# A frequency-domain analysis of inexact gradient methods

arXiv: https://arxiv.org/abs/1912.13494 · HTML: https://mlsanity.com/paper/1912.13494/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2019.

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

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- [Tight analyses of first-order methods with error feedback](https://mlsanity.com/paper/2506.05271/) (2025)
- [Exact Formulas for Finite-Time Estimation Errors of Decentralized Temporal Difference Learning with Linear Function Approximation](https://mlsanity.com/paper/2204.09801/) (2022)
- [From Inexact Gradients to Byzantine Robustness: Acceleration and Optimization under Similarity](https://mlsanity.com/paper/2602.03329/) (2026)

## Abstract (arXiv)

We study robustness properties of some iterative gradient-based methods for strongly convex functions, as well as for the larger class of functions with sector-bounded gradients, under a relative error model. Proofs of the corresponding convergence rates are based on frequency-domain criteria for the stability of nonlinear systems. Applications are given to inexact versions of gradient descent and the Triple Momentum Method. To further emphasize the usefulness of frequency-domain methods, we derive improved analytic bounds for the convergence rate of Nesterov's accelerated method (in the exact setting) on strongly convex functions.