# Beyond Discretization: Learning the Optimal Solution Path

arXiv: https://arxiv.org/abs/2410.14885 · HTML: https://mlsanity.com/paper/2410.14885/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 18 Oct 2024.

## Builds on (most cited first, 11 of 11)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [SGD and Hogwild! Convergence Without the Bounded Gradients Assumption](https://mlsanity.com/paper/1802.03801/) (2018)
- [Aiming towards the minimizers: fast convergence of SGD for overparametrized problems](https://mlsanity.com/paper/2306.02601/) (2023)
- [On Convergence-Diagnostic based Step Sizes for Stochastic Gradient Descent](https://mlsanity.com/paper/2007.00534/) (2020)
- [Which Lp norm is the fairest? Approximations for fair facility location across all "p"](https://mlsanity.com/paper/2211.14873/) (2022)
- [Safe Grid Search with Optimal Complexity](https://mlsanity.com/paper/1810.05471/) (2018)
- [Continuation Path with Linear Convergence Rate](https://mlsanity.com/paper/2112.05104/) (2021)
- [New Methods for Parametric Optimization via Differential Equations](https://mlsanity.com/paper/2306.08812/) (2023)

## Abstract (arXiv)

Many applications require minimizing a family of optimization problems indexed by some hyperparameter λ ∈ Lambda to obtain an entire solution path. Traditional approaches proceed by discretizing Lambda and solving a series of optimization problems. We propose an alternative approach that parameterizes the solution path with a set of basis functions and solves a single stochastic optimization problem to learn the entire solution path. Our method offers substantial complexity improvements over discretization. When using constant-step size SGD, the uniform error of our learned solution path relative to the true path exhibits linear convergence to a constant related to the expressiveness of the basis. When the true solution path lies in the span of the basis, this constant is zero. We also prove stronger results for special cases common in machine learning: When λ ∈ [-1, 1] and the solution path is nu-times differentiable, constant step-size SGD learns a path with ε uniform error after at most O(ε^frac11-nu log(1/ε)) iterations, and when the solution path is analytic, it only requires Oleft(log2(1/ε)loglog(1/ε)right). By comparison, the best-known discretization schemes in these settings require at least O(ε-1/2) discretization points (and even more gradient calls). Finally, we propose an adaptive variant of our method that sequentially adds basis functions and demonstrates strong numerical performance through experiments.