Warm-Starting with ReHLine

This tutorial explains how to use warm-starting with ReHLine, a Python library for regression with hinge loss, to enhance the efficiency of solving similar optimization problems.

Introduction

Warm-starting is a technique used to accelerate the convergence of optimization algorithms by initializing them with a solution from a previous run. This is particularly beneficial when you have a sequence of related problems to solve.

Setup

Before you begin, ensure you have the necessary packages installed. You need the rehline library, which is used for regression with hinge loss, and numpy for numerical operations. Install these packages using pip if you haven’t already:

pip install rehline numpy

Simulating the Dataset

We first create a random dataset for classification:

import numpy as np

n, d, C = 1000, 3, 0.5
X = np.random.randn(n, d)
beta0 = np.random.randn(d)
y = np.sign(X.dot(beta0) + np.random.randn(n))

  • n is the number of samples.

  • d is the number of features.

  • C is a regularization parameter.

  • X is the feature matrix.

  • y is the target vector, generated as a sign function of a linear combination of features plus some noise.

Using ReHLine Solver

The ReHLine_solver is tested first with a cold start and then with a warm start:

from rehline._base import ReHLine_solver

U = -(C*y).reshape(1,-1)
V = (C*np.array(np.ones(n))).reshape(1,-1)
res = ReHLine_solver(X, U, V)  # Cold start
res_ws = ReHLine_solver(X, U, V, Lambda=res.Lambda)  # Warm start

  • Cold Start: The solver starts from scratch without any prior information.

  • Warm Start: The solver uses the solution from the cold start (res.Lambda) as the initial point for the next run.

Using ReHLine Class

The ReHLine class is used to fit a model:

from rehline import ReHLine

clf = ReHLine(verbose=1)
clf.C = C
clf.U = -y.reshape(1,-1)
clf.V = np.array(np.ones(n)).reshape(1,-1)
clf.fit(X)  # Cold start

clf.C = 2*C
clf.warm_start = 1
clf.fit(X)  # Warm start

  • Cold Start: The class is fitted with the initial data.

  • Warm Start: The class is fitted again with a different regularization parameter (2*C), using the previous solution as a starting point.

Using plqERM_Ridge

Finally, the plqERM_Ridge class is tested similarly:

from rehline import plqERM_Ridge

clf = plqERM_Ridge(loss={'name': 'svm'}, C=C, verbose=1)
clf.fit(X=X, y=y)  # Cold start

clf.C = 2*C
clf.warm_start = 1
clf.fit(X=X, y=y)  # Warm start