mlsanityData to 7 Oct 2026

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

arXiv 2507.03404 ↗First posted 4 Jul 2025cs.LG math.OCcited by 2

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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.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 31 of its 71 references that are in the map. First posted on arXiv 4 Jul 2025.

Builds on

31 of 71 references.

References · 31

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  11. 2014Global convergence of the Heavy-ball method for convex optimizationcited by 60
  12. 2023High-dimensional analysis of double descent for linear regression with random projectionscited by 47
19 more
  1. 2021Last iterate convergence of SGD for Least-Squares in the Interpolation regimecited by 41
  2. 2020Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Modelcited by 39
  3. 2015From Averaging to Acceleration, There is Only a Step-sizecited by 24
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  5. 2019A Universally Optimal Multistage Accelerated Stochastic Gradient Methodcited by 17
  6. 2015Exact Worst-case Performance of First-order Methods for Composite Convex Optimizationcited by 16
  7. 2018Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterationscited by 15
  8. 2021A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossipcited by 15
  9. 2019Implicit Regularization of Accelerated Methods in Hilbert Spacescited by 14
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  11. 2023Provable non-accelerations of the heavy-ball methodcited by 11
  12. 2022Tight Convergence Rate Bounds for Optimization Under Power Law Spectral Conditionscited by 9
  13. 2022Accelerated SGD for Non-Strongly-Convex Least Squarescited by 9
  14. 2025Dimension-adapted Momentum Outscales SGDcited by 7
  15. 2024Acceleration by Random Stepsizes: Hedging, Equalization, and the Arcsine Stepsize Schedulecited by 5
  16. 2025A Simplified Analysis of SGD for Linear Regression with Weight Averagingcited by 4
  17. 2024SGD with memory: fundamental properties and stochastic accelerationcited by 2
  18. 2022Acceleration of Gossip Algorithms through the Euler-Poisson-Darboux Equationcited by 1
  19. 2024Strong Convergence of FISTA Iterates under H"olderian and Quadratic Growth Conditionscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Towards Robust Scaling Laws for Optimizerscited by 4
  2. 2026Power-Law Spectrum of the Random Feature Modelcited by 3