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An optimization technique that chooses a step size along a given direction to reduce the objective function.
DESCRIPTION
Line search methods complement direction-finding algorithms by selecting how far to move along the chosen descent direction. They range from exact line search to inexact schemes using conditions like Armijo and Wolfe. Backtracking line search is widely used for its simplicity and robustness. Proper step-size selection improves convergence speed and stability in gradient-based optimization for machine learning. Implementations often combine line search with methods such as conjugate gradients or quasi-Newton.
ALTERNATIVE DEFINITIONS
Wikipedia
In mathematical optimization, a line search method is an iterative procedure to choose a step size that determines how far to move along a given search direction.
Highlights step size selection along a search direction in iterative optimization.
MathWorks
Line search algorithms determine a step size that satisfies specific criteria to ensure sufficient decrease of the objective.
Emphasizes sufficient decrease conditions used in practice.
Boyd & Vandenberghe
Backtracking line search chooses the step length by repeatedly reducing it until a sufficient decrease condition is satisfied.
Defines backtracking as a practical inexact line search strategy.
RELATED TERMS
Gradient Descent Wolfe Conditions Armijo Rule Backtracking Newton’s Method Conjugate Gradient Trust Region Stride Strong Wolfe Conditions Convex Optimization
COMPETING TERMS
Trust-Region Methods
RELATIONS
- Broader: Search
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer MLOps Engineer Software Developer
EXTERNAL SOURCES
- Wikipedia - Line search
- Convex Optimization (Boyd & Vandenberghe) - Canonical textbook describing line search
- Northwestern Optimization - Educational overview of line search
- MathWorks - Practical line search algorithms
TOP QUESTIONS
Why is Line Search used in gradient-based optimization?
It stabilizes convergence by selecting step sizes that reduce the objective sufficiently, avoiding overshooting or slow progress from poor fixed step choices.
Category: Relevancy
What is the Armijo condition in Line Search?
It ensures sufficient decrease in the objective for a given step size, forming the basis of backtracking line search.
Category: Definition
How do Wolfe conditions extend Armijo?
Wolfe conditions add a curvature requirement, balancing sufficient decrease with appropriate directional derivative behavior.
Category: Distinction
When should Trust-Region methods be preferred over Line Search?
Trust-region may perform better in highly nonlinear problems or ill-conditioned scenarios by explicitly controlling step regions.
Category: Opinion
How does Line Search interact with Newton’s Method?
Damped Newton often uses line search to scale the Newton step when far from the optimum or with nonconvex objectives.
Category: Integration
What are common mistakes with Line Search implementations?
Incorrect gradient calculations, failing to update step sizes properly, and overly strict conditions that cause unnecessary function evaluations.
Category: Mistakes
Is exact Line Search practical in machine learning?
Rarely, due to high cost; inexact methods like backtracking and Wolfe conditions are favored for efficiency.
Category: Opinion
How to tune Line Search parameters?
Adjust initial step size, reduction factors, and condition constants; monitor function evaluations and convergence rate.
Category: HowTo
What is a striking analogy for Line Search?
Like walking downhill: you choose direction first, then decide how big a step to take to reduce altitude safely.
Category: Example