Convex optimization

See also: Machine learning terms

Introduction

Convex optimization is a subfield of mathematical optimization that deals with the minimization (or maximization) of convex functions over convex sets. In the context of machine learning, convex optimization plays a crucial role in finding the best model parameters, given a particular training dataset and a loss function. This field has gained significant attention in recent years, as it provides reliable and efficient algorithms for solving various machine learning problems, including linear regression, logistic regression, and support vector machines.

Convex Functions and Convex Sets

Convex Functions

A real-valued function f(x) is said to be convex if, for all points x and y in its domain and any t in the range [0, 1], the following inequality holds:

f(tx + (1-t)y) <= tf(x) + (1-t)f(y)

In other words, a function is convex if the line segment connecting any two points on the graph of the function lies above the graph itself. Convex functions have important properties, such as a unique global minimum, which make them amenable to optimization algorithms.

Convex Sets

A set S in a Euclidean space is convex if, for any two points x and y in S and any t in the range [0, 1], the point tx + (1-t)y also belongs to S. Intuitively, a set is convex if the line segment connecting any two points within the set is entirely contained within the set.

Convex Optimization Problems

A convex optimization problem is defined as minimizing (or maximizing) a convex objective function subject to convex constraints. Formally, the problem can be expressed as:

minimize f(x) subject to g_i(x) <= 0, for i = 1, ..., m and h_j(x) = 0, for j = 1, ..., p

where f(x) is a convex objective function, g_i(x) are convex inequality constraints, and h_j(x) are affine equality constraints.

Convex Optimization in Machine Learning

Machine learning involves learning a model from a given dataset by minimizing a loss function that quantifies the discrepancy between the model's predictions and the true target values. In many cases, the loss function and the constraints are convex, which allows the application of convex optimization techniques to find the optimal model parameters. Some notable machine learning algorithms that employ convex optimization include:

• Linear regression
• Logistic regression
• Support vector machines
• Lasso regression

Explain Like I'm 5 (ELI5)

Imagine you have a bowl, and you drop a marble inside it. The marble will naturally roll to the lowest point in the bowl. In convex optimization, we want to find the lowest point in a "bowl-shaped" curve (called a convex function) because that's where our machine learning model works the best. Convex optimization is like a set of rules that helps the marble find the lowest point in the bowl quickly and efficiently.