academic Works

Combinatorial problems have a notorious reputation for being computationally hard to solve efficiently and, as a result, understanding the underlying properties proves useful in many real life scenarios.

I propose a probabilistic approach which approximates the solution to the perfect sum problem by approximating the distribution of potential sums. Since this problem is an extension of the subset sum, our approximation also grants some probabilistic insight into the solution for the subset sum problem. We harness distributional approximations to model the number of subsets which sum to a certain size. 

These approximations can be computed in O(n) complexity, and can increase in accuracy with the size of the input data making it useful for large-scale combinatorial problems.

What sorts of factors actually contribute to higher educational attainment? It has been argued that ”education attainment is a continuous process in which the education achievement of the prior stage affects the later-stage achievement both cumulatively and probabilistically”. In fact, our entire education system is built on trying to quantify education achievement and performance via standardized measures such as GPA and standardized test scores. These measurements significantly affect the ability of the student to pursue further, higher level education. As a result, investigating factors which affect these educational ”success” scores inherently gives us insight into the factors which contribute to educational attainment later down the line.

In 100 AD, a Roman astronomer by the name of Ptolemy noticed that the planets in the night sky all seemed to revolve around the Earth in a cyclical-like pattern. Of course, today we know that what he saw was not the planets revolving around the Earth, but the Earth and other planets revolving together around the Sun. Despite this, Ptolemy's ingenious methods allowed him to model planetary motions in the night sky quite accurately.

We attempted to recreate these geometrically elegant predictions of ptolemaic era astronomy using modern day machine learning methods. I focused on using Fourier features as input for regression and neural network based models. These models turned out to perform the best on unseen test data.

Scarily enough, modern Reinforcement Learning methods have enabled computers to learn to play certain games quite well, sometimes even better than humans. Among these methods is Proximal Policy Optimization (PPO2) developed by OpenAI which has demonstrated an efficient approach to this complex field, with good performance. However, several of these method suffers overfitting and fail to generalize well. A variety of methods address this, including mixture regularization, however, these approaches mainly deal with modifying the input data instead of addressing network architecture.

In this project, we take a novel approach to Reinforcement Learning generalization and regularization using partially decoupled policy-value networks implemented in the PPO2 algorithm.

Many of calculus' and statistics' most important theorems leverage a measure theoretic approach (Fubini Theorem, Strong Law of Large Numbers).

Here, I've tried to present an intuitive examination of the fundamental properties of the Lebesgue Measure, arguably the most important and useful measure for statistics, followed by some formal proofs.