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ADMM

Implemented ADMM for solving convex optimization problems such as Lasso, Ridge regression

Introduction

Alternating Direction Method of Multiplier is framework for solving objecting function with divide-and-conquer approach.

ADMM works in two steps

  1. Divide
    a. Break down original problem into small problems
    b. Distribute these small problem to processors / computing resources
    c. Every processor solves the smaller problem
  2. Conquer
    a. Combine solution from <N> processors into one

Our work

We implemented ADMM in PyTorch framework Click here to view for Lasso and Ridge regression.

Results

Lasso Solver

Lasso Loss Lasso Prediction
ADMM Lasso Loss ADMM vs Scikit Lasso Solver

Ridge regression

Ridge regression Loss Ridge regression Prediction
ADMM Ridge regression Loss ADMM vs Scikit ridge regression Solver

ADMM vs Newton vs Gradient Descent

Contour Plot of ADMM vs GD vs Newtons method for Lasso Problem

Contour plot does shows that ADMM reaches to the optimal solution fast and then takes smaller steps as it reaches to near to the solution. Hence, it confirms that ADMM is middle solution to many problems which can solve problems nearly as fast as newton and is not just restricted to quadratic problems.

Issues

  1. ADMM needs distributed infrastructure to scale to general problems
  2. Gradient of individual small problems needs to be known in order to divide the problem
  3. How to divide the problem into smaller problems? a. This is reason behind we need to manually devise the smaller problem and then scale for parallalization

Slides

Click here to view Presentation

Report

Click here to view Report

Other resources

  1. My Convex Optimization assignments - https://github.com/bhushan23/Convex-Optimization
  2. Reading material - By Professor Steven Boyd - http://web.stanford.edu/~boyd/admm.html
  3. Implementation - By Niru Maheswaranathan - https://github.com/nirum/ADMM
  4. General Convex optimization problems implementation - By Stanford Convex Optimization group - https://github.com/cvxgrp/cvxpy

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Implemented ADMM for solving convex optimization problems such as Lasso, Ridge regression

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