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

arXiv:2610.13590 · math.OC, cs.LG · Submitted 2026-10-09 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets".

Jane: The gist The authors prove an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets,

Tom: First, who's behind it and why it matters.

Title and authors: Jane: So we've talked about what they proved generally—that randomization doesn't help the order of convergence. Now let's look at what this specific paper, "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets," actually does in detail.

Tom: It sets up a very specific scenario involving a hard family of instances based on coordinate permutations. They define an objective function f(x) that is one-strongly convex, and they characterize the geometry of these sets using Lemma two point two, where every set S pi is one-strongly convex and its diameter is 2C(two - sqrt two) <ref:2610.13590#pg1>.

Lu: That geometric setup is key because it defines the space they are working in. They are focusing on an objective function defined as f(x) = one/two x - M one squared, where M depends on a parameter rho <ref:2610.13590#pg1>.

Meng: So this isn't just any smooth convex set; it’s a very specific, hard-to-solve one they are using to test the limits of the method.

Tom: And then they establish the information-theoretic lower bound derivation by bounding the expected decrease of a combinatorial statistic through which optimizer variance is controlled. The key estimate there bounds this as E

t - t+one T R t: 6m.

Jane: That statistic is what they use to control the conditional optimizer variance, and they show that by choosing d to be a large constant multiple of T+one you're left with substantial conditional optimizer variance <ref:2610.13590#pg1>.

Lu: And that variance gets converted into optimization error via a common optimal value argument, leading them to the final distributional lower bound: E,U

f(T) - f: c diam(S) squared (T+one) squared <ref:2610.13590#pg1>.

Meng: This is what I mean when I say they are setting up a distribution of instances that they know are hard. It’s not just some random problem; it’s a constructed family based on coordinate permutations.

The paper's summary: Tom: Let's summarize the main contribution of "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets." They show that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points.

Jane: So they prove this by providing a hard distribution of instances and establishing a corresponding distributional lower bound that then by Yao’s principle gives them a randomized lower bound.

Lu: The core mechanism involves following the general information-theoretic approach from Braun and Pokutta, where they bound the expected information gain in each iteration. They use a combinatorial statistic through which they lower-bound the conditional optimizer variance.

Meng: They retain the feasible sets, common quadratic objective, optimizer and thresholding formula from Grimmer and Liu for converting that variance into loss. But their departure is the fixed uniform prior over their feasible sets, which they call our hard distribution.

Tom: In contrast to a resisting oracle setup where you can respond adversarially to the queries of the algorithm, in this case we can no longer respond adversarially to.

Jane: They also characterize the exact LMO transcripts using Lemma three point one, which shows that the pair (p, z) determines lambda and tau, every active rank pi i, and inactive lower bounds. Plus Lemma three point two describes the form of every conditional posterior under a fixed seed and transcript.

Lu: They define a combinatorial statistic to account for row-dependent restrictions arising from the compatible assignments. And Lemma four point one provides a lower bound on rank variance under nested lower bounds, showing sum i Var

sqrt r (i): two/12m.

Meng: So they are getting very granular here, looking at the variance in the ranks under these constraints. It’s a lot of detail to prove that this specific distribution yields that lower bound.

The paper's improvements: Tom: The paper suggests a few things for improvement or for clarifying the results. One major thing is showing that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points.

Jane: They achieve this by providing a hard distribution of instances and establishing a corresponding distributional lower bound that then by Yao’s principle provides a randomized lower bound, which they obtain as E

f(T) - f: = (Ldiam(S) squared (T+one) two) <ref:2610.13590#pg1>.

Lu: They also establish constant probability of success. Theorem two point nine shows that under the specified assumptions, the probability of getting a gap at least c zero Ldiam(S) squared (T+one) squared is at least three/four.

Meng: And they have a stopping time tail bound too, Corollary two point one zero shows that if the probability of returning an output whose gap is at least the threshold in Theorem two point nine is at most eta, then T three/four - eta.

Tom: They also show that there are instances requiring T = ((Ldiam(S) two/epsilon) one/two) queries to guarantee an expected gap at most epsilon in Corollary two point eight.

Jane: And they do touch on initialization sensitivity with the analysis of Halbey et al.'s work, showing that the sensitivity of exact-line-search backward construction leads to a linearized precision requirement B lin(T, eta) = two T eta = T + two(one/eta) + O(T) <ref:2610.13590#pg1>.

Lu: And they also prove something about initialization distributions. Theorem A.four shows that the set of initializations whose trajectories never reach p and eventually satisfy the slow-envelope condition has two-dimensional Lebesgue measure zero.

Meng: That means any initialization distribution absolutely continuous with respect to area assigns probability zero to this set, which is pretty strong for ensuring practical success in a randomized setting.

Conclusion: Tom: So, to wrap up on "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets," the main point is that randomization cannot improve the corresponding order of convergence. The paper establishes an (Ldiam(S) two/T two) distributional lower bound <ref:2610.13590#pg1>.

Jane: This means that even with randomization, you can't get better than that rate in terms of objective error for unrestricted first-order LMO methods over strongly convex sets. It sets a clear boundary on what is achievable.

Lu: The implication is that the restriction being made—the uniform prior fixed before the seed—is crucial because it allows them to establish this hard distribution, which then leads to a randomized lower bound via Yao’s principle.

Meng: For deployment, knowing this means you can't rely on randomization to magically jump up the convergence rate beyond what is theoretically possible for this setup. It gives you a concrete floor.

Tom: Yeah, and they also show that deterministic methods can be converted into general LMO-based methods using a resisting oracle, which is different from the randomized lower bounds where the instance is fixed independently of the realized seed.

Jane: So, in short, this paper establishes a distributional lower bound of (Ldiam(S) two/T two) for unrestricted first-order LMO methods over strongly convex sets <ref:2610.13590#pg1>. It's a firm statement about the limits of what randomization can achieve here.

Lu: And it provides that hard distribution of instances, which is the key to translating that theoretical bound into a randomized result using Yao's principle.

Meng: That's all for this one. We have a lot more on arXiv, so we'll see you next time.

Sebastian Pokutta

Zuse Institute Berlin · TU Berlin

math.OC, cs.LG

Submitted: 2026-10-09

Updated: 2026-10-09

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The gist The authors prove an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets, demonstrating that randomization

Key concepts

LMO Oracle Model: This model describes optimization where each query returns the function value, gradient, and information about the next step's oracle call. It formalizes how an algorithm interacts with an external oracle that provides linear-like information about the objective function $f(x)$. The analysis focuses on how this specific type of query structure affects convergence speed.
Distributional Lower Bound: This is a mathematical proof showing that for a given set of problem instances, the expected performance (error) of any randomized algorithm is bounded from below. The authors establish a lower bound on the expected error $\mathbb{E}[f(\hat{x}_T) - f_{\star}]$ that holds over all possible random choices made by the algorithm.
Coordinate Permutation Hardness: The authors create a difficult family of instances by using permutations of coordinate weights. This construction ensures that the geometry of the set $S$ is sufficiently complex, leading to strong convexity properties and providing a hard setting where randomized methods struggle to achieve faster convergence.

Summary

The gist The authors prove an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets, demonstrating that randomization cannot improve the corresponding order of convergence and establishing a gap between deterministic and randomized bounds.

Model and Setup

The problem considered is solving min x∈S f(x), where f is an L-smooth convex function and S is a compact convex set. The oracle model involves queries that return yt ≐ (f(xt), ∇f(xt), zt+1) where zt+1 ≐ LMOs(pt). The authors define a hard family of instances based on coordinate permutations, where the hidden instance is determined by a permutation of known coordinate weights. This construction leads to an objective function f(x) ≐ 1/2‖x − M1‖22 where M is defined via a parameter rho. The geometry of these sets is characterized by Lemma 2.2, stating that every Spi is 1-strongly convex and diam(Spi) = 2C(2 − √2).

Information-Theoretic Lower Bound Derivation

The proof employs a general information-theoretic approach to bound the expected decrease of a combinatorial statistic Δ through which optimizer variance is controlled. The key estimate bounds the expected decrease in one query as E[Δt − Δt+1 ∣ TRt] ≤ 6m. Since the initial potential is quadratic in d, while the expected decrease per query is at most linear in d, choosing d to be a sufficiently large constant multiple of T + 1 leaves substantial conditional optimizer variance. This variance is converted into optimization error via a common optimal value argument. The final distributional lower bound is established as EΠ,U[f(̂xT) − f⋆] ≥ c diam(SΠ)2(T + 1)2.

Conditional Distributions and Variance Control

The exact LMO transcripts are characterized by Lemma 3.1, which shows that the pair (p, z) determines lambda, tau, every active rank pi, and inactive lower bounds. Lemma 3.2 describes the form of every conditional posterior under a fixed seed and transcript. The authors define a combinatorial statistic Δ(l) to account for row-dependent restrictions arising from the compatible assignments. Lemma 4.1 provides a lower bound on rank variance under nested lower bounds, showing ∑i Var[√rΣ(i)] ≥ Δ2/12m. Corollary 4.2 establishes that after fixing the seed, any fixed vector x feasible on every compatible instance satisfies E[f(x) − f(x(Π)⋆) ∣ h] ≥ ρ 2C 2Δ 2/48md.

Query Complexity and Probability Bounds

Corollary 2.8 shows that there are instances requiring T = Ω((Ldiam(S)2/epsilon)1/2) queries to guarantee expected gap at most ε. Theorem 2.9 establishes a constant-probability lower bound, showing that under the specified assumptions, PΠ,U[f(̂xT) − f(x(Π)⋆) ≥ c0 Ldiam(S)2(T + 1)2] ≥ 3/4. Corollary 2.10 provides a stopping-time tail bound, stating that if the probability of returning an output whose gap is at least the threshold in Theorem 2.9 is at most η T] ≥ 3/4 − η.

Sensitivity and Initialization Bounds

The analysis of Halbey et al. [2026a] shows that the sensitivity of exact-line-search backward construction leads to a linearized precision requirement Blin(T, η) ≐ log2 ΛT η = T + log2(1/η) + O(log T). Theorem A.4 proves that for exact-line-search Frank–Wolfe on the unit disk in R2, the set of initializations whose trajectories never reach p and eventually satisfy the slow-envelope condition has two-dimensional Lebesgue measure zero. This implies that an initialization distribution absolutely continuous with respect to area assigns probability zero to this set.

Conclusion on Method Class Distinction

The authors conclude that the LMO-span restriction is significant because deterministic methods can be converted into general LMO-based methods using a resisting oracle, whereas randomized lower bounds require an instance fixed independently of the realized seed. This distributional theorem rules out improvements for unrestricted randomized first-order LMO methods.

The paper establishes that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points and provides a hard distribution of instances and establishing a corresponding distributional lower bound that then by Yao’s principle [Yao, 1977] provides a randomized lower bound. The authors obtain E[f(̂xT) − f⋆] = Ω(Ldiam(S)2/T 2) for unrestricted first-order LMO methods over strongly convex sets.

How it works

The proof follows the general information-theoretic approach from Braun and Pokutta [2015], where they bound the (expected) information gain in each iteration. They use a combinatorial statistic Δ through which they lower-bound the conditional optimizer variance. The hidden instance is determined by a permutation of a known list of d coordinate weights. The key estimate bounds its expected decrease in one query as E[Δt − Δt+1 ∣ TRt] ≤ 6m.

Conditional Variance and Optimization Error

The authors define the following estimate to account for row-dependent restrictions from (22), which is Δ(l) ≐ m∑i(c i − 1) = m(m + 1)/2 − m∑il i. Lemma 4.1 provides a lower bound on rank variance under nested lower bounds, showing ∑i Var[√rΣ(i)] ≥ Δ2/12m. Corollary 4.2 establishes that after fixing the seed, any fixed vector x feasible on every compatible instance satisfies E[f(x) − f(x(Π)⋆) ∣ h] ≥ ρ 2C 2Δ 2/48md.

Expected Decrease in One Query

The authors bound the expected decrease of Δ in one LMO query by counting pairs of compatible assignments that differ by a single swap and produce different answers. The final result for the conditional average is E[Δ − Δ+] = 2 - 1/2 = 3/2.

Final Bounds

Theorem A.4 shows that the set of initializations whose trajectories never reach p and eventually satisfy the slow-envelope condition has two-dimensional Lebesgue measure zero. This implies an initialization distribution absolutely continuous with respect to area assigns probability zero to this set. Lemma A.8 supplies a point of E at every initial radius r0 ∈ (0, δ], by Lemma A.7. The rate lemma also gives t 2f(x t) = (r tt) squared → 9/16.

The authors establish a distributional lower bound for randomized first-order LMO methods over strongly convex sets, demonstrating that randomization cannot improve the corresponding order of convergence and establishing a gap between deterministic and randomized bounds. The paper establishes an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets.

The authors prove that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points and establishing a gap between deterministic and randomized bounds. The paper establishes an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. The authors prove that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points and establishing a gap between deterministic and randomized bounds. The paper establishes an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. The authors prove that randomization cannot improve the corresponding order of convergence, even without restrictions on query directions or intermediate evaluation points and establishing a gap between deterministic and randomized bounds. The paper establishes an Ω(Ldiam(S)2/T 2) distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets.

Improvements for AI systems

A randomized first-order LMO method can achieve an expected objective error of at least Ω(Ldiam(S)2/(T + 1)2) by utilizing a fixed uniform prior over coordinate permutations, which is superior to methods restricted to the LMO-span model that only guarantee a lower bound for LMO-span methods.
The improved system can achieve a constant probability of success, as Theorem 2.9 states: PΠ,U[f(̂xT) − f(x(Π)⋆) ≥ c0 Ldiam(S)2 (T + 1)2] ≥ 3/4.
The system can guarantee a stopping time tail bound, as Corollary 2.10 shows that for a threshold of at most η T] ≥ 3/4 − η.
The system can be designed to require optimal query complexity, as Corollary 2.8 establishes: There are instances requiring T = Ω((Ldiam(S)2/ε)1/2) queries to guarantee expected gap at most ε.
The system can utilize an information-theoretic approach by bounding the combinatorial statistic Δ through the estimate: E[Δt − Δt+1 ∣ TRt] ≤ 6m, where m is the number of coordinate–weight assignments not yet explicitly disclosed.
The system can leverage conditional distribution characterization, where conditional on the full transcript, including the seed, the induced reduced permutation Σ is uniformly distributed over the compatible assignments defined by Lemma 3.2.
The system can implement a mechanism to analyze initialization sensitivity using Theorem A.3, which bounds the trajectory error: ‖JT −1 ⋯ J0‖2 ≥ a2T(T + 1)−C.
The system can incorporate non-linear additive perturbation models, where Theorem A.5 provides a bound on the perturbed state distance: ‖̃zt − zt‖∞ ≤ ηr3t = O(η(t + 1)3).