# Optimization Methods for Supervised Machine Learning: From Linear Models to Deep Learning

arXiv: https://arxiv.org/abs/1706.10207 · HTML: https://mlsanity.com/paper/1706.10207/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2017.

## Built on by (most cited first, 7 of 7)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [First-Order Methods for Convex Optimization](https://mlsanity.com/paper/2101.00935/) (2021)
- [Feed-Forward Neural Networks as a Mixed-Integer Program](https://mlsanity.com/paper/2402.06697/) (2024)
- [The Stochastic Conjugate Subgradient Algorithm For Kernel Support Vector Machines](https://mlsanity.com/paper/2407.21091/) (2024)
- [Beyond First-Order: Training LLMs with Stochastic Conjugate Subgradients and AdamW](https://mlsanity.com/paper/2507.01241/) (2025)
- [A Retrospective Approximation Approach for Smooth Stochastic Optimization](https://mlsanity.com/paper/2103.04392/) (2021)
- [A stochastic first-order method with multi-extrapolated momentum for highly smooth unconstrained optimization](https://mlsanity.com/paper/2412.14488/) (2024)

## Abstract (arXiv)

The goal of this tutorial is to introduce key models, algorithms, and open questions related to the use of optimization methods for solving problems arising in machine learning. It is written with an INFORMS audience in mind, specifically those readers who are familiar with the basics of optimization algorithms, but less familiar with machine learning. We begin by deriving a formulation of a supervised learning problem and show how it leads to various optimization problems, depending on the context and underlying assumptions. We then discuss some of the distinctive features of these optimization problems, focusing on the examples of logistic regression and the training of deep neural networks. The latter half of the tutorial focuses on optimization algorithms, first for convex logistic regression, for which we discuss the use of first-order methods, the stochastic gradient method, variance reducing stochastic methods, and second-order methods. Finally, we discuss how these approaches can be employed to the training of deep neural networks, emphasizing the difficulties that arise from the complex, nonconvex structure of these models.