# Line Search for Convex Minimization

arXiv: https://arxiv.org/abs/2307.16560 · HTML: https://mlsanity.com/paper/2307.16560/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 7 references that are in the map.
- First posted on arXiv 31 Jul 2023.

## Builds on (most cited first, 1 of 1)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)

## Built on by (most cited first, 1 of 1)

- [Rethinking Early Stopping: Refine, Then Calibrate](https://mlsanity.com/paper/2501.19195/) (2025)

## Abstract (arXiv)

Golden-section search and bisection search are the two main principled algorithms for 1d minimization of quasiconvex (unimodal) functions. The first one only uses function queries, while the second one also uses gradient queries. Other algorithms exist under much stronger assumptions, such as Newton's method. However, to the best of our knowledge, there is no principled exact line search algorithm for general convex functions -- including piecewise-linear and max-compositions of convex functions -- that takes advantage of convexity. We propose two such algorithms: Delta-Bisection is a variant of bisection search that uses (sub)gradient information and convexity to speed up convergence, while Delta-Secant is a variant of golden-section search and uses only function queries. While bisection search reduces the x interval by a factor 2 at every iteration, Delta-Bisection reduces the (sometimes much) smaller x^*-gap Deltax (the x coordinates of Delta) by at least a factor 2 at every iteration. Similarly, Delta-Secant also reduces the x^*-gap by at least a factor 2 every second function query. Moreover, the y^*-gap Deltay (the y coordinates of Delta) also provides a refined stopping criterion, which can also be used with other algorithms. Experiments on a few convex functions confirm that our algorithms are always faster than their quasiconvex counterparts, often by more than a factor 2. We further design a quasi-exact line search algorithm based on Delta-Secant. It can be used with gradient descent as a replacement for backtracking line search, for which some parameters can be finicky to tune -- and we provide examples to this effect, on strongly-convex and smooth functions. We provide convergence guarantees, and confirm the efficiency of quasi-exact line search on a few single- and multivariate convex functions.