In (unconstrained) mathematical optimization, a backtracking line search is a line search method to determine the amount to move along a given search direction. Its use requires that the objective function is differentiable and that its gradient is known.

The method involves starting with a relatively large estimate of the step size for movement along the line search direction, and iteratively shrinking the step size (i.e., "backtracking") until a decrease of the objective function is observed that adequately corresponds to the amount of decrease that is expected, based on the step size and the local gradient of the objective function. A common stopping criterion is the Armijo–Goldstein condition.

Backtracking line search is typically used for gradient descent (GD), but it can also be used in other contexts. For example, it can be used with Newton's method if the Hessian matrix is positive definite.

Given a starting position ${\displaystyle \mathbf {x} }$ and a search direction ${\displaystyle \mathbf {p} }$, the task of a line search is to determine a step size ${\displaystyle \alpha >0}$ that adequately reduces the objective function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ (assumed ${\displaystyle C^{1}}$ i.e. continuously differentiable), i.e., to find a value of ${\displaystyle \alpha }$ that reduces ${\displaystyle f(\mathbf {x} +\alpha \,\mathbf {p} )}$ relative to ${\displaystyle f(\mathbf {x} )}$. However, it is usually undesirable to devote substantial resources to finding a value of ${\displaystyle \alpha }$ to precisely minimize ${\displaystyle f}$. This is because the computing resources needed to find a more precise minimum along one particular direction could instead be employed to identify a better search direction. Once an improved starting point has been identified by the line search, another subsequent line search will ordinarily be performed in a new direction. The goal, then, is just to identify a value of ${\displaystyle \alpha }$ that provides a reasonable amount of improvement in the objective function, rather than to find the actual minimizing value of ${\displaystyle \alpha }$.

The backtracking line search starts with a large estimate of ${\displaystyle \alpha }$ and iteratively shrinks it. The shrinking continues until a value is found that is small enough to provide a decrease in the objective function that adequately matches the decrease that is expected to be achieved, based on the local function gradient ${\displaystyle \nabla f(\mathbf {x} )\,.}$

Define the local slope of the function of ${\displaystyle \alpha }$ along the search direction ${\displaystyle \mathbf {p} }$ as ${\displaystyle m=\nabla f(\mathbf {x} )^{\mathrm {T} }\,\mathbf {p} =\langle \nabla f(\mathbf {x} ),\mathbf {p} \rangle }$ (where ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the dot product). It is assumed that ${\displaystyle \mathbf {p} }$ is a vector for which some local decrease is possible, i.e., it is assumed that ${\displaystyle m<0}$.

Based on a selected control parameter ${\displaystyle c\,\in \,(0,1)}$, the Armijo–Goldstein condition tests whether a step-wise movement from a current position ${\displaystyle \mathbf {x} }$ to a modified position ${\displaystyle \mathbf {x} +\alpha \,\mathbf {p} }$ achieves an adequately corresponding decrease in the objective function. The condition is fulfilled, see Armijo (1966), if ${\displaystyle f(\mathbf {x} +\alpha \,\mathbf {p} )\leq f(\mathbf {x} )+\alpha \,c\,m\,.}$

This condition, when used appropriately as part of a line search, can ensure that the step size is not excessively large. However, this condition is not sufficient on its own to ensure that the step size is nearly optimal, since any value of ${\displaystyle \displaystyle \alpha }$ that is sufficiently small will satisfy the condition.