Quentin Fruytier

Paper Submission: "Learning Mixtures of Experts with EM"

Mixtures of Experts (MoE) are Machine Learning models that involve partitioning the input space, with a separate "expert" model trained on each partition. Recently, MoE have become popular as components in today's large language models as a means to reduce training and inference costs. There, the partitioning function and the experts are both learnt jointly via gradient descent on the log-likelihood. In this paper we focus on studying the efficiency of the Expectation Maximization (EM) algorithm for the training of MoE models. We first rigorously analyze EM for the cases of linear or logistic experts, where we show that EM is equivalent to Mirror Descent with unit step size and a Kullback-Leibler Divergence regularizer. This perspective allows us to derive new convergence results and identify conditions for local linear convergence based on the signal-to-noise ratio (SNR). Experiments on synthetic and (small-scale) real-world data show that EM outperforms the gradient descent algorithm both in terms of convergence rate and the achieved accuracy.

Research Project: "A Case Against Using Elo or Glicko Algorithms for Rating Players in n vs n Games"

This course project investigates the intricate task of rating players in n vs n zero-sum games, particularly prevalent in the context of online esports titles where large player bases engage in diverse team-based matchups. We explore and evaluate three well-known rating algorithms: the Elo system, Glicko, and Glicko-2, each building upon its predecessor to account for additional variables. A simulation study is conducted to empirically demonstrate the inadequacies of these algorithms in n vs n scenarios where n > 1, highlighting the urgency for tailored solutions. Our findings reveal significant performance drawbacks in current rating systems, especially concerning the failure to consider the performance gap between players during a game. The necessity for a revamped system becomes evident, with a proposed introduction of a rating scale that incorporates game closeness to expedite accurate player rating. The Glicko rating's assumption of a Normal distribution is also challenged by empirical evidence, emphasizing the need for algorithmic adjustments. This project advocates for the development of a novel rating system tailored to n vs n games, emphasizing the incorporation of individual player performance metrics. Despite the challenges, existing performance rating systems in popular esports titles showcase promising avenues for future improvements in the dynamic landscape of player rating and matchmaking.

Master's Thesis : "A Review of theExpectation-Maximization Algorithm and its Applications to Mixture Models."

The Expectation-Maximization (EM) algorithm has long been recognized as a powerful tool for approximating the maximum likelihood estimator in parametric models with latent variables. This thesis provides a selective survey of the existing EM literature, spanning from its original formulation in the 1970s to its present-day developments, with the objective of creating a valuable resource for future research. By exploring the evolution of the EM algorithm, we present both earlier and recent results as well as practical applications in mixture models. Chapter 1 serves as a thorough introduction to the EM, contextualizing it within the broader framework of parameter estimation in parametric models with latent variables. In Chapter 2, we study the general convergence properties of the EM; in particular, we present conditions under which the algorithm’s fitted iterates converge inside a ball centered around the true parameter of the model. Meanwhile, in Chapter 3, we survey the existing literature on the EM algorithm as it relates to Gaussian mixture models and mixed linear regression models. Finally, in Chapter 4, we conclude with a discussion on important aspects such as initialization, SNR, parameterization, and new research directions for the EM algorithm. By collating the wealth of knowledge available on the EM algorithm, this thesis offers researchers a valuable reference for understanding, applying, and advancing the EM algorithm.

Traffic Prediction using DCRNN

This research project aims to analyse and propose improvements to Deep Learning methods for traffic prediction. Traffic Prediction problem is a difficult and important problem impacting multiple industries. Previous methods have struggled to adequatly model the spatial and temporal dependency as well as the non-stationarity of the problem to yield good long and short term predictions. Recently, Deep Learning has become the state of the art method for traffic prediction as it is unmatched for discovering hidden patterns in large datasets. More specifically, Recurrent Neural Networks (RNN), Graph Convolutions and Encoder-Decoder frameworks have led to very succesful tools such as DCRNN, 3D-TGCRNN and SLCNN. Here, I take a deeper look at DCRNN by studying the effect of feature space’s dimension on the METR-LA dataset and suggest improvements to the architecture.

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