# Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets

arXiv: https://arxiv.org/abs/2610.13590 · HTML: https://mlsanity.com/paper/2610.13590/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 23 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 11 of 11)

- [Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets](https://mlsanity.com/paper/1406.1305/) (2014)
- [The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle](https://mlsanity.com/paper/1309.5550/) (2013)
- [Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory](https://mlsanity.com/paper/1407.5144/) (2014)
- [Info-Greedy sequential adaptive compressed sensing](https://mlsanity.com/paper/1407.0731/) (2014)
- [Restarting Frank-Wolfe: Faster Rates Under H"olderian Error Bounds](https://mlsanity.com/paper/1810.02429/) (2018)
- [Lower Bounds for Frank-Wolfe on Strongly Convex Sets](https://mlsanity.com/paper/2602.04378/) (2026)
- [Lower Bounds for Linear Minimization Oracle Methods Optimizing over Strongly Convex Sets](https://mlsanity.com/paper/2602.22608/) (2026)
- [Curvature-Dependent Lower Bounds for Frank-Wolfe](https://mlsanity.com/paper/2605.10595/) (2026)
- [A Linearly Convergent Projection-Free Algorithm for Smooth Convex Sets](https://mlsanity.com/paper/2608.04321/) (2026)
- [Mastermind with a Linear Number of Queries](https://mlsanity.com/paper/2011.05921/) (2020)
- [Frank-Wolfe Beyond 1/t Convergence](https://mlsanity.com/paper/2604.28006/) (2026)

## Abstract (arXiv)

Recent results [arXiv:2602.04378, arXiv:2602.22608] have established worst-case objective-error lower bounds of order 1/T2 after T linear minimization oracle (LMO) calls, matching the upper bound [arXiv:1406.1305] in its dependence on T and thereby resolving a long-standing open question. Upon close inspection, however, both lower bounds are somewhat brittle. The first [arXiv:2602.04378] applies only to the standard Frank--Wolfe algorithm, and, as we show, its prescribed slow exact-line-search trajectories can amplify sufficiently small initialization errors exponentially, corresponding to roughly one additional bit of initialization accuracy per iteration. The second [arXiv:2602.22608] obtains lower bounds for any deterministic LMO-based first-order method via a resisting oracle that adversarially adapts to the queries of the algorithm. In both cases it is conceivable that randomization may circumvent these lower bounds; in particular, neither establishes a lower bound for unrestricted randomized first-order LMO methods. We close this gap by proving an Omega(Loperatornamediam(S)2/T2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. While our construction reuses the family of [arXiv:2602.22608], our proof characterizes the conditional distribution of the hidden permutation after adaptive queries and bounds the expected decrease of a combinatorial statistic controlling optimizer variance. We further show that the same error threshold holds with probability at least 3/4 and yields a stopping-time tail bound.