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Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds

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Pith's one-line read On manifolds with bounded positive curvature, a curvature-tuned Riemannian consensus step contracts disagreement at a linear rate, and this paper shows the same O(sqrt T) regret for decentralized online gradient descent and two-point bandit

desk verdict Solid extension of decentralized Riemannian OCO beyond Hadamard manifolds, but the stated and analyzed consensus step-sizes disagree; the central O(√T) claims look recoverable with a careful revision. read the letter →

arxiv 2509.07779 v2 pith:DRAXUPGL submitted 2025-09-09 math.OC cs.LGcs.MA

Emre Sahinoglu , Shahin Shahrampour This is my paper
classification math.OCcs.LGcs.MA MSC 90C2590C1568W27
keywords decentralizedoptimizationonlineRiemannianpositivecurvatureregretboundstwo-pointbanditgeodesicconvexityrandomizedsmoothingconsensus
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

This paper sets out to show that decentralized online optimization on curved spaces does not have to settle for worse regret rates once positive curvature is allowed. The authors prove that a specific Riemannian consensus step—each agent moves a tuned fraction of the way toward its neighbors along geodesics—contracts disagreement linearly even on manifolds with bounded positive sectional curvature, not just Hadamard manifolds. Building on that contraction, they show decentralized online Riemannian gradient descent achieves O(sqrt T) static regret under full gradient feedback, and the same O(sqrt T) rate under two-point bandit feedback using smoothed gradient estimators. The upshot: neither decentralization nor positive curvature needs to change the optimal dependence on the time horizon.

What carries the argument

The consensus step (1) is the load-bearing object: a curvature-aware averaging rule that replaces Euclidean weighted averages by exponential-map combinations. Its step size s is chosen to balance two opposing curvature effects—positive curvature weakens the convexity needed for contraction while negative curvature weakens smoothness—and the analysis uses geodesic law-of-cosines inequalities (Lemma II.4) together with a bound on how much exponential maps distort tangent-space distances (Lemma II.5) to obtain a uniform contraction rate rho < 1.

What would settle it

On a unit sphere (Kmax = 1), place three points at vertices of a small equilateral spherical triangle inside a ball of radius pi/8 (diameter < pi/2), apply the consensus update with s = C2/(2C1), and measure Var after one step; Theorem III.2 predicts a contraction ratio rho < 1. A configuration inside the stated diameter bound with ratio >= 1 would refute it. Likewise, running Algorithm 1 on the sphere with eta = 1/sqrt T and tracking cumulative regret should grow like sqrt T; a visibly superlinear trajectory would refute Theorem IV.2.

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Extended reading notes

Core claim

The central claim is that the consensus update x_i(s) = Exp_{y_i}(s sum_j w_ij Log_{y_i} y_j), with step size s = C2/(2C1), has variance contraction Var({x_i(s)}) <= rho Var({y_i}) with rho in (0,1) whenever the domain is a geodesically convex set of diameter below pi/(2 sqrt(Kmax)). This is Theorem III.2. The contraction coefficient depends on the network spectral gap, curvature bounds, and domain diameter through curvature-dependent constants. Armed with it, the paper proves Theorem IV.2: with gradient step size eta = O(1/sqrt T) and consensus step size s = alpha(1 - sigma_2(W))/(4 C1), decentralized online Riemannian gradient descent has regret at most D C5 sqrt T. Theorem V.2 then extend

Load-bearing premise

All iterates and the comparator must stay within a geodesically convex ball whose diameter is strictly less than pi/(2 sqrt(Kmax)); if the domain is larger or the positive curvature is too strong, the lower-bound constants that make consensus contract can lose their sign.

Editorial extensions

If this is right

  • Decentralized online Riemannian gradient descent achieves O(sqrt T) static regret on manifolds with bounded sectional curvature, including positive curvature, matching the Euclidean optimal rate in T.
  • Two-point bandit feedback carries the same O(sqrt T) rate, so zeroth-order queries do not asymptotically hurt beyond the full-gradient setting.
  • The consensus step is isolated as a stand-alone linear variance-reduction result, so other decentralized Riemannian algorithms that can control network error can reuse the contraction bound.
  • The regret matches centralized Riemannian and decentralized Euclidean rates, so curvature and decentralization do not introduce an extra asymptotic penalty under the stated assumptions.
  • When Kmax = 0, the results reduce to the Hadamard-manifold setting, giving a unified treatment of both negatively and positively curved spaces.

Reading between the lines

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

  • A test this paper does not run: if the consensus contraction is as general as claimed, offline decentralized Riemannian optimization with strongly geodesically convex objectives should inherit linear convergence on positively curved manifolds, since the hard part is exactly the variance-reduction lemma.
  • The diameter restriction D < pi/(2 sqrt(Kmax)) means the theory covers geodesic balls smaller than a quarter circumference; on spheres that is natural, but a natural next question is whether the rate survives domains crossing the convexity radius with additional projection care.
  • The bandit estimator has a dimensional factor d entering through the bound ||g^delta|| <= dL, so in high-dimensional manifolds the constant grows; tightening this dimensional dependence or proving a lower bound is a separate question.
  • The same consensus step might extend to time-varying networks if sigma_2(W) is replaced by a worst-case connectivity factor, but that extension is not analyzed here.
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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 / 4 minor

Summary. The paper studies decentralized online Riemannian optimization on manifolds with bounded sectional curvature, allowing positive curvature. The authors prove a linear variance-reduction property for the Riemannian consensus step (Theorem III.2), and then use it to establish O(sqrt T) static regret for full-information decentralized online Riemannian gradient descent (Theorem IV.2) and for the two-point bandit feedback setting (Theorem V.2). The analysis is based on curvature comparison inequalities and assumes the domain has diameter D < pi/(2 sqrt(Kmax)) when Kmax > 0. The central claims are that neither positive curvature nor decentralization changes the optimal-in-T regret rate.

Significance. If the main theorems are correct, this is a significant contribution to online Riemannian optimization: it removes the Hadamard-manifold restriction in decentralized settings and gives the first O(sqrt T) two-point bandit bound in this setup. The consensus step-size optimization and the subconvexity analysis of smoothed objectives are interesting, and the proofs are provided in detail with no fitted parameters or circular derivations. However, the stated theorems and the proofs use different consensus step-sizes, so the results are not yet established as written. The issue appears local and fixable, which supports a major revision rather than rejection.

major comments (2)
  1. [Section IV (Lemma IV.1, Theorem IV.2) and Section VII-D] The stated algorithm and the analyzed algorithm use different consensus step-sizes. Lemma IV.1 and Theorem IV.2 state s = alpha(4C1)^{-1}(1 - sigma2(W)), whereas Theorem III.2 proves linear variance reduction only for s = C2/(2C1). In the proof of Lemma IV.1, the contraction coefficient rho is used as computed in Theorem III.2 without re-derivation for the stated s; in the proof of Theorem IV.2, immediately after Eq. (17), the text says 'Since we have s = C2/(2C1) <= C2/C1' to conclude sum_i T_{i,5} <= 0. Consequently, the network-error bound (8), the full-information regret bound (9), and, by inheritance, the bandit regret bound (11) are not proven for the step-size stated in the theorems. The O(sqrt T) conclusion is plausibly recoverable by consistently setting s = C2/(2C1) or by re-doing the contraction analysis for the smaller stated s, but as written the central theorems do not appl
  2. [Section III (Theorem III.2) and Appendix VII-H] The contraction coefficient is mis-specified. Theorem III.2 states rho = 1 - C3^2(1 - sigma2(W)) / (4 C1 (1 + C4 D^2)^2), but the proof in Appendix VII-B derives rho = 1 - C2^2(1 - sigma2(W)) / (4 C1 (1 + C4 D^2)^2) by maximizing q(s) = s C2 - s^2 C1. The constant C3 is never defined; Appendix VII-H simply refers to Lemma 4 of [43]. Since rho enters the network-error bound and the constant C5, this is a load-bearing inconsistency. In addition, the equality in Lemma IV.1, 2 sqrt(n) eta L / (1 - rho) = 8 C1 sqrt(n) eta L / ((1 - sigma2(W))^2 alpha^2), does not follow from the definitions of alpha and rho given in the paper.
minor comments (4)
  1. [Appendix VII-H] Several constants are left as pointers to external references ('For C3 and C4, find the definition in Lemma 4 of [43]; for C8, find the definition in Lemma 3 of [43]'). For a self-contained proof these constants should either be stated explicitly or the cited statements reproduced with their hypotheses.
  2. [Section VI] The numerical experiments use consensus step-sizes s in {0.6, 0.8, 1}, which do not correspond to the theoretically analyzed step-size s = C2/(2C1) nor to the value stated in Lemma IV.1. A short remark explaining the relation, or clarifying that the experiments are illustrative, would improve the paper.
  3. [Theorem V.2 and its proof] The theorem statement says delta = O(1/sqrt T) and tau = O(delta), while the proof chooses delta = 1/T and tau = delta/(r theta). The statement and proof should be aligned. Also, the bound O(eta^{-1} + eta T + sqrt T) is redundant once eta = O(1/sqrt T); the proof actually yields O(eta^{-1} + eta T + 1).
  4. [Throughout] Minor typographical issues include 'proprty' in the conclusion, inconsistent accents on Frechet, and the cross-reference 'Section Proof VII-D'. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the consensus contraction and regret bounds are derived from external geometric lemmas and convexity assumptions; the noted consensus step-size mismatch is a non-circular theorem/proof consistency gap.

full rationale

The derivation chain is self-contained with respect to its inputs. The central consensus contraction (Theorem III.2) is proved from the external cosine-law inequalities in Lemma II.4 (cited to [42] and [38]) and the tangent-log distortion bound in Lemma II.5 (cited to [43] and [41]); the consensus step-size s=C2/(2C1) is then chosen by maximizing the quadratic q(s)=sC2-s^2*C1 in the bound, not fitted to the target regret value. The full-information regret bound (Theorem IV.2) and the bandit regret bound (Theorem V.2) decompose regret into network-error and local-optimization terms, each controlled by the same geometric inequalities and by the stated Lipschitz and geodesic-convexity assumptions; no parameter is calibrated to the comparator x* or to the regret expression. The authors' self-citations ([7], [8], [20], [21], [29]) appear in background or as prior decentralized-Euclidean/Stiefel work and are not the load-bearing support for the new consensus or regret results; the load-bearing geometric inequalities come from external, independently available sources. One non-circular consistency gap should be flagged: Lemma IV.1 and Theorem IV.2 state the consensus step-size as s=alpha*(4*C1)^{-1}*(1-sigma2(W)), while the proof of Theorem IV.2 (Appendix VII-D) explicitly invokes s=C2/(2C1) to conclude sum_i T_{i,5} <= 0. This means the stated algorithm step-size is not the one analyzed in that proof, but this is a theorem-statement/proof mismatch, not a reduction of the claimed result to its own inputs, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All assumptions are standard in Riemannian online optimization. The paper introduces no new particles, forces, or entities, and no data-fitted constants. The algorithm step-sizes are selected as explicit functions of curvature, diameter, spectral gap, and horizon T; they are not fit to any dataset.

assumptions (6)
  • domain assumption The communication matrix W is symmetric, doubly stochastic, and the network is connected, with sigma2(W) < 1.
    Assumption II.1, standard in decentralized optimization, is used throughout the consensus and network-error analysis.
  • domain assumption Sectional curvature in X is bounded, Kmin <= K <= Kmax, and if Kmax > 0 then diam(X) < pi/(2 sqrt(Kmax)).
    Assumption II.2 ensures unique geodesic convexity, positive c2 bounds, and validity of the projection and triangle lemmas.
  • domain assumption Local objectives f_i,t are geodesically convex and L-Lipschitz on X, and the comparator x* belongs to X.
    Assumption II.3 is used in the regret decomposition, gradient norm bounds, and the bandit smoothing analysis.
  • standard math Riemannian law-of-cosines inequalities (Lemma II.4) and log-map distance bounds (Lemma II.5) with constants c1, c2, C3, and C4 from [42], [43], and [38].
    These unproved background geometric bounds are invoked in Lemma III.1, Theorem III.2, and the regret proofs.
  • standard math Metric projection onto a geodesically convex set satisfies the variational inequality <Log_{P_X(y)}(y), Log_{P_X(y)}(x)> <= 0 for x in X.
    Used in Proof of Lemma IV.1 and the projection bounds; cited from [46] and Lemma 47 of [6].
  • standard math Perturbation estimates for parallel transport and exponential maps (Lemmas VII.2, VII.3, VII.5), based on Jacobi field bounds from [43] and [47].
    Used in Proof of Lemma V.1 to establish the delta L C6 g-subconvexity of smoothed objectives.

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Pith. "Pith review of Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds." pith.science (2026). https://pith.science/paper/DRAXUPGL

@misc{pith2026250907779,
  author       = {Pith},
  title        = {Pith review of: Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRAXUPGL}},
  note         = {Machine review of arXiv:2509.07779}
}
abstract

We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives.

Figures

Figures reproduced from arXiv: 2509.07779 by the authors.

Figure 1
Figure 1. Cumulative regret with gradient step-size [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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Forward citations

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