to determine the remaining entries of and additionally the point matrix producing . This is similar to the generic Low rank matrix completion problem, except on EDMs.
This is a follow up to my previous post: An Efron-Stein Like Lower bound for Variance, where I derived the following inequality.
In a homework of the Machine Learning Theory course that I am taking at UMass, we were asked to prove the Efron-Stein Inequality. It upper bounds the variance of arbitrary functions of iid random variables. While working on the proof, I found an intriguing inequality which has the structure of Efron-Stein, but is a lower bound.
This post is about finding the minimum and maximum eigenvalues and the corresponding eigenvectors of a matrix using Projected Gradient Descent. There are several algorithms to find the eigenvalues of a given matrix (See Eigenvalue algorithms). Although I could not find Projected Gradient Descent in this list, I feel it is a simple and intuitive algorithm.
You are an intern at a prestigious AI lab. Assume there are researchers in this lab. The lab is located at the origin of a metric space. The researchers live at various points in this space. Everyday, these researchers travel to the lab from their respective homes and back. Each researcher owns an eco-friendly motor bike which can carry at most 2 passengers. They decide to bike-pool to work such that the total distance traveled is minimized. They entrust you to come up with an algorithm which can find the optimal sharing scheme.
This post is about a simple yet very efficient Bayesian algorithm for minimizing cumulative regret in Episodic Fixed Horizon Markov Decision Processes(EFH-MDP). First introduced as a heuristic by Strens under the name of “Bayesian Dynamic Programming”, it was later shown by Osband et.al that it is in fact an efficient RL algorithm. They also gave it a more informative name - Posterior Sampling for Reinforcement Learning (PSRL).
This blog post is about Episodic Fixed Horizon Markov Decision Processes (EFH-MDP). MDPs are a very generic framework and can be used to model complex systems in manufacturing, control and robotics to state a few. These are also used to model environments in Reinforcement Learning (RL).
You are given coins which may be biased. Whenever you toss a coin and get a Heads, you score a point. Now consider these two objectives:
This blog post is about Bayesian Inference. It finds extensive use in several Machine learning algorithms and applications. Bayesian Inference, along with Frequentist Inference are the two main approaches to Statistical Inference.
This blog post is about the Multi Armed Bandit(MAB) problem and about the Exploration-Exploitation dilemma faced in reinforcement learning. MABs find applications in areas such as advertising, drug trials, website optimization, packet routing and resource allocation.
Backpropagation is an algorithm used to train neural networks, used along with an optimization routine such as gradient descent. Gradient descent requires access to the gradient of the loss function with respect to all the weights in the network to perform a weight update, in order to minimize the loss function. Backpropagation computes these gradients in a systematic way. Backpropagation along with Gradient descent is arguably the single most important algorithm for training Deep Neural Networks and could be said to be the driving force behind the recent emergence of Deep Learning.
The purpose of this post is to serve as a very basic introduction to Concentration Inequalities. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. These inequalities are used extensively in Machine Learning Theory for deriving PAC like results.
This post is about a neat application of Gödel numbering. This post is inspired by a question my friend was asked during an interview for Microsoft.
Linear Programming(LP) is a widely used technique in algorithm design and analysis, especially for getting good approximations for NP hard problems. The first step in LP would be to formulate the the problem as an optimization problem with linear constraints where the quantity we are trying to optimize is also linear. It is a good excersize to formulate simple problems as linear programs. Linear programming problems typically expressed in their canonical form look like this:
This post is about reconstructing the Minimum Spanning Tree(MST) of a graph when the weight of some edge changes.
This post will be about simple linear regression. I will first use calculus to derive an expression. Then I derive same expression in a more intuitive way using linear algebra.
This post will be about using Randomization for constructing Binary Search Trees (BST) whose worst case height must be . This post is inspired from a question I was asked in my B.Tech Grand Viva.
This is my first blog post. In this blog, I would be writing about Algorithms, Machine Learning, Optimization and Mathematics. I wanted to begin with something simple, so this post would be about a problem from Probability.