Interactive
The speed–robustness trade-off
Momentum makes gradient methods fast — and fragile. This page lets you feel both. Tune a first-order method on an ill-conditioned quadratic, corrupt its gradients, and watch what speed costs.
Start with the presets: gradient descent is slow but placid; the tuned heavy-ball method converges at the accelerated rate but rings like a struck bell. Then raise the momentum slider with the stepsize fixed and watch the readout — the rate ρ improves and then worsens, while the amplification J only ever grows. Past the sweet spot, every increment of speed is bought with fragility. That inequality is a theorem, not an accident of this example.
Then switch the gradient error to adversarial. The disturbance injected is not random: it is a decaying cosine at the algorithm’s own resonant frequency — the worst-case shape that H∞ analysis singles out — and a small amount of it does what much larger random noise cannot.
Interactive · the speed–robustness trade-off
- Rate ρ
- 0.775 per iteration — smaller is faster
- Amplification J
- 0.346 H₂ norm — smaller is more robust
- Iterations to 1e−8
- 39 with the current gradient error
ρ is the spectral radius of the iteration matrix and J is the exact H₂ norm, obtained from the discrete Lyapunov equation — both computed live, not fitted. Push the momentum up and watch ρ fall while J rises: past a point, every increment of speed is bought with fragility. The adversarial setting injects a decaying cosine at the algorithm’s resonant frequency, the shape worst-case analysis singles out.
Speed times robustness is bounded below: the frontier can be traced, and designed on, but not escaped.
The mathematics behind this page — the Pareto frontier, the risk-averse tuning, and the worst-case noise construction — is developed in SIAM J. Optim. 2020, Optim. Methods Softw. 2025, and Math. Oper. Res. 2025; the wider research program is on the research page.
To reference this page:
@misc{gurbuzbalaban2026playground,
title = {The speed--robustness trade-off, interactively},
author = {G{\"u}rb{\"u}zbalaban, Mert},
year = {2026},
howpublished = {\url{https://mert-g.org/playground/}}
}