Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds

# Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds
Source: [https://arxiv.org/html/2609.13646](https://arxiv.org/html/2609.13646)
Emre SahinogluShahin Shahrampour††thanks:This work is supported in part by NSF Award ECCS\-2240788 as well as NSF CAREER Award ECCS\-2442321\.††thanks:The authors are with the Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115\.Emails: \{cai\.zha; sahinoglu\.m;s\.shahrampour\}@northeastern\.edu\.

###### Abstract

This work addresses decentralized online Riemannian optimization on Hadamard manifolds\. Prior work under geodesic convexity \(g\-convexity\) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature\. Curvature may also enter the step size or contraction factor of tangent\-space Riemannian consensus schemes\. In this work, we relax the curvature dependence for a narrower class of horospherical convex \(h\-convex\) functions\. We study Distributed Riemannian Online Gradient Descent \(D\-ROGD\), which combines local Riemannian h\-subgradient updates with an implicit Fréchet\-mean consensus\. For h\-convex and strongly h\-convex local objectives, we establishO⁡\(T\)O\(\\sqrt\{T\}\)andO⁡\(log⁡T\)O\(\\log T\)static regret, respectively, matching the corresponding Euclidean rates with respect toTT, with network dependence governed solely by the spectral gap\. To our knowledge, these are the first curvature\-independent regret guarantees for decentralized online optimization on Hadamard manifolds\. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature\.

###### Index Terms:

Online optimization, Riemannian optimization, distributed optimization, Hadamard manifolds, horospherical convexity, regret\.

## IIntroduction

Many control and engineering problems naturally involve decision variables constrained to nonlinear geometric spaces\. Representative examples include covariance estimation and averaging over symmetric positive\-definite matrices\[[1](https://arxiv.org/html/2609.13646#bib.bib1)\], Riemannian system identification with structured positive\-definite variables\[[2](https://arxiv.org/html/2609.13646#bib.bib2)\], and feedback\-policy synthesis over submanifolds induced by control constraints\[[3](https://arxiv.org/html/2609.13646#bib.bib3)\]\. Riemannian optimization provides a framework for such problems, enabling optimization directly on the underlying manifold while respecting the geometry of the decision space\[[4](https://arxiv.org/html/2609.13646#bib.bib4),[5](https://arxiv.org/html/2609.13646#bib.bib5)\]\.

On the other hand, the sequential and distributed nature of many control, estimation, and learning systems motivates online and decentralized extensions of Riemannian optimization\. In the online setting, the objective varies over time and is revealed only after a decision is made\[[6](https://arxiv.org/html/2609.13646#bib.bib6),[7](https://arxiv.org/html/2609.13646#bib.bib7)\], while in the decentralized setup, data and/or computation are distributed across a network of agents\[[8](https://arxiv.org/html/2609.13646#bib.bib8)\]\. Combining these two settings, we consider agents that sequentially update their \(manifold\-valued\) decisions, observe private local losses, and communicate only with neighboring agents, with the goal of achieving a small regret, defined relative to the best fixed collective decision in hindsight\. While this framework is well understood in the Euclidean domain, its Riemannian counterpart introduces additional geometric challenges\.

Unlike in the Euclidean setting, curvature can affect decentralized online Riemannian optimization through two distinct mechanisms\. On the optimization side, standard analyses under g\-convexity rely on a Riemannian cosine inequality containing the factor\|κ\|​D​coth⁡\(\|κ\|​D\)\\sqrt\{\|\\kappa\|\}D\\coth\(\\sqrt\{\|\\kappa\|\}D\), which requires a finite sectional\-curvature lower bound and deteriorates with both curvatureκ\\kappaand domain diameterDD\[[9](https://arxiv.org/html/2609.13646#bib.bib9)\]\. On the network side, curvature\-aware tangent\-space consensus schemes may require curvature\-dependent step sizes and contraction factors\[[10](https://arxiv.org/html/2609.13646#bib.bib10),[11](https://arxiv.org/html/2609.13646#bib.bib11),[12](https://arxiv.org/html/2609.13646#bib.bib12)\]\. Implicit Fréchet\-mean consensus avoids this dependence on Hadamard manifolds\[[10](https://arxiv.org/html/2609.13646#bib.bib10)\], but curvature\-independent error guarantees compatible with the diminishing step sizes required forO⁡\(log⁡T\)O\(\\log T\)regret have not been established\. Thus, curvature\-independent regret bounds require controlling both optimization progress and network error without curvature\-dependent terms\.

We address these two sources of curvature dependence through complementary geometric mechanisms\. On the optimization side, we adopt horospherical convexity \(h\-convexity\), which provides curvature\-free inequalities through Busemann functions\[[13](https://arxiv.org/html/2609.13646#bib.bib13),[14](https://arxiv.org/html/2609.13646#bib.bib14)\]\. On the network side, we employ implicit weighted Fréchet\-mean consensus, whose contraction depends only on network connectivity and requires no curvature\-dependent consensus step size\[[10](https://arxiv.org/html/2609.13646#bib.bib10)\]\. We then develop a curvature\-independent network\-error analysis for arbitrary step size schedules, accommodating both the h\-convex and strongly h\-convex online regimes without requiring a finite lower bound on the sectional curvature\. This curvature independence comes at the cost of a more restrictive objective class and a consensus step that requires solving a Fréchet\-mean subproblem in each round\. Our main contributions are as follows:

- •Curvature\-independent regret\.We propose Distributed Riemannian Online Gradient Descent \(D\-ROGD\) and establishO⁡\(T\)O\(\\sqrt\{T\}\)andO⁡\(log⁡T\)O\(\\log T\)static regret for h\-convex and strongly h\-convex losses, respectively \(Theorems[III\.2](https://arxiv.org/html/2609.13646#S3.Thmtheorem2)and[III\.3](https://arxiv.org/html/2609.13646#S3.Thmtheorem3)\), without requiring a finite lower bound on the sectional curvature\. The rates match their Euclidean counterparts inTT, with network dependence governed by the spectral gap\[[15](https://arxiv.org/html/2609.13646#bib.bib15),[16](https://arxiv.org/html/2609.13646#bib.bib16)\]\. To our knowledge, these are the first curvature\-free regret guarantees for decentralized online optimization on Hadamard manifolds\.
- •Network error under arbitrary step sizes\.We establish a curvature\-independent bound on the network error induced by implicit weighted Fréchet\-mean consensus under arbitrary step size schedules \(Lemma[III\.1](https://arxiv.org/html/2609.13646#S3.Thmtheorem1)\), with network dependence governed solely by the spectral gap\. This bound remains valid for diminishing step sizes, includingηt=1/\(μ​t\)\\eta\_\{t\}=1/\(\\mu t\)in the strongly h\-convex analysis\. While this result is essential for our regret bound, it may also be of independent interest\.
- •Numerical validation\.We evaluate D\-ROGD on hyperbolic embeddings of real hierarchical data while varying both manifold curvature and network connectivity\. The experiments exhibit regret growth consistent with the theoretical rates, show no systematic deterioration as curvature increases, and confirm that poorer network connectivity leads to higher regret, as predicted by the spectral\-gap dependence of the theoretical bounds\.

### I\-ALiterature Review

Riemannian Online Optimization and Curvature Independence\. Online optimization over Riemannian manifolds has been developed primarily under g\-convexity, extending classical regret analysis to curved decision spaces\[[6](https://arxiv.org/html/2609.13646#bib.bib6),[7](https://arxiv.org/html/2609.13646#bib.bib7)\]\. Subsequent works have considered zeroth\-order feedback and tracking\[[17](https://arxiv.org/html/2609.13646#bib.bib17)\], projection\-free online optimization\[[18](https://arxiv.org/html/2609.13646#bib.bib18)\], and optimistic methods with dynamic regret\[[19](https://arxiv.org/html/2609.13646#bib.bib19)\]\. A persistent feature of this literature is that standard g\-convex analyses rely on geometric comparison inequalities whose constants depend on a finite lower bound on the sectional curvature\[[9](https://arxiv.org/html/2609.13646#bib.bib9)\], motivating recent efforts toward curvature\-independent online guarantees\. One route, proposed by Roux et al\.\[[20](https://arxiv.org/html/2609.13646#bib.bib20)\], retains g\-convexity and removes geometric constants through inexact implicit updates, at the price of replacing standard explicit first\-order steps with approximately solved implicit subproblems\. A complementary structural route is provided by h\-convexity, which uses Busemann functions to obtain curvature\-free comparison inequalities on Hadamard manifolds\[[13](https://arxiv.org/html/2609.13646#bib.bib13),[21](https://arxiv.org/html/2609.13646#bib.bib21)\]\. Building on this framework, Sahinoglu and Shahrampour\[[14](https://arxiv.org/html/2609.13646#bib.bib14)\]establish curvature\-independentO⁡\(T\)O\(\\sqrt\{T\}\)andO⁡\(log⁡T\)O\(\\log T\)regret for online optimization with h\-convex and strongly h\-convex objectives, respectively\. These results, however, concern a single online learner and leave open the question of how to preserve curvature independence in the decentralized setting, where network disagreement introduces an additional geometric challenge\.

Decentralized Riemannian Optimization and Consensus\.Distributed online Riemannian optimization has so far been developed primarily under g\-convexity\. Chen and Sun\[[10](https://arxiv.org/html/2609.13646#bib.bib10)\]analyze dynamic regret on Hadamard manifolds, while Sahinoglu and Shahrampour\[[11](https://arxiv.org/html/2609.13646#bib.bib11)\]extend distributed online optimization beyond Hadamard manifolds to settings with zeroth\-order feedback\. More recently, Cai et al\.\[[12](https://arxiv.org/html/2609.13646#bib.bib12)\]establishO⁡\(log⁡T\)O\(\\log T\)regret for strongly g\-convex objectives using a network\-error analysis that accommodates diminishing step sizes\. Despite these advances, curvature\-dependent quantities remain either in the optimization analysis under g\-convexity or in the consensus mechanism\. In contrast, we combine h\-convexity with implicit weighted Fréchet\-mean consensus and establish a curvature\-independent disagreement bound for arbitrary step\-size schedules\. This allows both the optimization and network components of the regret analysis to remain independent of sectional\-curvature bounds\.

## IIPreliminaries

We work on a Hadamard manifoldℳ\\mathcal\{M\}, i\.e\., a complete, simply connected Riemannian manifold with nonpositive sectional curvature\. Forx∈ℳx\\in\\mathcal\{M\}, letTx​ℳT\_\{x\}\\mathcal\{M\}denote the tangent space,⟨⋅,⋅⟩x\\langle\\cdot,\\cdot\\rangle\_\{x\}the Riemannian metric with induced norm‖v‖x:=⟨v,v⟩x\\\|v\\\|\_\{x\}:=\\sqrt\{\\langle v,v\\rangle\_\{x\}\}, where the subscript is omitted when clear, andd⁡\(⋅,⋅\)d\(\\cdot,\\cdot\)the induced geodesic distance\. For brevity, we also write\|x​y\|:=d⁡\(x,y\)\|xy\|:=d\(x,y\)forx,y∈ℳx,y\\in\\mathcal\{M\}\. A set𝒳⊆ℳ\\mathcal\{X\}\\subseteq\\mathcal\{M\}is g\-convex if the geodesic segment joining any two points in𝒳\\mathcal\{X\}lies entirely in𝒳\\mathcal\{X\}\. The exponential mapExpx:Tx​ℳ→ℳ\\operatorname\{Exp\}\_\{x\}:T\_\{x\}\\mathcal\{M\}\\rightarrow\\mathcal\{M\}and logarithmic mapLogx:ℳ→Tx​ℳ\\operatorname\{Log\}\_\{x\}:\\mathcal\{M\}\\rightarrow T\_\{x\}\\mathcal\{M\}are globally well\-defined on Hadamard manifolds, see\[[4](https://arxiv.org/html/2609.13646#bib.bib4),[5](https://arxiv.org/html/2609.13646#bib.bib5)\]for further background\.

### II\-AProblem Formulation

#### II\-A1Decentralized Online Riemannian Optimization

Considernnagents communicating over a network described by a symmetric, doubly stochastic matrixW=\(wi​j\)∈ℝn×nW=\(w\_\{ij\}\)\\in\\mathbb\{R\}^\{n\\times n\}, wherewi​j\>0w\_\{ij\}\>0fori≠ji\\neq jonly if agentsiiandjjare neighbors\[[10](https://arxiv.org/html/2609.13646#bib.bib10),[8](https://arxiv.org/html/2609.13646#bib.bib8)\]\. At each roundt∈\[T\]:=\{1,…,T\}t\\in\[T\]:=\\\{1,\\ldots,T\\\}, agentiiselects a decisionxi,tx\_\{i,t\}from a common feasible set𝒳⊆ℳ\\mathcal\{X\}\\subseteq\\mathcal\{M\}and receives first\-order feedback atxi,tx\_\{i,t\}\. The global loss is defined as

ft​\(x\):=1n​∑i=1nfi,t​\(x\)\.f\_\{t\}\(x\):=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}f\_\{i,t\}\(x\)\.\(1\)We evaluate the collective performance of the agents through static regret, which measures the cumulative gap between the agents’ decisions and the best fixed collective decision in hindsight, both evaluated under the global loss sequence\. Specifically,

Reg⁡\(T\):=1n​∑i=1n∑t=1Tft​\(xi,t\)−∑t=1Tft​\(x∗\),\\operatorname\{Reg\}\(T\):=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\_\{i,t\}\)\-\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x^\{\\ast\}\),\(2\)wherex∗∈arg⁡min⁡∑t=1Tx∈𝒳⁡ft​\(x\)x^\{\\ast\}\\in\\arg\\min\_\{x\\in\\mathcal\{X\}\}\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x\)\. The goal is to design a decentralized algorithm that achieves sublinear regret,Reg⁡\(T\)=o⁡\(T\)\\operatorname\{Reg\}\(T\)=o\(T\), using only local loss information and neighbor\-to\-neighbor communication\.

To perform consensus on the manifold, we use the weighted Fréchet mean as the intrinsic analogue of Euclidean averaging\. Given points\{zj\}j=1n⊂ℳ\\\{z\_\{j\}\\\}\_\{j=1\}^\{n\}\\subset\\mathcal\{M\}and weightswj≥0w\_\{j\}\\geq 0with∑j=1nwj=1\\sum\_\{j=1\}^\{n\}w\_\{j\}=1, their weighted Fréchet mean is defined as

z¯w:=arg⁡minz∈ℳ​\{∑j=1nwj​d2​\(z,zj\)\},\\bar\{z\}\_\{w\}:=\\arg\\min\_\{z\\in\\mathcal\{M\}\}\\left\\\{\\sum\_\{j=1\}^\{n\}w\_\{j\}d^\{2\}\(z,z\_\{j\}\)\\right\\\},\(3\)which is uniquely defined on a Hadamard manifold\[[22](https://arxiv.org/html/2609.13646#bib.bib22)\]\. In the decentralized setting, agentiiuses the weights\{wi​j\}j=1n\\\{w\_\{ij\}\\\}\_\{j=1\}^\{n\}to aggregate its neighbors’ iterates through this minimization, yielding an implicit consensus update\[[10](https://arxiv.org/html/2609.13646#bib.bib10)\]\. Unlike linear averaging in Euclidean spaces, the update is defined through an optimization problem over the manifold\.

For equal weights, we denote the Fréchet mean of a collection\{zi\}i=1n⊂ℳ\\\{z\_\{i\}\\\}\_\{i=1\}^\{n\}\\subset\\mathcal\{M\}byz¯:=arg⁡minz∈ℳ​1n​∑i=1nd2​\(z,zi\)\\bar\{z\}:=\\arg\\min\_\{z\\in\\mathcal\{M\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d^\{2\}\(z,z\_\{i\}\)and define the corresponding Fréchet variance as

VF​\(\{zi\}i=1n\):=1n​∑i=1nd2​\(z¯,zi\)\.V\_\{F\}\(\\\{z\_\{i\}\\\}\_\{i=1\}^\{n\}\):=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d^\{2\}\(\\bar\{z\},z\_\{i\}\)\.\(4\)The Fréchet variance serves as a geometric measure of disagreement among the agents and will be used to characterize the contraction of the implicit consensus step\.

#### II\-A2Horospherical Convexity

A functionf:ℳ→ℝf:\\mathcal\{M\}\\rightarrow\\mathbb\{R\}is geodesically convex \(g\-convex\) if its restriction to every geodesic is convex in the Euclidean sense\[[9](https://arxiv.org/html/2609.13646#bib.bib9),[23](https://arxiv.org/html/2609.13646#bib.bib23)\]\. H\-convexity is a stronger notion on Hadamard manifolds that uses Busemann functions as the analogue of affine functions in Euclidean convexity; we refer to\[[13](https://arxiv.org/html/2609.13646#bib.bib13),[14](https://arxiv.org/html/2609.13646#bib.bib14)\]for their construction and geometric interpretation\. For a unit\-speed geodesic rayγ:\[0,∞\)→ℳ\\gamma:\[0,\\infty\)\\to\\mathcal\{M\}, its Busemann function is defined asBγ​\(x\):=limt→∞\(d⁡\(x,γ⁡\(t\)\)−t\)\.B\_\{\\gamma\}\(x\):=\\lim\_\{t\\to\\infty\}\\bigl\(d\(x,\\gamma\(t\)\)\-t\\bigr\)\.Following\[[13](https://arxiv.org/html/2609.13646#bib.bib13),[14](https://arxiv.org/html/2609.13646#bib.bib14)\], fory∈ℳy\\in\\mathcal\{M\}andv∈Ty​ℳ∖\{0\}v\\in T\_\{y\}\\mathcal\{M\}\\setminus\\\{0\\\}, letBy,v:=‖v‖​BγB\_\{y,v\}:=\\\|v\\\|B\_\{\\gamma\}, whereγ⁡\(0\)=y\\gamma\(0\)=yandγ˙\(0\)=−v/∥v∥\\dot\{\\gamma\}\(0\)=\-v/\\\|v\\\|; we setBy,0≡0B\_\{y,0\}\\equiv 0and we recall the following definition\.

###### Definition II\.1\(h\-convexity\)

A functionf:ℳ→ℝf:\\mathcal\{M\}\\rightarrow\\mathbb\{R\}is h\-convex if, for everyy∈ℳy\\in\\mathcal\{M\}, there existsv∈Ty​ℳv\\in T\_\{y\}\\mathcal\{M\}such that

f⁡\(x\)−f⁡\(y\)≥By,v​\(x\),∀x∈ℳ\.f\(x\)\-f\(y\)\\geq B\_\{y,v\}\(x\),\\qquad\\forall x\\in\\mathcal\{M\}\.\(5\)Such a vectorvvis called an h\-subgradient offfatyy, and we denote the set of all h\-subgradients offfatyyas∂hf⁡\(y\)\\partial^\{h\}f\(y\)\.

Building on h\-convexity, we also consider its strongly convex counterpart\[[13](https://arxiv.org/html/2609.13646#bib.bib13),[14](https://arxiv.org/html/2609.13646#bib.bib14)\]\.

###### Definition II\.2\(μ\\mu\-strong h\-convexity\)

A functionf:ℳ→ℝf:\\mathcal\{M\}\\rightarrow\\mathbb\{R\}isμ\\mu\-strongly h\-convex, forμ\>0\\mu\>0, if for everyy∈ℳy\\in\\mathcal\{M\}, there existsv∈Ty​ℳv\\in T\_\{y\}\\mathcal\{M\}such that, for allx∈ℳx\\in\\mathcal\{M\},

f⁡\(x\)−f⁡\(y\)≥Qy,vμ​\(x\):=−12​μ​‖v‖2\+μ2​d2​\(Expy⁡\(−1μ​v\),x\)\.f\(x\)\-f\(y\)\\geq Q\_\{y,v\}^\{\\mu\}\(x\):=\-\\frac\{1\}\{2\\mu\}\\\|v\\\|^\{2\}\+\\frac\{\\mu\}\{2\}d^\{2\}\\\!\\left\(\\operatorname\{Exp\}\_\{y\}\\\!\\left\(\-\\frac\{1\}\{\\mu\}v\\right\),x\\right\)\.\(6\)The set of all such vectors is denoted by∂μhf⁡\(y\)\\partial\_\{\\mu\}^\{h\}f\(y\)\. For notational convenience, we set∂0hf⁡\(y\):=∂hf⁡\(y\)\\partial\_\{0\}^\{h\}f\(y\):=\\partial^\{h\}f\(y\)\.

For fixedy∈ℳy\\in\\mathcal\{M\}andv∈Ty​ℳv\\in T\_\{y\}\\mathcal\{M\}, we havelimμ→0\+Qy,vμ​\(x\)=By,v​\(x\)\\lim\_\{\\mu\\to 0^\{\+\}\}Q\_\{y,v\}^\{\\mu\}\(x\)=B\_\{y,v\}\(x\)for allx∈ℳx\\in\\mathcal\{M\}, so the h\-convex inequality is recovered in the limit\[[13](https://arxiv.org/html/2609.13646#bib.bib13)\]\. Moreover, h\-convexity is a subclass of g\-convexity; the two notions coincide in Euclidean spaces, while the inclusion can be strict on Hadamard manifolds\[[13](https://arxiv.org/html/2609.13646#bib.bib13)\]\. Despite this stronger requirement, the class contains important geometric objectives: in particular,d⁡\(⋅,z\)d\(\\cdot,z\)is h\-convex and12​d2​\(⋅,z\)\\frac\{1\}\{2\}d^\{2\}\(\\cdot,z\)is11\-strongly h\-convex, corresponding to geometric median and Fréchet mean problems, respectively\[[13](https://arxiv.org/html/2609.13646#bib.bib13)\]\.

### II\-BAssumptions

We impose the following assumptions for the analysis\.

###### Assumption II\.3\(Feasible set\)

The feasible set𝒳⊆ℳ\\mathcal\{X\}\\subseteq\\mathcal\{M\}is compact and g\-convex with diameter bounded byDD, i\.e\.,d⁡\(x,y\)≤Dd\(x,y\)\\leq Dfor allx,y∈𝒳x,y\\in\\mathcal\{X\}\.

The g\-convexity of𝒳\\mathcal\{X\}on a Hadamard manifold ensures that the metric projection𝒫𝒳​\(z\):=arg⁡minx∈𝒳⁡d⁡\(x,z\)\\mathcal\{P\}\_\{\\mathcal\{X\}\}\(z\):=\\arg\\min\_\{x\\in\\mathcal\{X\}\}d\(x,z\)is uniquely defined and nonexpansive, i\.e\.,d⁡\(𝒫𝒳​\(z\),𝒫𝒳​\(y\)\)≤d⁡\(z,y\)d\(\\mathcal\{P\}\_\{\\mathcal\{X\}\}\(z\),\\mathcal\{P\}\_\{\\mathcal\{X\}\}\(y\)\)\\leq d\(z,y\)\[[24](https://arxiv.org/html/2609.13646#bib.bib24)\]\. Moreover, since𝒳\\mathcal\{X\}is g\-convex, the weighted Fréchet mean of points in𝒳\\mathcal\{X\}also belongs to𝒳\\mathcal\{X\}\[[25](https://arxiv.org/html/2609.13646#bib.bib25)\], so minimizing the Fréchet objective in Eq\. \([3](https://arxiv.org/html/2609.13646#S2.E3)\) overℳ\\mathcal\{M\}or𝒳\\mathcal\{X\}is equivalent\.

###### Assumption II\.4\(Network connectivity\)

The communication matrixW∈ℝn×nW\\in\\mathbb\{R\}^\{n\\times n\}is symmetric and doubly stochastic, and satisfiesσ2​\(W\)∈\[0,1\)\\sigma\_\{2\}\(W\)\\in\[0,1\), whereσ2​\(W\)\\sigma\_\{2\}\(W\)denotes the second\-largest singular value ofWW\.

The quantity1−σ2​\(W\)1\-\\sigma\_\{2\}\(W\)is the spectral gap and quantifies network connectivity; larger gaps correspond to faster consensus\[[10](https://arxiv.org/html/2609.13646#bib.bib10),[11](https://arxiv.org/html/2609.13646#bib.bib11)\]\.

###### Assumption II\.5\(Local losses\)

There existsr\>0r\>0such that for everyi∈\[n\]i\\in\[n\]andt∈\[T\]t\\in\[T\], the local lossfi,tf\_\{i,t\}isLL\-Lipschitz on𝒳r:=\{x∈ℳ:d⁡\(x,𝒳\)≤r\}\\mathcal\{X\}\_\{r\}:=\\\{x\\in\\mathcal\{M\}:d\(x,\\mathcal\{X\}\)\\leq r\\\}, i\.e\.,\|fi,t​\(x\)−fi,t​\(y\)\|≤L​d​\(x,y\)\|f\_\{i,t\}\(x\)\-f\_\{i,t\}\(y\)\|\\leq L\\,d\(x,y\)for allx,y∈𝒳rx,y\\in\\mathcal\{X\}\_\{r\}\.

Wheneverfi,tf\_\{i,t\}is h\-convex, this assumption bounds every h\-subgradient on𝒳\\mathcal\{X\}\. Fixy∈𝒳y\\in\\mathcal\{X\}and a nonzerov∈∂hfi,t​\(y\)v\\in\\partial^\{h\}f\_\{i,t\}\(y\), and setxs:=Expy​\(s​v/‖v‖\)x\_\{s\}:=\\mathrm\{Exp\}\_\{y\}\\\!\\left\(s\\,v/\\\|v\\\|\\right\)with0<s≤r0<s\\leq r, so thatxs∈𝒳rx\_\{s\}\\in\\mathcal\{X\}\_\{r\}\. SinceBy,v​\(xs\)=s​‖v‖B\_\{y,v\}\(x\_\{s\}\)=s\\\|v\\\|by the definition ofBy,vB\_\{y,v\}, Definition[II\.1](https://arxiv.org/html/2609.13646#S2.Thmtheorem1)and Lipschitzness give

s​‖v‖≤fi,t​\(xs\)−fi,t​\(y\)≤L​s,s\\\|v\\\|\\;\\leq\\;f\_\{i,t\}\(x\_\{s\}\)\-f\_\{i,t\}\(y\)\\;\\leq\\;Ls,hence‖v‖≤L\\\|v\\\|\\leq L\. Because∂μhfi,t​\(y\)⊆∂hfi,t​\(y\)\\partial^\{h\}\_\{\\mu\}f\_\{i,t\}\(y\)\\subseteq\\partial^\{h\}f\_\{i,t\}\(y\)\[[13](https://arxiv.org/html/2609.13646#bib.bib13), Proposition 2\(i\)\], the same bound holds in the strongly h\-convex case\. The neighborhood condition is needed because the geodesic direction defining an h\-subgradient at a boundary point of𝒳\\mathcal\{X\}need not remain in𝒳\\mathcal\{X\}\. Assumption[II\.5](https://arxiv.org/html/2609.13646#S2.Thmtheorem5)therefore controls both the displacement of the local update and the terms arising in the regret analysis\. The convexity requirement is imposed separately in the main results: we consider h\-convex losses in Theorem[III\.2](https://arxiv.org/html/2609.13646#S3.Thmtheorem2)andμ\\mu\-strongly h\-convex losses in Theorem[III\.3](https://arxiv.org/html/2609.13646#S3.Thmtheorem3)\. Together with the compactness of𝒳\\mathcal\{X\}, Lipschitz continuity guarantees the existence of a best fixed decisionx∗x^\{\\ast\}defined above\.

### II\-CAuxiliary Geometric Lemmas

We collect here the geometric inequalities used in the regret analysis\. Lemmas[II\.6](https://arxiv.org/html/2609.13646#S2.Thmtheorem6)and[II\.7](https://arxiv.org/html/2609.13646#S2.Thmtheorem7)provide the inequalities underlying the h\-convex and strongly h\-convex analyses, respectively, while Lemmas[II\.8](https://arxiv.org/html/2609.13646#S2.Thmtheorem8)and[II\.9](https://arxiv.org/html/2609.13646#S2.Thmtheorem9)characterize the averaging and contraction properties of weighted Fréchet\-mean consensus\.

###### Lemma II\.6\(\[[13](https://arxiv.org/html/2609.13646#bib.bib13), Lemma 2\]\)

For anyx,y,p∈ℳx,y,p\\in\\mathcal\{M\}, it holds that

d2​\(p,x\)≤d2​\(p,y\)\+d2​\(x,y\)\+2​By,−Logy​\(p\)​\(x\)\.d^\{2\}\(p,x\)\\leq d^\{2\}\(p,y\)\+d^\{2\}\(x,y\)\+2\\,B\_\{y,\-\\mathrm\{Log\}\_\{y\}\(p\)\}\(x\)\.\(7\)

###### Lemma II\.7\(\[[14](https://arxiv.org/html/2609.13646#bib.bib14), Lemma 2\.4\]\)

Let△​a​b​c\\triangle abcbe a geodesic triangle and letpplie on the geodesic segmentb​cbc\. Then

\|a​b\|2​\|p​c\|\+\|a​c\|2​\|p​b\|≥\(\|p​a\|2\+\|p​b\|​\|p​c\|\)​\|b​c\|\.\|ab\|^\{2\}\|pc\|\+\|ac\|^\{2\}\|pb\|\\geq\\big\(\|pa\|^\{2\}\+\|pb\|\\,\|pc\|\\big\)\|bc\|\.\(8\)

###### Lemma II\.8\(\[[22](https://arxiv.org/html/2609.13646#bib.bib22), Theorem 6\.2\]\)

Lety¯w\\bar\{y\}\_\{w\}be the weighted Fréchet mean of\{yi\}i=1n\\\{y\_\{i\}\\\}\_\{i=1\}^\{n\}with weights\{wi\}i=1n\\\{w\_\{i\}\\\}\_\{i=1\}^\{n\}\. For any g\-convex functionf:ℳ→ℝf:\\mathcal\{M\}\\rightarrow\\mathbb\{R\},

f⁡\(y¯w\)≤∑i=1nwi​f​\(yi\)\.f\(\\bar\{y\}\_\{w\}\)\\leq\\sum\_\{i=1\}^\{n\}w\_\{i\}f\(y\_\{i\}\)\.\(9\)

###### Lemma II\.9\(\[[10](https://arxiv.org/html/2609.13646#bib.bib10), Lemma 3\]\)

Let\{yi\}i=1n⊂ℳ\\\{y\_\{i\}\\\}\_\{i=1\}^\{n\}\\subset\\mathcal\{M\}, and letxix\_\{i\}denote the weighted Fréchet mean of\{yj\}j=1n\\\{y\_\{j\}\\\}\_\{j=1\}^\{n\}with weights\{wi​j\}j=1n\\\{w\_\{ij\}\\\}\_\{j=1\}^\{n\}, whereW=\(wi​j\)W=\(w\_\{ij\}\)is symmetric and doubly stochastic\. Then

VF​\(\{xi\}i=1n\)≤σ22​\(W\)​VF​\(\{yi\}i=1n\)\.V\_\{F\}\(\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\}\)\\leq\\sigma\_\{2\}^\{2\}\(W\)\\,V\_\{F\}\(\\\{y\_\{i\}\\\}\_\{i=1\}^\{n\}\)\.\(10\)

Importantly, this contraction depends only on the communication matrix and requires no sectional\-curvature parameter\.

## IIIAlgorithm and Main Results

### III\-ADistributed Riemannian Online Gradient Descent

The proposed Distributed Riemannian Online Gradient Descent \(D\-ROGD\) alternates a local Riemannian h\-subgradient update with an implicit weighted Fréchet\-mean consensus step\. At roundtt, agentiicomputes

∇i,t∈\{∂hfi,t​\(xi,t\),when​fi,t​is h\-convex,∂μhfi,t​\(xi,t\),when​fi,t​isμ\-strongly h\-convex,\\nabla\_\{i,t\}\\in\\begin\{cases\}\\partial^\{h\}f\_\{i,t\}\(x\_\{i,t\}\),&\\text\{when \}f\_\{i,t\}\\text\{ is h\-convex\},\\\\ \\partial^\{h\}\_\{\\mu\}f\_\{i,t\}\(x\_\{i,t\}\),&\\text\{when \}f\_\{i,t\}\\text\{ is $\\mu$\-strongly h\-convex\},\\end\{cases\}takes a projected h\-subgradient step to obtainyi,t\+1y\_\{i,t\+1\}, and then computesxi,t\+1x\_\{i,t\+1\}by weighted Fréchet\-mean consensus, as summarized in Algorithm[1](https://arxiv.org/html/2609.13646#alg1)\.

Algorithm 1Distributed Riemannian Online Gradient Descent \(D\-ROGD\)1:Input:feasible set

𝒳⊆ℳ\\mathcal\{X\}\\subseteq\\mathcal\{M\}, horizon

TT, step sizes

\{ηt\}\\\{\\eta\_\{t\}\\\}, matrix

WW, common initial point

x1∈𝒳x\_\{1\}\\in\\mathcal\{X\}, where

xi,1=x1x\_\{i,1\}=x\_\{1\}for all

i∈\[n\]i\\in\[n\]\.

2:for

t=1t=1to

TTdo

3:foreach agent

i=1,…,ni=1,\\dots,n\(in parallel\)do

4:Observe

∇i,t∈∂μhfi,t​\(xi,t\)\\nabla\_\{i,t\}\\in\\partial^\{h\}\_\{\\mu\}f\_\{i,t\}\(x\_\{i,t\}\)\(

μ=0\\mu=0in the h\-convex case\)\.

5:

yi,t\+1=P𝒳​\(Expxi,t​\(−ηt​∇i,t\)\)y\_\{i,t\+1\}=P\_\{\\mathcal\{X\}\}\\big\(\\mathrm\{Exp\}\_\{x\_\{i,t\}\}\(\-\\eta\_\{t\}\\nabla\_\{i,t\}\)\\big\)\.

6:

xi,t\+1=arg​minz∈𝒳⁡\{∑j=1nwij​d2​\(yj,t\+1,z\)\}x\_\{i,t\+1\}=\\argmin\_\{z\\in\\mathcal\{X\}\}\\left\\\{\\sum\_\{j=1\}^\{n\}w\_\{ij\}\\,d^\{2\}\(y\_\{j,t\+1\},z\)\\right\\\}\.

7:endfor

8:endfor

For the analysis, letx¯t\\bar\{x\}\_\{t\}denote the Fréchet mean of\{xi,t\}i=1n\\\{x\_\{i,t\}\\\}\_\{i=1\}^\{n\}\. We decompose the static regret as

Reg⁡\(T\)\\displaystyle\\mathrm\{Reg\}\(T\)=1n​∑i=1n∑t=1T\(ft​\(xi,t\)−fi,t​\(xi,t\)\)⏟\(I\): network error\\displaystyle=\\underbrace\{\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{t=1\}^\{T\}\\big\(f\_\{t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x\_\{i,t\}\)\\big\)\}\_\{\\text\{\(I\): network error\}\}\+1n​∑i=1n∑t=1Tfi,t​\(xi,t\)−∑t=1Tft​\(x∗\)⏟\(II\): optimization error\.\\displaystyle\\quad\+\\underbrace\{\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{t=1\}^\{T\}f\_\{i,t\}\(x\_\{i,t\}\)\-\\sum\_\{t=1\}^\{T\}f\_\{t\}\(x^\{\\ast\}\)\}\_\{\\text\{\(II\): optimization error\}\}\.\(11\)Term \(I\) captures the network disagreement, whereas term \(II\) captures the local optimization error\.

### III\-BCurvature\-Independent Network Error

The following lemma bounds the disagreement among the agents under arbitrary step size schedules in terms of the network spectral gap\. This curvature\-independent estimate is the main tool for controlling the network\-error term in \([11](https://arxiv.org/html/2609.13646#S3.E11)\)\.

###### Lemma III\.1\(Network Error\)

Let Assumptions[II\.3](https://arxiv.org/html/2609.13646#S2.Thmtheorem3),[II\.4](https://arxiv.org/html/2609.13646#S2.Thmtheorem4)and[II\.5](https://arxiv.org/html/2609.13646#S2.Thmtheorem5)hold\. Then, for any nonnegative step size sequenceηt\\eta\_\{t\}, the iterates of Algorithm[1](https://arxiv.org/html/2609.13646#alg1)satisfy

1n​∑t=1T∑i=1nd⁡\(xi,t,x¯t\)≤L​σ2​\(W\)1−σ2​\(W\)​∑k=1T−1ηk\.\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{n\}d\(x\_\{i,t\},\\bar\{x\}\_\{t\}\)\\leq\\frac\{L\\sigma\_\{2\}\(W\)\}\{1\-\\sigma\_\{2\}\(W\)\}\\sum\_\{k=1\}^\{T\-1\}\\eta\_\{k\}\.\(12\)

###### Proof:

Letσ:=σ2​\(W\)\\sigma:=\\sigma\_\{2\}\(W\)and defineat:=VF​\(\{xi,t\}i=1n\)1/2a\_\{t\}:=V\_\{F\}\(\\\{x\_\{i,t\}\\\}\_\{i=1\}^\{n\}\)^\{1/2\}\. By Cauchy–Schwarz,1n​∑i=1nd⁡\(xi,t,x¯t\)≤at\.\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d\(x\_\{i,t\},\\bar\{x\}\_\{t\}\)\\leq a\_\{t\}\.Applying Lemma[II\.9](https://arxiv.org/html/2609.13646#S2.Thmtheorem9)and using the definition of the Fréchet variance,

at≤σ​VF​\(\{yi,t\}i=1n\)1/2≤σ​\(1n​∑i=1nd2​\(x¯t−1,yi,t\)\)1/2\.a\_\{t\}\\leq\\sigma V\_\{F\}\(\\\{y\_\{i,t\}\\\}\_\{i=1\}^\{n\}\)^\{1/2\}\\leq\\sigma\\left\(\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d^\{2\}\(\\bar\{x\}\_\{t\-1\},y\_\{i,t\}\)\\right\)^\{1/2\}\.\(13\)
By triangle inequality onℳ\\mathcal\{M\}and Minkowski’s inequality, \( 1 n ∑\_i=1^n d^2\( ¯ x \_t\-1,y\_i,t\) \)^1/2 ≤a\_t\-1 \+ \( 1 n ∑\_i=1^n d^2\(x\_i,t\-1,y\_i,t\) \)^1/2\. Moreover, by the nonexpansiveness of𝒫𝒳\\mathcal\{P\}\_\{\\mathcal\{X\}\}and the bound‖∇i,t−1‖≤L\\\|\\nabla\_\{i,t\-1\}\\\|\\leq L, we haved⁡\(xi,t−1,yi,t\)≤ηt−1​‖∇i,t−1‖≤L​ηt−1\.d\(x\_\{i,t\-1\},y\_\{i,t\}\)\\leq\\eta\_\{t\-1\}\\\|\\nabla\_\{i,t\-1\}\\\|\\leq L\\eta\_\{t\-1\}\.Therefore,at≤σ​at−1\+σ​L​ηt−1a\_\{t\}\\leq\\sigma a\_\{t\-1\}\+\\sigma L\\eta\_\{t\-1\}\.

Since the agents are initialized at a common point,a1=0a\_\{1\}=0\. Unrolling the recursion givesat≤L​∑k=1t−1σt−k​ηk\.a\_\{t\}\\leq L\\sum\_\{k=1\}^\{t\-1\}\\sigma^\{t\-k\}\\eta\_\{k\}\.Therefore,

1n​∑t=1T∑i=1nd⁡\(xi,t,x¯t\)≤∑t=1Tat≤L​∑k=1T−1ηk​∑t=k\+1Tσt−k\.\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{n\}d\(x\_\{i,t\},\\bar\{x\}\_\{t\}\)\\leq\\sum\_\{t=1\}^\{T\}a\_\{t\}\\leq L\\sum\_\{k=1\}^\{T\-1\}\\eta\_\{k\}\\sum\_\{t=k\+1\}^\{T\}\\sigma^\{t\-k\}\.\(14\)
Using∑t=k\+1Tσt−k≤σ/\(1−σ\)\\sum\_\{t=k\+1\}^\{T\}\\sigma^\{t\-k\}\\leq\\sigma/\(1\-\\sigma\)forσ∈\[0,1\)\\sigma\\in\[0,1\)yields

1n​∑t=1T∑i=1nd⁡\(xi,t,x¯t\)≤L​σ1−σ​∑k=1T−1ηk,\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{n\}d\(x\_\{i,t\},\\bar\{x\}\_\{t\}\)\\leq\\frac\{L\\sigma\}\{1\-\\sigma\}\\sum\_\{k=1\}^\{T\-1\}\\eta\_\{k\},\(15\)which proves the claim\. ∎

To bound the network\-error term, add and subtractft​\(x¯t\)f\_\{t\}\(\\bar\{x\}\_\{t\}\)andfi,t​\(x¯t\)f\_\{i,t\}\(\\bar\{x\}\_\{t\}\)and obtainft​\(xi,t\)−fi,t​\(xi,t\)=ft​\(xi,t\)−ft​\(x¯t\)\+ft​\(x¯t\)−fi,t​\(x¯t\)\+fi,t​\(x¯t\)−fi,t​\(xi,t\)\.f\_\{t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x\_\{i,t\}\)=f\_\{t\}\(x\_\{i,t\}\)\-f\_\{t\}\(\\bar\{x\}\_\{t\}\)\+f\_\{t\}\(\\bar\{x\}\_\{t\}\)\-f\_\{i,t\}\(\\bar\{x\}\_\{t\}\)\+f\_\{i,t\}\(\\bar\{x\}\_\{t\}\)\-f\_\{i,t\}\(x\_\{i,t\}\)\.Averaging overii, the term∑i\(ft​\(x¯t\)−fi,t​\(x¯t\)\)\\sum\_\{i\}\\left\(f\_\{t\}\(\\bar\{x\}\_\{t\}\)\-f\_\{i,t\}\(\\bar\{x\}\_\{t\}\)\\right\)vanishes sinceft=1n​∑i=1nfi,tf\_\{t\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}f\_\{i,t\}\. Moreover, bothftf\_\{t\}andfi,tf\_\{i,t\}areLL\-Lipschitz, so\(I\)≤2​Ln​∑t=1T∑i=1nd⁡\(xi,t,x¯t\)\.\\text\{\(I\)\}\\leq\\frac\{2L\}\{n\}\\sum\_\{t=1\}^\{T\}\\sum\_\{i=1\}^\{n\}d\(x\_\{i,t\},\\bar\{x\}\_\{t\}\)\.Applying Lemma[III\.1](https://arxiv.org/html/2609.13646#S3.Thmtheorem1)yields

\(I\)≤2​L2​σ2​\(W\)1−σ2​\(W\)​∑t=1T−1ηt\.\\text\{\(I\)\}\\leq\\frac\{2L^\{2\}\\sigma\_\{2\}\(W\)\}\{1\-\\sigma\_\{2\}\(W\)\}\\sum\_\{t=1\}^\{T\-1\}\\eta\_\{t\}\.\(16\)

### III\-CMain Results

We now combine the network\-error bound in \([16](https://arxiv.org/html/2609.13646#S3.E16)\) with bounds on the optimization\-error term \(II\) to establish the regret guarantees for h\-convex and strongly h\-convex objectives\.

###### Theorem III\.2\(Regret for h\-convex objectives\)

Let Assumptions[II\.3](https://arxiv.org/html/2609.13646#S2.Thmtheorem3)–[II\.5](https://arxiv.org/html/2609.13646#S2.Thmtheorem5)hold and supposefi,tf\_\{i,t\}is h\-convex for everyi∈\[n\]i\\in\[n\],t∈\[T\]t\\in\[T\]\. Then Algorithm[1](https://arxiv.org/html/2609.13646#alg1)with constant step sizeηt=η=1/T\\eta\_\{t\}=\\eta=1/\\sqrt\{T\}satisfies

Reg⁡\(T\)≤\(2​L2​σ2​\(W\)1−σ2​\(W\)\+L2\+D22\)​T\.\\mathrm\{Reg\}\(T\)\\leq\\Big\(\\frac\{2L^\{2\}\\,\\sigma\_\{2\}\(W\)\}\{1\-\\sigma\_\{2\}\(W\)\}\+\\frac\{L^\{2\}\+D^\{2\}\}\{2\}\\Big\)\\,\\sqrt\{T\}\.\(17\)

###### Proof:

Letzi,t:=Expxi,t⁡\(−η​∇i,t\)z\_\{i,t\}:=\\operatorname\{Exp\}\_\{x\_\{i,t\}\}\(\-\\eta\\nabla\_\{i,t\}\)\. By Definition[II\.1](https://arxiv.org/html/2609.13646#S2.Thmtheorem1)withv=∇i,tv=\\nabla\_\{i,t\},fi,t​\(xi,t\)−fi,t​\(x∗\)≤−Bxi,t,∇i,t​\(x∗\)f\_\{i,t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x^\{\\ast\}\)\\leq\-B\_\{x\_\{i,t\},\\nabla\_\{i,t\}\}\(x^\{\\ast\}\)\. Since−Logxi,t⁡\(zi,t\)=η​∇i,t\-\\operatorname\{Log\}\_\{x\_\{i,t\}\}\(z\_\{i,t\}\)=\\eta\\nabla\_\{i,t\}, the positive homogeneity of the scaled Busemann function gives−Bxi,t,∇i,t​\(x∗\)=−1η​Bxi,t,−Logxi,t⁡\(zi,t\)​\(x∗\)\-B\_\{x\_\{i,t\},\\nabla\_\{i,t\}\}\(x^\{\\ast\}\)=\-\\frac\{1\}\{\\eta\}B\_\{x\_\{i,t\},\-\\operatorname\{Log\}\_\{x\_\{i,t\}\}\(z\_\{i,t\}\)\}\(x^\{\\ast\}\)\. Applying Lemma[II\.6](https://arxiv.org/html/2609.13646#S2.Thmtheorem6)withy=xi,ty=x\_\{i,t\},p=zi,tp=z\_\{i,t\}, andx=x∗x=x^\{\\ast\}yields

fi,t​\(xi,t\)−fi,t​\(x∗\)≤12​η​\[d2​\(zi,t,xi,t\)\+d2​\(x∗,xi,t\)−d2​\(zi,t,x∗\)\]\.f\_\{i,t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x^\{\\ast\}\)\\leq\\frac\{1\}\{2\\eta\}\\left\[d^\{2\}\(z\_\{i,t\},x\_\{i,t\}\)\+d^\{2\}\(x^\{\\ast\},x\_\{i,t\}\)\-d^\{2\}\(z\_\{i,t\},x^\{\\ast\}\)\\right\]\.\(18\)
Sinced⁡\(zi,t,xi,t\)=η​‖∇i,t‖≤η​Ld\(z\_\{i,t\},x\_\{i,t\}\)=\\eta\\\|\\nabla\_\{i,t\}\\\|\\leq\\eta L, andx∗∈𝒳x^\{\\ast\}\\in\\mathcal\{X\}, the nonexpansiveness of𝒫𝒳\\mathcal\{P\}\_\{\\mathcal\{X\}\}givesd⁡\(yi,t\+1,x∗\)=d⁡\(𝒫𝒳​\(zi,t\),𝒫𝒳​\(x∗\)\)≤d⁡\(zi,t,x∗\)d\(y\_\{i,t\+1\},x^\{\\ast\}\)=d\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{X\}\}\(z\_\{i,t\}\),\\mathcal\{P\}\_\{\\mathcal\{X\}\}\(x^\{\\ast\}\)\\right\)\\leq d\(z\_\{i,t\},x^\{\\ast\}\)\. Hence,

fi,t​\(xi,t\)−fi,t​\(x∗\)≤\\displaystyle f\_\{i,t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x^\{\\ast\}\)\\leq\{\}η​L22\+12​η​\[d2​\(x∗,xi,t\)−d2​\(x∗,xi,t\+1\)\]\\displaystyle\\frac\{\\eta L^\{2\}\}\{2\}\+\\frac\{1\}\{2\\eta\}\\left\[d^\{2\}\(x^\{\\ast\},x\_\{i,t\}\)\-d^\{2\}\(x^\{\\ast\},x\_\{i,t\+1\}\)\\right\]\+12​η​\[d2​\(x∗,xi,t\+1\)−d2​\(x∗,yi,t\+1\)\]\.\\displaystyle\+\\frac\{1\}\{2\\eta\}\\left\[d^\{2\}\(x^\{\\ast\},x\_\{i,t\+1\}\)\-d^\{2\}\(x^\{\\ast\},y\_\{i,t\+1\}\)\\right\]\.\(19\)
Sinced2​\(⋅,x∗\)d^\{2\}\(\\cdot,x^\{\\ast\}\)is g\-convex on a Hadamard manifold, Lemma[II\.8](https://arxiv.org/html/2609.13646#S2.Thmtheorem8)applied to the weighted Fréchet\-mean consensus step givesd2​\(x∗,xi,t\+1\)≤∑j=1nwi​j​d2​\(x∗,yj,t\+1\)d^\{2\}\(x^\{\\ast\},x\_\{i,t\+1\}\)\\leq\\sum\_\{j=1\}^\{n\}w\_\{ij\}d^\{2\}\(x^\{\\ast\},y\_\{j,t\+1\}\)\. Summing overiiand using the double stochasticity ofWWyields

∑i=1nd2​\(x∗,xi,t\+1\)≤∑i=1nd2​\(x∗,yi,t\+1\)\.\\sum\_\{i=1\}^\{n\}d^\{2\}\(x^\{\\ast\},x\_\{i,t\+1\}\)\\leq\\sum\_\{i=1\}^\{n\}d^\{2\}\(x^\{\\ast\},y\_\{i,t\+1\}\)\.\(20\)Therefore, by averaging \([19](https://arxiv.org/html/2609.13646#S3.Ex4)\) overiiand using \([20](https://arxiv.org/html/2609.13646#S3.E20)\), the last term in \([19](https://arxiv.org/html/2609.13646#S3.Ex4)\) vanishes\. Also, another term telescopes overttand we get

\(II\)≤η​L22​T\+12​η​n​∑i=1n\[d2​\(x∗,xi,1\)−d2​\(x∗,xi,T\+1\)\]\\displaystyle\\leq\\frac\{\\eta L^\{2\}\}\{2\}T\+\\frac\{1\}\{2\\eta n\}\\sum\_\{i=1\}^\{n\}\\left\[d^\{2\}\(x^\{\\ast\},x\_\{i,1\}\)\-d^\{2\}\(x^\{\\ast\},x\_\{i,T\+1\}\)\\right\]\(21\)≤η​L22​T\+D22​η\.\\displaystyle\\leq\\frac\{\\eta L^\{2\}\}\{2\}T\+\\frac\{D^\{2\}\}\{2\\eta\}\.\(22\)Combining this bound with \([16](https://arxiv.org/html/2609.13646#S3.E16)\) and settingη=1/T\\eta=1/\\sqrt\{T\}yields the result\. ∎

###### Theorem III\.3\(Regret for strongly h\-convex objectives\)

Let Assumptions[II\.3](https://arxiv.org/html/2609.13646#S2.Thmtheorem3)–[II\.5](https://arxiv.org/html/2609.13646#S2.Thmtheorem5)hold and supposefi,tf\_\{i,t\}isμ\\mu\-strongly h\-convex for everyi∈\[n\]i\\in\[n\],t∈\[T\]t\\in\[T\]\. Then Algorithm[1](https://arxiv.org/html/2609.13646#alg1)with the diminishing step\-size scheduleηt=1/\(μ​t\)\\eta\_\{t\}=1/\(\\mu t\)satisfies

Reg⁡\(T\)≤1μ​\(2​L2​σ2​\(W\)1−σ2​\(W\)\+L22\)​\(1\+log⁡T\)\.\\operatorname\{Reg\}\(T\)\\leq\\frac\{1\}\{\\mu\}\\left\(\\frac\{2L^\{2\}\\sigma\_\{2\}\(W\)\}\{1\-\\sigma\_\{2\}\(W\)\}\+\\frac\{L^\{2\}\}\{2\}\\right\)\(1\+\\log T\)\.\(23\)

###### Proof:

Letui,t:=Expxi,t​\(−μ−1​∇i,t\)u\_\{i,t\}:=\\mathrm\{Exp\}\_\{x\_\{i,t\}\}\(\-\\mu^\{\-1\}\\nabla\_\{i,t\}\)andzi,t:=Expxi,t​\(−ηt​∇i,t\)z\_\{i,t\}:=\\mathrm\{Exp\}\_\{x\_\{i,t\}\}\(\-\\eta\_\{t\}\\nabla\_\{i,t\}\)\. Byμ\\mu\-strong h\-convexity and Definition[II\.2](https://arxiv.org/html/2609.13646#S2.Thmtheorem2),

fi,t​\(xi,t\)−fi,t​\(x∗\)≤12​μ​‖∇i,t‖2−μ2​d2​\(ui,t,x∗\)\.f\_\{i,t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x^\{\*\}\)\\;\\leq\\;\\tfrac\{1\}\{2\\mu\}\\\|\\nabla\_\{i,t\}\\\|^\{2\}\-\\tfrac\{\\mu\}\{2\}\\,d^\{2\}\(u\_\{i,t\},x^\{\*\}\)\.Sinceηt=1/\(μ​t\)≤1/μ\\eta\_\{t\}=1/\(\\mu t\)\\leq 1/\\mu, the pointzi,tz\_\{i,t\}lies on the geodesic segment fromxi,tx\_\{i,t\}toui,tu\_\{i,t\}, with\|ui,t​xi,t\|=μ−1​‖∇i,t‖\|u\_\{i,t\}x\_\{i,t\}\|=\\mu^\{\-1\}\\\|\\nabla\_\{i,t\}\\\|,\|zi,t​xi,t\|=ηt​‖∇i,t‖\|z\_\{i,t\}x\_\{i,t\}\|=\\eta\_\{t\}\\\|\\nabla\_\{i,t\}\\\|and\|zi,t​ui,t\|=\(μ−1−ηt\)​‖∇i,t‖\|z\_\{i,t\}u\_\{i,t\}\|=\(\\mu^\{\-1\}\-\\eta\_\{t\}\)\\\|\\nabla\_\{i,t\}\\\|\. Applying Lemma[II\.7](https://arxiv.org/html/2609.13646#S2.Thmtheorem7)to△​x∗​xi,t​ui,t\\triangle x^\{\*\}x\_\{i,t\}u\_\{i,t\}withp=zi,tp=z\_\{i,t\}and simplifying gives

d2​\(ui,t,x∗\)≥\\displaystyle d^\{2\}\(u\_\{i,t\},x^\{\*\}\)\\geq1μ​ηt​d2​\(zi,t,x∗\)\+1μ​\(1μ−ηt\)​‖∇i,t‖2\\displaystyle\\tfrac\{1\}\{\\mu\\eta\_\{t\}\}d^\{2\}\(z\_\{i,t\},x^\{\*\}\)\+\\tfrac\{1\}\{\\mu\}\\Big\(\\tfrac\{1\}\{\\mu\}\-\\eta\_\{t\}\\Big\)\\\|\\nabla\_\{i,t\}\\\|^\{2\}−1ηt​\(1μ−ηt\)​d2​\(x∗,xi,t\)\.\\displaystyle\-\\tfrac\{1\}\{\\eta\_\{t\}\}\\Big\(\\tfrac\{1\}\{\\mu\}\-\\eta\_\{t\}\\Big\)d^\{2\}\(x^\{\*\},x\_\{i,t\}\)\.Substituting this bound and usingd⁡\(zi,t,x∗\)≥d⁡\(yi,t\+1,x∗\)d\(z\_\{i,t\},x^\{\*\}\)\\geq d\(y\_\{i,t\+1\},x^\{\*\}\)by non\-expansiveness, we obtain

fi,t​\(xi,t\)−fi,t​\(x∗\)≤\\displaystyle\\textstyle f\_\{i,t\}\(x\_\{i,t\}\)\-f\_\{i,t\}\(x^\{\\ast\}\)\\textstyle\\leqηt2⋅‖∇i,t‖2−12​ηt⋅\|yi,t\+1​x∗\|2\\displaystyle\\textstyle\\frac\{\\eta\_\{t\}\}\{2\}\\cdot\\\|\\nabla\_\{i,t\}\\\|^\{2\}\-\\textstyle\\frac\{1\}\{2\\eta\_\{t\}\}\\cdot\\bigl\|\\,y\_\{i,t\+1\}\\,x^\{\*\}\\bigr\|^\{2\}\+\(12​ηt−μ2\)⋅\|x∗xi,t\|2\\displaystyle\+\\textstyle\\left\(\\frac\{1\}\{2\\eta\_\{t\}\}\-\\frac\{\\mu\}\{2\}\\right\)\\cdot\\bigl\|\\,x^\{\*\}\\,x\_\{i,t\}\\bigr\|^\{2\}=\\displaystyle\\textstyle=ηt2⋅‖∇i,t‖2\+12​ηt⋅\(\|xi,t\+1​x∗\|2−\|yi,t\+1​x∗\|2\)\\displaystyle\\textstyle\\frac\{\\eta\_\{t\}\}\{2\}\\cdot\\\|\\nabla\_\{i,t\}\\\|^\{2\}\+\\frac\{1\}\{2\\eta\_\{t\}\}\\cdot\\textstyle\\bigl\(\\bigl\|\\,x\_\{i,t\+1\}\\,x^\{\*\}\\bigr\|^\{2\}\-\\bigl\|\\,y\_\{i,t\+1\}\\,x^\{\*\}\\bigr\|^\{2\}\\bigr\)\+\(12​ηt−μ2\)⋅\|x∗xi,t\|2−12​ηt⋅\|xi,t\+1x∗\|2\\displaystyle\\textstyle\+\\left\(\\frac\{1\}\{2\\eta\_\{t\}\}\-\\frac\{\\mu\}\{2\}\\right\)\\cdot\\bigl\|\\,x^\{\*\}\\,x\_\{i,t\}\\bigr\|^\{2\}\-\\frac\{1\}\{2\\eta\_\{t\}\}\\cdot\\bigl\|\\,x\_\{i,t\+1\}\\,x^\{\*\}\\bigr\|^\{2\}Averaging overii, applying \([20](https://arxiv.org/html/2609.13646#S3.E20)\) again, and usingηt=1/\(μ​t\)\\eta\_\{t\}=1/\(\\mu t\), for which12​ηt−12​ηt−1−μ2=0\\tfrac\{1\}\{2\\eta\_\{t\}\}\-\\tfrac\{1\}\{2\\eta\_\{t\-1\}\}\-\\tfrac\{\\mu\}\{2\}=0,T≥t≥2T\\geq t\\geq 2, the distance terms telescope and yield

\(II\)≤L22​∑t=1Tηt≤L22​μ​\(log⁡T\+1\)\.\\text\{\(II\)\}\\leq\\frac\{L^\{2\}\}\{2\}\\sum\_\{t=1\}^\{T\}\\eta\_\{t\}\\leq\\frac\{L^\{2\}\}\{2\\mu\}\\big\(\\log T\+1\\big\)\.\(24\)Combining \([16](https://arxiv.org/html/2609.13646#S3.E16)\) and \([24](https://arxiv.org/html/2609.13646#S3.E24)\) withηt=1/\(μ​t\)\\eta\_\{t\}=1/\(\\mu t\)and∑t=1T1/\(μ​t\)≤1μ​\(1\+log⁡T\)\\sum\_\{t=1\}^\{T\}1/\(\\mu t\)\\leq\\tfrac\{1\}\{\\mu\}\(1\+\\log T\)gives the stated bound\. ∎

## IVNumerical Experiments

We evaluate D\-ROGD on real hierarchical data embedded in hyperbolic space, varying manifold curvature and network connectivity\. Fixing the intrinsic data diameter allows us to assess curvature dependence at a common geometric scale111The code is available at:[https://github\.com/ZhanyCai\-opt/decentralized\-online\-riemannian\-curvature\-independent](https://github.com/ZhanyCai-opt/decentralized-online-riemannian-curvature-independent)\.\.

Data and manifold\.From the WordNet noun taxonomy\[[26](https://arxiv.org/html/2609.13646#bib.bib26)\]we take the sub\-hierarchy rooted atmammal\.n\.01, retainN=300N=300synsets, and embed the tree’s all\-pairs shortest\-path distances into the1010\-dimensional Poincaré ball\[[27](https://arxiv.org/html/2609.13646#bib.bib27)\]by stress minimization\. For eachκ∈\{0\.25,1,4,9,16\}\\kappa\\in\\\{0\.25,1,4,9,16\\\}the embedding is rescaled radially inℍκ10\\mathbb\{H\}^\{10\}\_\{\\kappa\}, the Poincaré ball of curvature−κ\-\\kappa, so that the intrinsic geodesic diameter remains fixed atD=4D=4while the curvature varies\. The resulting curvature–diameter parameter isκ​D∈\{2,4,8,12,16\}\\sqrt\{\\kappa\}D\\in\\\{2,4,8,12,16\\\}, which appears in the comparison constants of g\-convex analyses\[[9](https://arxiv.org/html/2609.13646#bib.bib9)\]\.

Network and protocol\.We usen=8n=8agents on a ring with Metropolis weights, each assigned a distinct semantic cluster of concepts bykk\-means on the embedding, so that local objectives are heterogeneous\. At roundttagentiidraws a concept uniformly from its cluster and observes a perturbed copySi,tS\_\{i,t\}\. We runT=2000T=2000rounds over three seeds\. The feasible set𝒳\\mathcal\{X\}is a geodesic ball containing the data; the consensus step, i\.e\., line 6 of Algorithm[1](https://arxiv.org/html/2609.13646#alg1), is solved by Karcher iteration to tolerance10−910^\{\-9\}\[[25](https://arxiv.org/html/2609.13646#bib.bib25)\], andx∗x^\{\\ast\}by projected Riemannian gradient descent over alln​TnTobservations in𝒳\\mathcal\{X\}\.

Objectives\.The h\-convex case is the online distributed geometric median,fi,t​\(x\)=d⁡\(x,Si,t\)f\_\{i,t\}\(x\)=d\(x,S\_\{i,t\}\)withηt=1/t\\eta\_\{t\}=1/\\sqrt\{t\}which mimics the1/T1/\\sqrt\{T\}scaling of Theorem[III\.2](https://arxiv.org/html/2609.13646#S3.Thmtheorem2)without requiring prior knowledge of the horizon; the strongly h\-convex case is the online distributed Fréchet mean,fi,t​\(x\)=12​d2​\(x,Si,t\)f\_\{i,t\}\(x\)=\\tfrac\{1\}\{2\}d^\{2\}\(x,S\_\{i,t\}\)withηt=1/\(μ​t\)\\eta\_\{t\}=1/\(\\mu t\),μ=1\\mu=1\. For the geometric median, we use the canonical h\-subgradient∇i,t=−Logxi,t\(Si,t\)/d\(xi,t,Si,t\)\\nabla\_\{i,t\}=\-\\mathrm\{Log\}\_\{x\_\{i,t\}\}\(S\_\{i,t\}\)/d\(x\_\{i,t\},S\_\{i,t\}\)whenxi,t≠Si,tx\_\{i,t\}\\neq S\_\{i,t\}and zero otherwise\.

Results\.Fig\.[1](https://arxiv.org/html/2609.13646#S4.F1)shows that the cumulative regret grows consistently with the predictedO⁡\(T\)O\(\\sqrt\{T\}\)andO⁡\(log⁡T\)O\(\\log T\)rates for the h\-convex and strongly h\-convex objectives, respectively, across all tested curvatures\. Fig\.[2](https://arxiv.org/html/2609.13646#S4.F2)\(a\) plots the normalized regret againstκ​D\\sqrt\{\\kappa\}D\. While the g\-convex comparison factorζ⁡\(κ,D\)=κ​D​coth⁡\(κ​D\)\\zeta\(\\kappa,D\)=\\sqrt\{\\kappa\}D\\coth\(\\sqrt\{\\kappa\}D\)grows substantially with curvature, the normalized regret exhibits only mild variation in the h\-convex case and decreases in the strongly h\-convex case, showing no systematic curvature\-dependent deterioration comparable to that induced byζ⁡\(κ,D\)\\zeta\(\\kappa,D\)\. Fig\.[2](https://arxiv.org/html/2609.13646#S4.F2)\(b\) varies the network topology and shows increasing regret asσ2​\(W\)\\sigma\_\{2\}\(W\)approaches one, consistent with the spectral\-gap dependenceσ2​\(W\)/\(1−σ2​\(W\)\)\\sigma\_\{2\}\(W\)/\(1\-\\sigma\_\{2\}\(W\)\)in the theoretical bounds\. Taken together, these results are consistent with the theoretical separation between curvature\-independent optimization guarantees and spectral\-gap\-dependent network effects\.

Fig\. 1:Cumulative network regret of D\-ROGD on the embedded WordNet hierarchy data, withD=4D=4andn=8n=8\. Left: h\-convex objective \(geometric median\)\. Right: strongly h\-convex objective \(Fréchet mean\)\.Fig\. 2:\(a\) Regret constants atT=2000T=2000againstκ​D\\sqrt\{\\kappa\}D:Reg⁡\(T\)/T\\mathrm\{Reg\}\(T\)/\\sqrt\{T\}\(h\-convex\) andReg⁡\(T\)/log⁡T\\mathrm\{Reg\}\(T\)/\\log T\(strongly h\-convex\)\. The dashed curve is the g\-convex curvature factorζ⁡\(κ,D\)=κ​D​coth⁡\(κ​D\)\\zeta\(\\kappa,D\)=\\sqrt\{\\kappa\}D\\coth\(\\sqrt\{\\kappa\}D\), scaled to start at the leftmost h\-convex point so that slopes, not heights, are comparable\. \(b\) Regret of the h\-convex objective on complete, Erdős–Rényi \(p=0\.5p=0\.5\), ring and path graphs at two curvatures\.
## VConclusions

We studied distributed online Riemannian optimization on Hadamard manifolds beyond the standard g\-convexity framework, analyzing h\-convex and strongly h\-convex objectives\. We proposed D\-ROGD, which couples a local Riemannian h\-subgradient step with a Fréchet\-mean consensus step, and established curvature\-independent regret bounds ofO⁡\(T\)O\(\\sqrt\{T\}\)andO⁡\(log⁡T\)O\(\\log T\)for h\-convex and strongly h\-convex objectives, respectively\. The bounds match the Euclidean rates inTTand depend on the network only through the spectral\-gap factorσ2​\(W\)/\(1−σ2​\(W\)\)\\sigma\_\{2\}\(W\)/\(1\-\\sigma\_\{2\}\(W\)\), thereby overcoming the curvature\-dependent constants that arise in decentralized online g\-convex analyses\. Future directions include dynamic regret and time\-varying networks, bandit feedback, and applications in distributed control and large\-scale learning on curved spaces\.

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