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Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper establishes curvature-independent O(√T) and O(log T) regret bounds for Riemannian online gradient descent on Hadamard manifolds, provided the losses are h-convex or strongly h-convex, respectively.

desk verdict A short, mostly sound paper that gives the first online regret analysis for h-convex functions, but the headline O(√T) bound rides on two unproven imports from a preprint; worth refereeing, not desk-rejecting. read the letter →

arxiv 2509.11236 v1 pith:B7JAEDXD submitted 2025-09-14 cs.LG math.OCstat.ML

Emre Sahinoglu , Shahin Shahrampour This is my paper
classification cs.LGmath.OCstat.ML MSC 52A4153C2290C2568W27
keywords h-convexityBusemannfunctionsHadamardmanifoldsonlinegradientdescentregretboundscurvature-independentTyler'sM-estimatorFréchetmean
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that online gradient descent on curved spaces need not pay a penalty for curvature. It proves that for h-convex losses—functions built from Busemann functions, the manifold analogue of affine planes—the cumulative regret is O(√T), and for µ-strongly h-convex losses it improves to O(log T). These rates exactly match Euclidean online convex optimization, with constants depending only on the Lipschitz constant, the set diameter, and the strong-convexity parameter, not on the sectional curvature. A sympathetic reader should care because prior guarantees for geodesically convex functions degraded with curvature, and this result identifies a structural condition under which that degradation disappears.

What carries the argument

The central object is the Busemann function B_{p,v}(x)=lim_{t→∞}(d(γ(t),x)−t), which generalizes affine functions to Hadamard manifolds. A function is h-convex if at every point y it is bounded below by some scaled Busemann function B_{y,v}, and it is µ-strongly h-convex if it is bounded below by a quadratic expression Q_{y,v}^µ(x)=−‖v‖²/(2µ)+(µ/2)d²(Exp_y(−v/µ),x). These definitions let the regret proof replace the Euclidean inner-product expansion with Busemann-function identities and a quadratic inequality from the cited preprint [9]. For the strong case, a newly proved Stewart-type inequality for geodesic triangles (Lemma 2.4) is used to telescope the distance terms. The ROGD update itse

What would settle it

Take a known h-convex function on a Hadamard manifold, such as f(Σ)=log(a^T Σ^{-1}a) on SPD(n), and directly verify whether the inequality f(x)−f(y) ≥ B_{y,grad f(y)}(x) holds for all x,y in the feasible set; a single counterexample would invalidate the first step of Theorem 3.3's proof. More generally, one could search for an h-convex function whose Busemann subgradient at some point is not collinear with the Riemannian gradient.

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Extended reading notes

Core claim

The paper's central claim is that the Riemannian Online Gradient Descent algorithm achieves Euclidean-level static regret guarantees on Hadamard manifolds when the loss functions are h-convex. Theorem 3.3 gives O(√T) regret for h-convex losses with step size 1/√T, and Theorem 3.5 gives O(log T) regret for µ-strongly h-convex losses with step size 1/(µt). Both bounds are independent of the sectional curvature, unlike earlier g-convex analyses in [31] that carried curvature-dependent constants. The proof uses the outer characterization of convexity via Busemann functions: the defining inequality f(x)−f(y) ≥ B_{y,v}(x) is combined with a quadratic estimate for Busemann functions (Lemma 2 of [9]

Load-bearing premise

The load-bearing premise is that the tangent vector appearing in the definitions of h-convexity and strong h-convexity is always the Riemannian gradient of the loss; the paper relies on a cited preprint for this identification and does not prove it itself.

Editorial extensions

If this is right

  • If h-convexity holds, online Riemannian optimization on Hadamard manifolds achieves O(√T) static regret with no curvature-dependent constants, matching the Euclidean baseline.
  • For µ-strongly h-convex objectives, the same algorithm achieves O(log T) regret, matching the Euclidean strongly-convex rate.
  • The results apply directly to concrete problems on the SPD manifold: online Tyler's M-estimation (h-convex) and online Fréchet mean estimation (1-strongly h-convex), both demonstrated numerically.
  • Since no lower bound on sectional curvature is assumed, the guarantees hold uniformly across all Hadamard manifolds, including those with unbounded negative curvature.
  • The curvature-free bound suggests that curvature was an artifact of the g-convexity analysis rather than an intrinsic obstacle to fast online learning on manifolds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's proof structure suggests that extending ROGD to stochastic or decentralized settings is likely to preserve the curvature-independent rates, because the curvature terms were removed via Busemann-function inequalities rather than gradient clipping or projection weighting.
  • If h-convexity is indeed the right lens, then the practical bottleneck shifts to computing Busemann-type subgradients for a given loss; for many problems such as Tyler's M-estimator these have simple closed forms, but for general losses the gradient identification remains a nontrivial geometric question.
  • The Stewart-type triangle inequality (Lemma 2.4) may find independent use in other geometric optimization and convex analysis proofs on Hadamard spaces, as a clean substitute for curvature-dependent comparison theorems.
  • A concrete testable extension would be to check whether the same curvature-free regret bounds survive under bandit (gradient-free) feedback, where one must estimate Busemann subgradients from function values; the paper does not address this case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies Riemannian online gradient descent (ROGD) on Hadamard manifolds under a notion of horospherical convexity (h-convexity) defined via Busemann functions. It claims O(√T) static regret for h-convex losses and O(log T) static regret for µ-strongly h-convex losses, with constants that do not depend on sectional curvature. The proof of Theorem 3.3 imports a quadratic Busemann inequality from the preprint [9], and the proof of Theorem 3.5 uses an in-paper Stewart-type inequality for Hadamard triangles (Lemma 2.4). Experiments on SPD(n) with the affine-invariant metric test online Tyler M-estimation and online Fréchet mean computation.

Significance. If the imported technical results hold, the paper provides a meaningful extension of online Riemannian optimization from geodesic convexity to h-convexity, removing curvature-dependent constants from the regret bounds and matching Euclidean rates. The in-paper algebra is sound: Lemma 2.4 is correctly proved, and the telescoping argument in Theorem 3.5 checks out. The strong h-convexity definition is a natural Busemann-analogue of Euclidean strong convexity. The main weakness is not the internal derivation but a load-bearing dependence on two unstated results from the unpublished preprint [9] and an implicit identification of the subgradient vector with the Riemannian gradient. These issues are fixable within the scope of the paper.

major comments (2)
  1. [Section III.C, proof of Theorem 3.3] The proof uses two results from [9] without stating them: (i) Proposition 2(iii), which identifies the vector in Definition 2.2 with the Riemannian gradient g_t, and (ii) Lemma 2, which gives the quadratic estimate -B_{x_t,g_t}(x*) = -B_{x_t,ηg_t}(x*)/η ≤ (|x̃_{t+1}x_t|^2 + |x*x_t|^2 - |x*x̃_{t+1}|^2)/(2η). The entire O(√T) guarantee depends on these imported statements. Since [9] is an arXiv preprint and the lemma is not reproduced, the bound cannot be verified from the manuscript. If Lemma 2 of [9] has any additional hypothesis (e.g., a restriction on η||g_t||, a differentiability assumption, or a requirement on x*), the displayed inequality may fail. Please state Lemma 2 in full with its hypotheses and either prove it or give a precise, verifiable reference; do the same for the gradient-identification result.
  2. [Section II.B / Algorithm 1 and Assumption 3.2] Algorithm 1 requires the Riemannian gradient g_t = grad f_t(x_t), but Assumption 3.2 only assumes Lipschitz continuity. Definitions 2.2 and 2.3 guarantee only the existence of some tangent vector v at each point y satisfying the relevant inequality, not differentiability or uniqueness. The proofs of Theorems 3.3 and 3.5 implicitly set v = g_t and use ||g_t|| ≤ L without stating or proving that this is the vector supplied by the h-convexity definition. Either add differentiability to Assumption 3.2, or reformulate the update in terms of a horospherical subgradient v_t and prove that the chosen v_t satisfies the same norm bound. As written, the chain from definitions to regret bounds has a gap.
minor comments (6)
  1. [Section II.A] The regret definition is written as Regret_T := ∑ f_t(x_t) − argmin_{x∈X} ∑ f_t(x); the argmin should be inside the sum, i.e., Regret_T := ∑ f_t(x_t) − min_{x∈X} ∑ f_t(x), or introduce x* = argmin and write Regret_T := ∑ (f_t(x_t) − f_t(x*)).
  2. [Section III.C, proof of Theorem 3.5] The proof divides by |x_t x̃_{t+1}| = η_t ||g_t||. When ||g_t|| = 0, this factor is zero. The case g_t = 0 should be handled separately or by a continuity argument.
  3. [Definition 2.3] The parameter µ is used without stating µ > 0. It should be stated explicitly, as the strong-convexity inequality only makes sense for positive µ.
  4. [Section IV] The theory for h-convex losses (Theorem 3.3) uses a constant step-size η = 1/√T, whereas the experiments use the decreasing schedule η_t = η/√t. No regret bound is proved for this schedule. For the strongly h-convex experiment, values of η larger than 1 violate the condition η_t ≤ 1/µ at early rounds if µ = 1. Please either extend the theory to the schedules used or adjust the experiments/statements to match the proved step-size conditions.
  5. [Section IV] The paper states that the Tyler loss and the squared-distance loss are h-convex / 1-strongly h-convex, citing [9] informally. Since these claims are used to interpret the experiments, please give precise statements or exact references (proposition numbers) in [9], or prove them in an appendix.
  6. [General] The text says 'h-convexity forms a subclass of g-convexity' but gives no proof or precise statement. A citation to [9] is given, but the exact proposition would be helpful, especially because the paper relies on this relationship for motivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the regret bounds follow from the h-convexity definitions plus external Busemann-function lemmas; no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is not circular. Theorem 3.3 uses Definition 2.2 plus imported facts from the external preprint [9] (Proposition 2(iii) and Lemma 2) to convert Busemann-function lower bounds into a quadratic telescope; the rest of the proof is standard non-expansiveness of projection and the Lipschitz bound. Theorem 3.5 uses Definition 2.3, the in-paper Lemma 2.4 (proved from the Hadamard cosine law), a triangle-inequality argument, and a telescoping sum with step size η_t=1/(µt). None of these steps assumes the target regret bound. The constants L, D, and µ are assumptions, not fitted parameters, and no computed quantity is renamed as a prediction. The self-citation [20] appears only in a literature-review sentence and is not load-bearing. The main external dependencies are technical lemmas from [9] about Busemann functions and gradients; they are not equivalent to the regret conclusion, and their unproved status is a robustness/correctness concern, not circularity. Therefore no circularity pattern is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the h-convexity framework imported from [9] (which supplies the Busemann-function machinery), plus standard assumptions of bounded diameter, Lipschitzness, and differentiability. The paper contributes a new definition of strong h-convexity and a new geometric lemma, but both are used as assumptions or proved internally. No free parameters are fitted to data in the theory.

assumptions (7)
  • domain assumption Each loss f_t is h-convex (Definition 2.2) or µ-strongly h-convex (Definition 2.3)
    The main theorems are conditional on the loss sequence belonging to these classes. The strong h-convexity definition is introduced in this paper.
  • domain assumption The decision set X is a g-convex subset with bounded diameter D (Assumption 3.1)
    Ensures projection onto X is single-valued and non-expansive and gives a finite bound on |x_1 x*|.
  • domain assumption Each f_t is L-Lipschitz (Assumption 3.2)
    Provides the uniform gradient norm bound ||g_t|| ≤ L used to sum the gradient terms.
  • domain assumption Differentiability of each f_t and existence of the Riemannian gradient g_t
    The algorithm computes gradf_t(x_t); differentiability is not stated as an assumption but is required.
  • standard math Busemann subgradient inequality and homogeneity B_{x,cv}=cB_{x,v}, and the quadratic potential estimate in Lemma 2 of [9]
    Cited from the recent preprint [9]; these are the external tools that turn h-convexity into a regret inequality. Not proven in the present paper.
  • standard math Non-expansiveness of the metric projection onto a g-convex set in a Hadamard manifold
    Used to replace the unprojected iterate with x_{t+1} in both proofs; standard in CAT(0) analysis.
  • standard math Cosine law for Hadamard spaces (cited to [2])
    Used in the proof of Lemma 2.4; standard comparison result in non-positively curved spaces.

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Pith. "Pith review of Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives." pith.science (2026). https://pith.science/paper/B7JAEDXD

@misc{pith2026250911236,
  author       = {Pith},
  title        = {Pith review of: Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7JAEDXD}},
  note         = {Machine review of arXiv:2509.11236}
}
abstract

We study online Riemannian optimization on Hadamard manifolds under the framework of horospherical convexity (h-convexity). Prior work mostly relies on the geodesic convexity (g-convexity), leading to regret bounds scaling poorly with the manifold curvature. To address this limitation, we analyze Riemannian online gradient descent for h-convex and strongly h-convex functions and establish $O(\sqrt{T})$ and $O(\log(T))$ regret guarantees, respectively. These bounds are curvature-independent and match the results in the Euclidean setting. We validate our approach with experiments on the manifold of symmetric positive definite (SPD) matrices equipped with the affine-invariant metric. In particular, we investigate online Tyler's $M$-estimation and online Fr\'echet mean computation, showing the application of h-convexity in practice.

Figures

Figures reproduced from arXiv: 2509.11236 by the authors.

Figure 1
Figure 1. Cumulative regret plots with step-size a) ηt = η√ t for online covariance estimation and b) ηt = η t for online Frechet mean estimation. ´ endowed with this metric has non-positive sectional curvature, guaranteeing uniqueness of geodesics and the validity of convexity notions such as g-convexity and h-convexity. These properties make SPD(n) a particularly well-suited setting for testing online Riemannian optimizatio… view at source ↗

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Cited by 1 Pith paper

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