# On the Effectiveness of the z-Transform Method in Quadratic Optimization

arXiv: https://arxiv.org/abs/2507.03404 · HTML: https://mlsanity.com/paper/2507.03404/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 71 references that are in the map.
- First posted on arXiv 4 Jul 2025.

## Builds on (most cited first, 30 of 31)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes](https://mlsanity.com/paper/1805.10074/) (2018)
- [A Continuous-Time View of Early Stopping for Least Squares](https://mlsanity.com/paper/1810.10082/) (2018)
- [4+3 Phases of Compute-Optimal Neural Scaling Laws](https://mlsanity.com/paper/2405.15074/) (2024)
- [Harder, Better, Faster, Stronger Convergence Rates for Least-Squares Regression](https://mlsanity.com/paper/1602.05419/) (2016)
- [Global convergence of the Heavy-ball method for convex optimization](https://mlsanity.com/paper/1412.7457/) (2014)
- [High-dimensional analysis of double descent for linear regression with random projections](https://mlsanity.com/paper/2303.01372/) (2023)
- [Last iterate convergence of SGD for Least-Squares in the Interpolation regime](https://mlsanity.com/paper/2102.03183/) (2021)
- [Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model](https://mlsanity.com/paper/2006.08212/) (2020)
- [From Averaging to Acceleration, There is Only a Step-size](https://mlsanity.com/paper/1504.01577/) (2015)
- [Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law](https://mlsanity.com/paper/2505.19227/) (2025)
- [A Universally Optimal Multistage Accelerated Stochastic Gradient Method](https://mlsanity.com/paper/1901.08022/) (2019)
- [Exact Worst-case Performance of First-order Methods for Composite Convex Optimization](https://mlsanity.com/paper/1512.07516/) (2015)
- [Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations](https://mlsanity.com/paper/1805.08531/) (2018)
- [A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip](https://mlsanity.com/paper/2106.07644/) (2021)
- [Implicit Regularization of Accelerated Methods in Hilbert Spaces](https://mlsanity.com/paper/1905.13000/) (2019)
- [Provable non-accelerations of the heavy-ball method](https://mlsanity.com/paper/2307.11291/) (2023)
- [Dimension-adapted Momentum Outscales SGD](https://mlsanity.com/paper/2505.16098/) (2025)
- [A view of mini-batch SGD via generating functions: conditions of convergence, phase transitions, benefit from negative momenta](https://mlsanity.com/paper/2206.11124/) (2022)
- [Tight Convergence Rate Bounds for Optimization Under Power Law Spectral Conditions](https://mlsanity.com/paper/2202.00992/) (2022)
- [Accelerated SGD for Non-Strongly-Convex Least Squares](https://mlsanity.com/paper/2203.01744/) (2022)
- [Acceleration by Random Stepsizes: Hedging, Equalization, and the Arcsine Stepsize Schedule](https://mlsanity.com/paper/2412.05790/) (2024)
- [A Simplified Analysis of SGD for Linear Regression with Weight Averaging](https://mlsanity.com/paper/2506.15535/) (2025)
- [SGD with memory: fundamental properties and stochastic acceleration](https://mlsanity.com/paper/2410.04228/) (2024)
- [Acceleration of Gossip Algorithms through the Euler-Poisson-Darboux Equation](https://mlsanity.com/paper/2202.10742/) (2022)

## Built on by (most cited first, 3 of 3)

- [Towards Robust Scaling Laws for Optimizers](https://mlsanity.com/paper/2602.07712/) (2026)
- [Power-Law Spectrum of the Random Feature Model](https://mlsanity.com/paper/2603.14578/) (2026)
- [Asymptotic Risk of Learning-Rate Schedules for Least-Squares Regression with Discretely Distributed Features](https://mlsanity.com/paper/2610.16518/) (2026)

## Abstract (arXiv)

The z-transform of a sequence is a classical tool used within signal processing, control theory, computer science, and electrical engineering. It allows for studying sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its z-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.