On the Effectiveness of the z-Transform Method in Quadratic Optimization
Abstract
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.
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BibTeX
@misc{bach2026effectiveness,
title = {{On the Effectiveness of the z-Transform Method in Quadratic Optimization}},
author = {Francis Bach},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05143189v3},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Effectiveness of the z-Transform Method in Quadratic Optimization
AU - Francis Bach
PY - 2026
DA - 2026/09/04
PB - HAL
UR - https://hal.science/hal-05143189v3
AB - 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.
ER -