This repository hosts the code submitted as part my Individual Project for the MSc in Artificial Intelligence (MSc AI) at Imperial College London. All code is written in Python3.

Mirror Descent and Interacting Mirror Descent for almost dimension-free optimisation on non-Euclidean spaces

An investigation of Mirror Descent and Interacting particle Mirror Descent optimisation on different problems.

Many fundamental optimisation problems in machine learning and artificial intelligence have at the core of their frameworks the search of an optimal set parameters that lead to the minimisation or maximisation of a differentiable convex or non-convex objective function over either constrained or unconstrained feasible sets. A plethora of these problems utilise iterative methods that compute the gradient of the objective function at each iteration and take a step towards the steepest ascent or descent. A widely used optimisation method is Stochastic Gradient Descent. SGD is typically a well-performing algorithm for optimisation over Euclidean vector spaces. However, SGD does not always manage to converge to the optimum due to scaling of the dimensionality of the problem on non-Euclidean spaces, or due to noisy computations of the gradient and frequently proves to be expensive if the optimisation is constrained. In this project we study an other almost dimension-free iterative optimisation algorithm called Mirror Descent and its variant Interacting particles Mirror Descent on six convex frameworks with convex objective functions. Our aim is to observe the performance and convergence properties of these methods and also compare them with the standard Stochastic Gradient Descent method. We show that the mirror descent algorithms can outperform gradient descent in cases of optimisation over constrained sets with certain geometry, such as the probability simplex or the matrix spectrahedron. Additionally, we demonstrate the benefits in terms of convergence when allowing particles to interact in the case of Mirror Descent.

Getting Started

These instructions will get you a copy of the project up and running on your local machine for development and testing purposes. See deployment for notes on how to deploy the project on a live system.

Dependencies

Several Python packages are required to be installed in order to run this code on your local machine:

numpy
torch
scipy
sklearn
matplotlib
tqdm
networkx

Installing

Most of the packages can be intalled using pip, pip3, or conda if you are using Anaconda. For example you can run on your bash the following line to install pytorch:

pip install torch torchvision

Please refer to the website of each package to find explicitly what you need for your system.

Run Optimisation

The main functions are in the following files which they contain the optimisation algorithms for each problem:

/The Traffic Assignment Problem/run_opt.py

/Linear System/run_opt.py

/Mini-Batch Optimisation/run_opt.py

/Laplacian Kmodes for Clustering/run_opt.py

/Connected Subgraph Detection on SDPs/startOpt/startOpt.py

/Langevin Diffuisions/run_opt.py

Authors

• George Yiasemis - Initial work

License

This project is licensed under the MIT License - see the LICENSE.md file for details.

Acknowledgments

I would like to acknowledge:

• My supervisor Dr Panos Parpas of the Department of Computing at Imperial College London for his counsel and recommendations.
• My family and friends for the encouragement and support during this difficult and challenging period.
• All the Academic staff of the Department of Computing at Imperial College for tha amazing knowledge they offered me this year.

References

The main references of these project are:

• Anastasia Borovykh, Nikolas Kantas, Panos Parpas, and Grigorios Pavliotis. On stochastic mirror descent with interacting particles: convergence properties and variance reduction, 07 2020.

• Sébastien Bubeck. Convex optimization: Algorithms and complexity, 2014.

• Weiran Wang and Miguel A. Carreira-Perpiñán. The laplacian k-modes algorithm for clustering, 2014.

• Sinho Chewi, Thibaut Le Gouic, Chen Lu, Tyler Maunu, Philippe Rigollet, and Austin J. Stromme. Exponential ergodicity of mirror-langevin diffusions, 2020.

• Cem Aksoylar, Lorenzo Orecchia, and Venkatesh Saligrama. Connected subgraph detection with mirror descent on SDPs. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 51–59, International Convention Centre, Sydney, Australia, 06–11 Aug 2017. PMLR.