# Abuse ΘϜ Notation

### Fixed point iteration for finding (I_n+UV)^{-1}

Let $B=I_n+UV$, where $U \in \mathbb{R}^{n \times m}$ and $V \in \mathbb{R}^{m \times n}$ such that $m>>n$. The goal is to find $B^{-1}$. Using the Sherman–Morrison–Woodbury formula, we have:

### Online Combinatorial Optimization using Hedge in Linear Time

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 $T$ 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.

### Submodularity and the Lovász extension

This blog is primarily about an easier way of understanding the Lovasz extension of submodular functions.

### Exp3: or How to do well in an exam with 0 preparation under bandit feedback?

This is a continuation of my previous blog post on Hedge. Here we consider Bandit feedback instead of full information.

### Hedge or: How to do well in an exam with 0 preparation?

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.

### How to distinguish between two kinds of 7s?

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.

### Learning the Mode under Bandit Feedback

Consider the following problem. There is an unknown discrete probability distribution on the alphabet $[K]$. Let this distribution be $p=[p_1,p_2,..,p_K]$. The mode of this distribution is $k^\star = \arg \max_{k \in [K]} p_k$ and let $p^\star = p_{k^\star}$. You have access to a noisy oracle. The goal is to query this oracle for $T$ rounds such that the following regret is minimized.

### Solving a Non-Convex problem by solving many Convex problems

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.

### Euclidean Distance Matrix Completion has No Spurious Local Minima?

Let $X$ be a $n\times k$ real matrix. It can be interpreted as $n$ points, $x_1,x_2,..x_n \in \mathbb{R}^k$. The Euclidean Distance Matrix(EDM) $D$ is $[D]_{i,j} = \|x_i-x_j\|_2^2$. In the EDM completion problem, we are given some entries of $D$ which are randomly sampled according to a probability $p$. The goal is to determine the remaining entries of $D$ and additionally the point matrix $X$ producing $D$. This is similar to the generic Low rank matrix completion problem, except on EDMs.

### An Efron-Stein Like Lower bound for Variance

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.

### Projected Gradient Descent - Max(Min) Eigenvalues(vectors)

This post is about finding the minimum and maximum eigenvalues and the corresponding eigenvectors of a matrix $A$ 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.

### Perfect Matchings : Optimal Bike-pooling for a Common Destination

You are an intern at a prestigious AI lab. Assume there are $n$ 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.

### Posterior Sampling for RL

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).

### Markov Decision Processes

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).

### Thompson Sampling vs Pure Exploration Thompson Sampling

You are given $N$ coins which may be biased. Whenever you toss a coin and get a Heads, you score a point. Now consider these two objectives:

### Multi Armed Bandits and Exploration Strategies

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.

### A Derivation of Backpropagation in Matrix Form

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.

### Die rolls and Concentration Inequalities

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.

### Finding anagrams using Godel numbering and Randomization

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.

### Formulating Linear Programs

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:

### Minimum Spanning Tree - Changing edge weights

This post is about reconstructing the Minimum Spanning Tree(MST) of a graph when the weight of some edge changes.

### Linear Regression with and without Calculus

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 $O(\log n)$. This post is inspired from a question I was asked in my B.Tech Grand Viva.