Research | ver0

Research Interests

In my research I use partial differential equations (PDEs), optimal control theory, and graph theory to analyze machine learning algorithms and investments, and give rigorous performance guarantees. I am a problem solver: I typically start with an application in mind, and learn and develop whatever methods I need in order to come up with a solution to the problem at hand. So far, I have bridged select ideas from viscosity solutions of PDEs, optimal control theory, and graph theory, to target problems in data science and investments. I have innovated and introduced continuous approaches to solve discrete problems from `prediction with expert advice' and `semi-supervised learning.' Moreover, my work proves that these approaches are asymptotically optimal.

A unifying theme for most of my research is adversarial two-player games, which arise in prediction problems as well as in p-Laplacian approaches to semi-supervised learning. Another common thread is investment algorithms, which appear in robo-advising as well as in prediction with expert advice. My broader research goal is to use techniques from PDEs and ODEs to analyze and understand discrete problems from machine learning, graphs, and finance. Select applications are algorithm boosting, advertisement placing, stock investments, sports betting, self-driving car software, and data classification.

Publications

J. Calder and N. Drenska. Consistency of Semi-Supervised Learning, Stochastic Tug-of-War Games, and the p-Laplacian. (submitted)
D. Mosaphir, J. Calder, and N. Drenska. Numerical Solution of a PDE Arising from Prediction with Expert Advice. (in preparation)
J. Calder and N. Drenska. Asymptotically Optimal Strategies for Online Prediction with History-Dependent Experts. accepted, Journal of Fourier Analysis and Applications, 27, article 20, 2020, https://doi.org/10.1007/s00041-021-09815-4
N. Drenska and J. Calder. Online Prediction with History-Dependent Experts: The General Case. Communications on Pure and Applied Mathematics (CPAM), 2022, https://doi.org/10.1002/cpa.22049
N. Drenska and R. V. Kohn. A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts. Communications on Pure and Applied Mathematics (CPAM), 2022, https://doi.org/10.1002/cpa.22071
N Drenska and R.V. Kohn. Prediction with Expert Advice: a PDE Perspective. Journal of Nonlinear Science, 30(1):137-173, 2020, https://doi.org/10.1007/s00332-019-09570-3
N Drenska. A PDE Approach to a Prediction Problem Involving Randomized Strategies. PhD thesis, New York University, 2017

Recent talks

Nadia Drenska's Machine Learning Journey
- The Johns Hopkins University, April 2024
- Louisiana State University, April 2024
Semi-Supervised Learning with the p-Laplacian in Geometric Methods in Machine Learning and Data Analysis
- Numerical PDEs: Analysis, Algorithms, and Data Challenges, ICERM, March 2024
- ICIAM, August 2023
Optimal Investment: Robo-Advising Under Small Changes of Risk Aversion
- Joint Mathematics Meetings 2023
A PDE Interpretation of Prediction with Expert Advice
- NJIT, January 2022
- JMU Artificial Intelligence and Machine Learning Seminar Series, April 2021
- Joint Mathematics Meetings, January 2021
- OneWorld Machine Learning, December 2020
- Optimal Control, Optimal Transport, and Data Science, IMA, November 2020
- LMS Bath Symposium, August 2020
Johns Hopkins Applied Mathematics and Statistics Colloquium, October 2021
WPI Colloquium, January 2021
Two PDE Approaches to A Problem from Prediction with Expert Advice
- IPAM, UCLA, May 2020
- Analysis and Applied Mathematics Seminar, UIC, February 2020
PDE Approaches to Two Problems from Prediction with Expert Advice
- Applied Interdisciplinary Mathematics Seminar, UMich 2019
A PDE Approach to Some Randomized-Strategy Two-Player Games
- IMA Data Science Seminar, UMN 2018
- Materials Working Group, NYU 2016
A PDE Approach to Prediction with Expert Advice
- WPI STEM Faculty Launch, WPI, 2016
- RPI Applied Math Days, RPI, 2016
- SIAM Conference on Analysis of PDEs, Scottsdale AZ, 2015
  - Received SIAM Student Travel Award
- Machine Learning Seminar, NYU, 2015
- Materials Working Group, NYU, 2015

Online

This is a link to my adviser Robert V Kohn's presentation of joint work with Kangping Zhu and me at Waves, Spectral Theory, and Applications conference, 2015.

Awards

Moses Greenfield Prize for Outstanding Interdisciplinary Studies, Courant Institute, NYU, 2016

McCracken Scholarship, NYU,

Rohn Truell Prize for Outstanding Undergraduate Student, DAM, Brown University, 2012

Sarah Dyer Barnes Scholarship, Brown University, 2011 - 2012

Sigma Xi, Brown University, 2012

Henry Parker Manning Prize Examination, First Prize, Brown University, 2011

First and Second Prizes on Math and Physics Olympiads, Bulgaria 2005-2007