# Continuous-time Lower Bounds for Gradient-based Algorithms

arXiv: https://arxiv.org/abs/2002.03546 · HTML: https://mlsanity.com/paper/2002.03546/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

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

- [On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems](https://mlsanity.com/paper/2107.08225/) (2021)
- [Towards a Systems Theory of Algorithms](https://mlsanity.com/paper/2401.14029/) (2024)
- [Accelerated First-Order Optimization under Nonlinear Constraints](https://mlsanity.com/paper/2302.00316/) (2023)
- [Distributed Event-Based Learning via ADMM](https://mlsanity.com/paper/2405.10618/) (2024)
- [Controlling Participation in Federated Learning with Feedback](https://mlsanity.com/paper/2411.19242/) (2024)
- [Convergence Properties of Natural Gradient Descent for Minimizing KL Divergence](https://mlsanity.com/paper/2504.19259/) (2025)
- [A Systems-Theoretic View on the Convergence of Algorithms under Disturbances](https://mlsanity.com/paper/2512.17598/) (2025)

## Abstract (arXiv)

This article derives lower bounds on the convergence rate of continuous-time gradient-based optimization algorithms. The algorithms are subjected to a time-normalization constraint that avoids a reparametrization of time in order to make the discussion of continuous-time convergence rates meaningful. We reduce the multi-dimensional problem to a single dimension, recover well-known lower bounds from the discrete-time setting, and provide insight into why these lower bounds occur. We present algorithms that achieve the proposed lower bounds, even when the function class under consideration includes certain nonconvex functions.