Convex Optimization Is Free When Accuracy Is Expensive
Abstract
This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like δ−γ in the accuracy δ. When γ>2, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on γ, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy δ by an unbiased estimator of it, whose variance σ2 becomes a second, independently priced dial: the cost of one call drops from δ−γ to δ2−γσ−2. Plain inexact gradient descent driven by that oracle reaches loss ε at expected compute Θ(ε−γ) in the convex case, against Θ(ε−(γ+1)) for the same method run at a fixed accuracy: randomization buys a full power of ε. Under μ-strong convexity the exponent halves, to ε−γ/2, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.
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BibTeX
@misc{paing2026convex,
title = {{Convex Optimization Is Free When Accuracy Is Expensive}},
author = {Arthur Paing and Arthur Jacot},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35418},
eprint = {2609.35418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35418},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Convex Optimization Is Free When Accuracy Is Expensive
AU - Arthur Paing
AU - Arthur Jacot
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35418
UR - https://arxiv.org/abs/2609.35418
AB - This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like δ^−γ in the accuracy δ. When γ>2, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on γ, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy δ by an unbiased estimator of it, whose variance σ^2 becomes a second, independently priced dial: the cost of one call drops from δ^−γ to δ^2−γσ^−2. Plain inexact gradient descent driven by that oracle reaches loss ε at expected compute Θ(ε^−γ) in the convex case, against Θ(ε^−(γ+1)) for the same method run at a fixed accuracy: randomization buys a full power of ε. Under μ-strong convexity the exponent halves, to ε^−γ/2, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.
ER -