Title: Riemannian Projection-free Online Learning
Venue: NeurIPS 2023
Accepted: yes
Keywords: Online learning

Riemannian optimization

projection-free optimization
Abstract: The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), 
for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or 
when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization
 subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on 
 Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions 
 are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization 
 on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and 
 when a separation oracle is available, our algorithms achieve $O(T^{\frac{1}{2}})$, $O(T^{\frac{3}{4}})$ and $O(T^{\frac{1}{2}})$ adaptive regret guarantees in the full
  information setting, the bandit setting with one-point feedback and the bandit setting with two-point feedback, respectively. When a linear optimization oracle is 
  available, we obtain regret rates of $O(T^{\frac{3}{4}})$ for geodesically convex losses 
and $O(T^{\frac{2}{3}}\log T)$ for strongly geodesically convex losses.

Riemannian Projection-free Online Learning

Abstract

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting , and while there has been growing interest in optimization on Riemannian manifolds , there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle . For geodesically convex losses, and when a separation oracle is available, our algorithms achieve O ( T 1 / 2 ) , O ( T 3 / 4 ) and O ( T 1 / 2 ) adaptive regret guarantees in the full information setting, the bandit setting with one-point feedback and the bandit setting with two-point feedback, respectively. When a linear optimization oracle is available, we obtain regret rates of O ( T 3 / 4 ) for geodesically convex losses and O ( T 2 / 3 log T ) for strongly geodesically convex losses.

1 Introduction

Online convex optimization (OCO) offers a framework for modeling sequential decision-making problems (Hazan et al., 2016). The standard setting depicts the learning process as a zero-sum game between a learner and an adversary. At round t , the learner selects a decision x t from a convex set K and observes the encountered convex loss function f t . The learner's goal is to minimize regret , defined as

In Euclidean space, OCO boasts a robust theoretical foundation and numerous real-world applications, such as online load balancing (Molinaro, 2017), optimal control (Li et al., 2019), revenue maximization (Lin et al., 2019), and portfolio management (Jézéquel et al., 2022). The standard approach for OCO is online gradient descent (OGD), which performs

where Π K represents the orthogonal projection onto K , ensuring the sequence { x t } T t =1 remains feasible. However, the projection operation can be computationally expensive in high-dimensional or complex feasible sets. Projection-free online learning provides a reasonable way to handle this situation. At the heart of this approach is the understanding that, in many cases, the complexity of

implementing an optimization oracle can be significantly lower than that of the orthogonal projection. As a result, both practical and theoretical interests lie in replacing the projection operation with these optimization oracles. The two most well-known projection-free optimization oracles are:

Whereas much attention has been given to the development of sub-linear regret guarantees using optimization oracles in Euclidean space (Hazan & Kale, 2012; Levy & Krause, 2019; Hazan & Minasyan, 2020; Wan et al., 2022; Garber & Kretzu, 2022; Mhammedi, 2022a; Wan et al., 2023), there is considerably less research on projection-free optimization on Riemannian manifolds . There are numerous scenarios where the space of feasible parameters is not a convex subset of Euclidean space, but instead has a manifold structure with a Riemannian metric; examples include non-negative PCA (Montanari & Richard, 2015), K -means clustering (Carson et al., 2017) and computing WassersteinBarycenters (Weber & Sra, 2022b). However, there has been significantly less research on efficiently computing projections or projection-free optimization in the Riemannian setting, even for highlystructured geodesically convex (gsc-convex) feasible sets like:

where each h i ( x ) is gsc-convex. In this paper we focus on projection-free optimization on Riemannian manifolds in the online setting. A Riemannian version of Online Gradient Descent (R-OGD) was given by Wang et al. (2021):

where K is now a gsc-convex set, and Π K is the metric projection onto K . This method achieves sub-linear regret guarantees for Riemannian OCO. The metric projection is often the most expensive operation, and it is not always clear how to implement it on a manifold. In the present work, we propose the use of two alternative operations: a separation oracle (SO) or a linear optimization oracle (LOO) on the Riemannian manifold, with definitions deferred to Section 3. It is important to note that, in Euclidean space, both oracles rely on the definition of a hyperplane. In the realm of Riemannian manifolds, there are two natural extensions of a hyperplane: one is the sub-level set of a Busemann function, known as a horosphere (Bridson & Haefliger, 2013); the other relies on the inverse exponential map, as outlined in (2). While a horosphere is gsc-convex, the corresponding separation oracle exists if every boundary point of the feasible gsc-convex set has a locally supporting horosphere (Borisenko, 2002). The existence of a separation oracle is assured for any gsc-convex sets on Hadamard manifolds (Silva Louzeiro et al., 2022), if defined via the inverse exponential map. Hence, we adopt this latter definition. It is worth noting that this object is typically non-convex (Kristály et al., 2016), leading to geometric complexities that necessitate careful management. Also, our work builds upon the findings of Garber & Kretzu (2022) and inherits the adaptive regret guarantee for gsc-convex losses, as defined by Hazan & Seshadhri (2009):

Our main contributions are summarized in Table 1. More specifically, and

Given a separation oracle, we attain adaptive regret bounds of O ( T 1 / 2 ) , O ( T 3 / 4 ) and O ( T 1 / 2 ) for gsc-convex losses in the full information setting, the bandit convex optimization setting with onepoint feedback 1 , and the bandit convex optimization setting with two-point feedback, respectively.

Assuming access to a linear optimization oracle, we provide algorithms that enjoy O ( T 3 / 4 ) adaptive regret for gsc-convex losses and O ( T 2 / 3 log T ) regret for strongly gsc-convex losses.

We highlight some key differences between convex sets on a Hadamard manifold and in Euclidean space. In particular, shrinking a gsc-convex set towards an interior point does not preserve convexity, and the Minkowski functional on a Hadamard manifold is non-convex. These results, which may not be well-known within the machine learning community, could be of independent interest.

The technical challenges of this paper can be divided into two parts.

Firstly, for the separation oracle, we need to bound the "thickness" of part of the feasible set cut by the separating hyperplane. While in Euclidean space, this can be achieved through a convex combination argument (Garber & Kretzu, 2022), the task becomes challenging on manifolds due to the varying metric at different points and the non-convex nature of the separating hyperplane. Fortunately, this problem can be addressed using the Jacobi field comparison technique. Also, unlike in Euclidean space, one cannot directly construct a separation oracle for (1 -δ ) K using a separation oracle for K . This poses significant challenges in the bandit setting. Nonetheless, we have identified a novel solution to this issue in the two-point feedback setting.

Secondly, in Euclidean space, the linear optimization oracle is typically invoked by online FrankWolfe (OFW) (Hazan & Kale, 2012; Kretzu & Garber, 2021) to achieve no-regret online learning. The analysis of OFW relies on the fact that the Hessian of a linear function is zero everywhere. But on Hadamard manifolds, such functions' existence implies that the manifold has zero sectional curvature everywhere (Kristály et al., 2016). The algorithms in Garber & Kretzu (2022) do not require affinity and serve as a starting point for our results. However, the analysis still needs to be conducted carefully due to the non-convexity of the separating hyperplane on manifolds.

2 Related Work

In this section, we briefly review previous work on projection-free online learning in Euclidean space as well as online and projection-free optimization on Riemannian manifolds.

2.1 Projection-free OCO in Euclidean Space

Linear Optimization Oracle. The pioneering work of Hazan & Kale (2012) first introduced an online variant of the Frank-Wolfe algorithm (OFW) and achieved O ( T 3 / 4 ) regret for convex functions. Hazan & Minasyan (2020) proposed a randomized algorithm that leverages smoothness to achieve O ( T 2 / 3 ) expected regret. The insightful analysis of Wan & Zhang (2021) and Kretzu & Garber (2021) demonstrated that OFW indeed attains O ( T 2 / 3 ) regret for strongly convex functions. Garber & Kretzu (2022) showed that it is possible to achieve O ( T 3 / 4 ) adaptive regret and O ( T 2 / 3 log T ) regret for convex and strongly convex functions, respectively. Mhammedi (2022b) illustrated how to obtain ˜ O ( T 2 / 3 ) regret for convex functions, where ˜ O ( · ) hides logarithmic terms. Our results are in line with those of Garber & Kretzu (2022) and inherit the adaptive regret guarantee.

Separation Oracle and Membership Oracle. Levy & Krause (2019) demonstrated that it is possible to achieve O ( √ T ) and O (log T ) regret bounds for convex and strongly convex functions when the feasible set is a sublevel set of a smooth and convex function. Mhammedi (2022a) generalized the idea of Levy & Krause (2019) and showed how to obtain O ( √ T ) and O (log T ) regret guarantees for general convex sets. Garber & Kretzu (2022) provided algorithms that ensure O ( √ T ) and O ( T 3 / 4 ) adaptive regret guarantees for convex losses in the full information and bandit settings, respectively.

2.2 Online and Projection-free Optimization on Manifolds

Online Optimization on Manifolds. Bécigneul & Ganea (2019) demonstrated that a series of adaptive optimization algorithms can be implemented on a product of Riemannian manifolds, with each factor manifold being assigned a learning rate. Antonakopoulos et al. (2020) proposed using Follow the Regularized Leader with a strongly gsc-convex regularizer to achieve O ( √ T ) regret when the loss satisfies Riemannian Lipschitzness. Wang et al. (2021) introduced Riemannian OGD (R-OGD) and showed regret guarantees in full information and bandit convex optimization settings. Hu et al. (2023) considered achieving optimistic and dynamic regret on Riemannian manifolds.

Projection-free Optimization on Manifolds. Rusciano (2018) provided a non-constructive cutting hyperplane method on Hadamard manifolds. By comparison, our algorithms are constructive and deterministic. Weber & Sra (2022b) proposed Riemannian Frank-Wolfe (RFW) for gsc-convex optimization and showed some practical applications on the manifold of SPD matrices. In a subsequent work, Weber & Sra (2022a) generalized RFW to the stochastic and non-convex setting. We use RFW as a subroutine to invoke the linear optimization oracle and establish sub-linear regret guarantees. Hirai et al. (2023) implemented the interior point method on Riemannian manifolds and used a self-concordant barrier to enforce the constraint.

3 Preliminaries and Assumptions

In this section, we lay the groundwork for our study by presenting an overview of Riemannian manifolds, the separation oracle and the linear optimization oracle within these spaces. Additionally, we establish key definitions and assumptions that will be integral to the following sections.

Riemannian Manifolds. We provide key notations in Riemannian geometry that will be employed throughout this paper. Readers looking for a more comprehensive treatment are encouraged to consult Petersen (2006); Lee (2018). Our proof also relies on the concept of the Jacobi field , and we provide some backgrounds in Appendix E.1. We consider an n -dimensional smooth manifold M equipped with a Riemannian metric g . This metric confers a point-wise inner product ⟨ u , v ⟩ x at every point x ∈ M , where u , v are vectors in the tangent space T x M at x . This tangent space, a vector space of dimension n , encompasses all vectors tangent to x . The Riemannian metric also determines the norm of a tangent vector u as: ∥ u ∥ := √ ⟨ u , u ⟩ . Ageodesic γ ( t ) : [0 , c ] →M is a piecewise smooth curve with a constant velocity that locally minimizes the distance between its endpoints, say, x and y . The Riemannian distance between these two points is given by d ( x , y ) := ∫ c 0 ∥ ˙ γ ( t ) ∥ dt = ∫ c 0 ∥ ˙ γ (0) ∥ dt . It's important to note that the Riemannian distance remains invariant under reparameterizations of γ ( t ) . Consider a geodesic γ ( t ) : [0 , 1] → M with γ (0) = x , γ (1) = y and ˙ γ (0) = v . The exponential map Exp x ( v ) transforms v ∈ T x M to y ∈ M , and the inverse exponential map Exp -1 x y performs the inverse operation, mapping y ∈ M to v ∈ T x M . The inverse exponential map also offers a handy way to express the Riemannian distance: d ( x , y ) = ∥ Exp -1 x y ∥ .

The sectional curvature at a point x ∈ M is contingent on two-dimensional subspaces of T x M , and describes the curvature near x . Generally, geodesics diverge on manifolds with negative sectional curvature, converge on manifolds with positive sectional curvature, and manifolds with zero sectional curvature are locally isometric to Euclidean space. In line with Zhang & Sra (2016); Wang et al. (2021), we primarily explore Hadamard manifolds, which are simply connected manifolds with non-positive curvature that admit a unique global minimizing geodesic between any pair of points. A set K ⊆ M is geodesically convex (gsc-convex) if it includes the geodesic connecting x and y for any x , y ∈ K . A µ -strongly gsc-convex (or gsc-convex, when µ = 0 ) function f : M→ R fulfills

for any x , y ∈ M , where ∇ f ( x ) ∈ T x M is the Riemannian gradient.

Optimization Oracles on Riemannian Manifolds. In this part, we introduce the concept of a separation oracle and a linear optimization oracle on Riemannian manifolds.

A separation oracle, given a point y not in the gsc-convex set K , returns a non-convex separating hyperplane that satisfies the following condition:

where g ∈ T y M . Even with the non-convexity of the separating hyperplane, for certain gsc-convex sets like K = { x | max 1 ≤ i ≤ m h i ( x ) ≤ 0 } where each h i ( x ) is gsc-convex, a separation oracle can be efficiently implemented by Lemma 18 and Remark 4. 2

On the other hand, a linear optimization oracle is responsible for solving the following problem:

on a gsc-convex set K , where x 0 ∈ K and g ∈ T x 0 M . Although this objective is not gsc-convex, it can still be solved in closed form for certain problems. Examples include computing the geometric mean and the Bures-Wasserstein barycenter on the manifold of SPD matrices (Weber & Sra, 2022b).

In this paper, we rely on a series of definitions and assumptions, which we introduce here for clarity and reference in subsequent sections.

Assumption 1. The manifold M is Hadamard with sectional curvature bounded below by κ , so the sectional curvature of M lies in the interval [ κ, 0] .

Assumption 2. The manifold M is a homogeneous Hadamard manifold with sectional curvature bounded below by κ . For every t ∈ [ T ] , the inequality | f t ( x ) | ≤ M holds. For the bandit setting with two-point feedback, we additionally require M to be symmetric.

Assumption 3. The set K ⊆ M is a gsc-convex decision set and satisfies B p ( r ) ⊆ K ⊆ B p ( R ) . Here, B p ( r ) represents the geodesic ball centered at p ∈ K with radius r .

Assumption 4. For every t ∈ [ T ] and x ∈ B p ( R ) , the function f t ( x ) is gsc-convex (or strongly gsc-convex), and the norm of its gradient is bounded by G , i.e., ∥∇ f t ( x ) ∥ ≤ G .

Let us make two important comments. First, the homogeneity and the symmetry of M allows us to employ the unbiased estimator presented in Wang et al. (2023) for the bandit setting. It should be noted that homogeneous and symmetric Hadamard manifolds include two of the most commonly used ones: the hyperbolic space and the manifold of SPD matrices. Second, the projection onto a geodesic ball, denoted as Π B p ( R ) ( · ) , is considered projection-free as it can be computed to an ϵ precision using log( 1 / ϵ ) bisections 3 .

Definition 1. We define a geometric constant ζ as ζ := 2 R √ -κ coth(2 R √ -κ ) .

Definition 2. Fixing p ∈ K , for any c ∈ (0 , ∞ ) , we define c K = { Exp p ( c Exp -1 p x ) | x ∈ K} .

Definition 3. We call ˜ y ∈ M an infeasible projection of y ∈ M onto a simply connected closed set ˜ K ⊆ M if for every point z ∈ ˜ K , the inequality d (˜ y , z ) ≤ d ( y , z ) holds. We define O IP ( ˜ K , y ) as an infeasible oracle for ˜ K which, given any y , returns an infeasible projection of y onto ˜ K .

We note that, in Euclidean space where the sectional curvature is zero everywhere, we have ζ = lim κ → 0 2 R √ -κ coth(2 R √ -κ ) = 1 . We also observe that the definition of the infeasible projection in Garber & Kretzu (2022) requires ˜ K to be convex, which is indeed unnecessary. This distinction is essential because, in the case of the separation-oracle-based OCO, we construct an infeasible projection oracle onto ˜ K = (1 -δ ) K , which may be non-convex (Theorem 6).

4 Warm-up: the High-level Idea

We briefly illustrate the overarching strategy of achieving regret guarantees. Our basic algorithm is Algorithm 1, which generates a sequence { y t } T t =1 by R-OGD that does not necessarily fall within the feasible set. A key insight is that we can build an infeasible projection oracle using either a separation oracle or a linear optimization oracle, resulting in a sequence { ˜ y t } T t =1 that exhibits a desirable regret guarantee (as shown in Lemma 1). The design of the infeasible projection oracle rests on a straightforward fact: whenever y t deviates significantly from K , we can call upon either oracle to produce a descent direction and then apply Lemma 2 to gauge the progress. Additional error terms, arising from the fact that ˜ y t may not necessarily lie in K , can be quantified by leveraging the boundedness of the gradient in Assumption 4.

Algorithm 1: Infeasible Riemannian OGD

We have the following guarantee for Algorithm 1.

Lemma 1. (Proof in Appendix A.1) Assume ˜ y t ∈ B p ( R ) and let ∇ t = ∇ f t (˜ y t ) and K ⊆ B p ( R ) be a gsc-convex subset of M . Consider ˜ K ⊆ K as a simply connected and compact set, and O IP ( ˜ K , · ) be an infeasible projection oracle as in Definition 3.

Suppose all losses are gsc-convex on B p ( R ) . Fix some η > 0 and let η t = η for all t ≥ 1 . Algorithm 1 guarantees that the adaptive regret is upper-bounded by:

Suppose all losses are α -strongly gsc-convex on B p ( R ) for some α > 0 . Let η t = 1 αt for all t ≥ 1 . Algorithm 1 guarantees that the static regret is upper bounded by:

Remark 1. To apply Lemma 1, we need to ensure that ˜ y t ∈ B p ( R ) for any t ∈ [ T ] . In the case of a separation oracle, we have ˜ K = (1 -δ ) K for some δ ∈ (0 , 1) , and ˜ y t ∈ K ⊆ B p ( R ) by Lemma 4. With a linear optimization oracle, we have ˜ K = K , and ˜ y t ∈ B p ( R ) is guaranteed by Lemma 6.

Lemma 2. (Proof in Appendix A.2) Consider ˜ K ⊆ K ⊆ B p ( R ) as a simply connected and compact subset of M . If y / ∈ ˜ K and g ∈ T y M satisfies -〈 Exp -1 y z , g 〉 ≥ Q , where Q > 0 , then consider ˜ y = Exp y ( -γ g ) . For γ = Q ζC 2 and ∥ g ∥ ≤ C , assume d ( y , z ) ≤ 2 R , then we have

Unlike Garber & Kretzu (2022), in Lemma 2, we also do not require ˜ K to be gsc-convex.

5 Riemannian OCO with a Separation Oracle

In this section, we show how to use a separation oracle to construct an infeasible projection oracle and achieve sublinear regret guarantees. We note that we rely on an infeasible projection oracle onto (1 -δ ) K rather than directly onto K . While we have a separation oracle that results in 〈 -Exp -1 y x , g 〉 > 0 , using Lemma 2 on this separating hyperplane may lead to minuscule progress, given that Q can be arbitrarily small. Consequently, achieving sublinear regret with only O ( T ) oracle calls becomes unfeasible. In contrast, constructing an infeasible projection onto (1 -δ ) K always ensures meaningful progress, as quantified by Lemmas 3 and 4.

Algorithm 2: Infeasible Projection onto (1 -δ ) K with a Riemannian Separation Oracle

.

Lemma 3. (Proof in Appendix B.1) Let y ∈ B p ( R ) \ K and let g ∈ T y M be the output of the separation oracle for y . Then, under Assumptions 1 and 3, we have that 〈 -Exp -1 y z , g 〉 > δ ¯ r ∥ g ∥ for any z ∈ (1 -δ ) K , where

In Euclidean space, we can establish that ⟨ y -z , g ⟩ > δr ∥ g ∥ (Garber & Kretzu, 2022, Lemma 11). However, as indicated in Lemma 3, the result on manifolds is significantly worse with respect to R , given the exponential nature of sinh . It is an interesting line of inquiry to explore whether this dependence is unavoidable. 4

Based on Lemma 3, to implement an infeasible projection oracle onto (1 -δ ) K , the number of calls to the separation oracle is bounded in Lemma 4.

Lemma 4. (Proof in Appendix B.2) Under Assumptions 1 and 3. Let 0 < δ < 1 and set γ i = δ ¯ r ∥ g ∥ .

Algorithm 2 executes at most ζ ( d ( y 0 , (1 -δ ) K ) 2 -d ( y , (1 -δ ) K ) 2 ) δ 2 ¯ r 2 +1 iterations and returns y ∈ K such that d ( y , z ) 2 ≤ d ( y 0 , z ) 2 holds for any z ∈ (1 -δ ) K .

In the full information setting, with a separation oracle, infeasible R-OGD is shown in Algorithm 3.

We can show the following regret guarantee for Algorithm 3.

Theorem 1. (Proof in Appendix B.3) Under Assumptions 1, 3 and 4. Set η = 2 R G √ ζT and δ = 1 2 √ T , then the regret of Algorithm 3 is upper bounded by √

and the number of calls to the separation oracle is O ( T ) .

Moving on, we demonstrate how to achieve a sublinear regret guarantee in the bandit convex optimization setting. A major challenge is that, while in Euclidean space, we can construct a separation oracle on (1 -δ ) K using the separation oracle on K (Garber & Kretzu, 2022, Lemma 11.). On Hadamard manifolds, (1 -δ ) K can even be non-convex (Theorem 6), thus a separation oracle for (1 -δ ) K may not exist. For Riemannian BCO with one-point feedback, in Algorithm 4, we address this by resorting to a non-standard setting: we play ˜ z t ∈ K but we receive feedback at z t where z t , ˜ z t are nearby points. We present the algorithm and the corresponding regret guarantee in Algorithm 4 and Theorem 2.

Algorithm 4: One-point bandit convex optimization on manifolds with a separation oracle

Data: feasible gsc-convex set K , radii ( R,r ) , step-size η , squeeze parameters ( δ, δ ′ , τ ) , ˜ y 1 = p . for t = 1 , . . . , T do

˜ y t +1 ← Output of Algorithm 2 with K , radius r , squeeze parameter δ and initial point y t +1 . end

Theorem 2. (Proof in Appendix B.4) Under Assumptions 2, 3 and 4. Set η = T -1 2 , δ = T -1 4 , τ = T -1 4 . Then regret of Algorithm 4 is upper bounded by

and the number of calls to the separation oracle is O ( T ) .

Remark 2. An acute reader may notice a discrepancy between the step-size η = O ( T -1 2 ) in our work and the step-size η = O ( T -3 4 ) in Garber & Kretzu (2022, Theorem 15). It is important to highlight that our η is equivalent to nη δ ′ as per Garber & Kretzu (2022), ensuring that the parameters are in fact consistent. This reasoning is also applicable to Theorem 3.

Interestingly, in the context of the two-point feedback setting, we can get rid of the non-standard setting by adapting the algorithm. In Algorithm 5, we adhere to the loop invariants: x t ∈ β K and y t ∈ K . Each round involves constructing an unbiased gradient estimator g t at x t using the two-point feedback. Subsequently, we map g t to the tangent space of y t and execute online gradient descent in that space. A key advantage of this parallel transport is the ability to employ the separation oracle on K to construct an infeasible projection onto (1 -δ ) K . Upon a meticulous analysis, we observe that, at each round, an additional distortion arises from this parallel transport:

This distortion accumulates to O ( √ T ) when choosing δ ′ = O ( 1 √ T ) , ensuring the regret bound remains unaffected. The regret assurance of Algorithm 5 is detailed in Theorem 3.

Algorithm 5: Two-point bandit convex optimization on manifolds with a separation oracle

Data: feasible gsc-convex set K , radii ( R,r ) , step-size η , parameters ( δ, δ ′ , β ) :

Initialize x 1 ∈ β K , y 1 = Exp p ( Exp -1 p x 1 β ) // y 1 ∈ K

for

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end

Theorem 3. (Proof in Appendix B.5) Under Assumptions 2, 3 and 4. Set η = 1 , δ = 1 -β = T -1 2 , then the regret of Algorithm 5 is upper bounded by

and the number of calls to the separation oracle is O ( T ) .

6 Riemannian OCO with a Linear Optimization Oracle

In this section, we focus on performing projection-free OCO on Riemannian manifolds utilizing a linear oracle. The linear oracle is invoked inside the Riemannian Frank-Wolfe (RFW) algorithm (Weber & Sra, 2022b). We outline how to obtain a separating hyperplane by RFW, as detailed in Algorithm 6 and Lemma 5.

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Algorithm 6: Separating Hyperplane via RFW

Lemma 5. (Proof in Appendix C.1) Under Assumptions 1 and 3. For any y ∈ B p ( R ) , Algorithm 6 terminates after at most ζ ⌈ (27 R 2 /ϵ ) -2 ⌉ iterations and returns ˜ x ∈ K satisfies:

At least one of the following holds: d (˜ x , y ) 2 ≤ 3 ϵ or ∀ z ∈ K : 〈 Exp -1 y z , Exp -1 y ˜ x 〉 ≥ 2 ϵ .

Remark 3. Note that the second item of Lemma 5 provides a separating hyperplane between y and K . One of the challenges in its proof is to find an analog of the Euclidean identity ⟨ z -y , ˜ x -y ⟩ = ∥ ˜ x -y ∥ 2 2 -⟨ z -˜ x , y -˜ x ⟩ on manifolds, which initially appears to be a daunting task. However, a clever application of Lemma 30 (Appendix E) provides a solution.

Algorithm 7: Closer Infeasible Projection via LOO

Algorithm 7 demonstrates how to "pull" an initial point y 0 towards K using RFW, while Lemma 6 verifies that the output of Algorithm 7 is indeed an infeasible projection onto K .

Lemma 6. (Proof in Appendix C.2) Under Assumptions 1 and 3. Fix ϵ > 0 . Setting γ = 2 ϵ 2 ,

Given that Lemma 6 provides an infeasible projection oracle, we can combine Algorithms 1 and 7 to achieve sublinear regret by setting the error tolerance as ϵ = o (1) . However, RFW requires Θ( 1 / ϵ ) = ω (1) iterations in the worst-case scenario (Lemma 5), and the resulting algorithm necessitates ω ( T ) calls to the linear optimization oracle. Garber & Kretzu (2022) utilize a block trick to address this challenge: the time horizon T is broken into B blocks, and the infeasible projection is computed once in each block. We demonstrate that this trick can be implemented on Riemannian manifolds in Algorithm 8. We present the regret guarantees for gsc-convex and strongly gsc-convex losses in Theorem 4 and Theorem 5, respectively.

Theorem 4. (Proof in Appendix C.3) Under Assumptions 1, 3 and 4. Fixing η i and ϵ i as η = Rζ G T -3 4 and ϵ = 60 R 2 ζ 2 T -1 2 , respectively, for any i ∈ { 1 , . . . , T B } . Setting B = 5 T 1 / 2 . Then the regret of

Algorithm 8: Block OGD on manifolds with a linear optimization oracle

Algorithm 8 for gsc-convex losses is bounded by

and the number of calls to the linear optimization oracle is bounded by T .

Theorem 5. (Proof in Appendix C.4) Under Assumptions 1, 3 and 4. Suppose all losses { f t } T t =1 are α -strongly gsc-convex for some known α > 0 and T ≥ 3 B . Choosing ϵ i = ( 20 G α ( i +3) ) 2 ζ and

η i = 2 αiB for any i = 1 , . . . , T B . With B = ( αR G ) 2 3 T 2 / 3 and assume T ≥ 3 B , the regret guarantee of Algorithm 8 is bounded by

And the number of total calls to the linear optimization oracle is bounded by ζT .

7 Conclusion and Perspective

This paper pioneers the exploration of projection-free online optimization on Riemannian manifolds. The primary technical challenges originate from the non-convex nature of the Riemannian hyperplane and the variable metric. These challenges are tackled effectively with the aid of the Jacobi field comparison, enabling us to establish a spectrum of sub-linear regret guarantees. Interested readers may question the difficulty of generalizing these techniques from Hadamard manifolds to CAT ( κ ) manifolds. Some hints toward this generalization are provided in Appendix E.3.

There are several promising directions for future research. First, there exists the potential to refine the regret bounds, particularly for strongly gsc-convex losses via a separation oracle. In the context of the separation oracle, reducing the dependence on the number of calls about the diameter of the decision set would be an intriguing objective. Moreover, devising an efficient method to optimize the linear optimization oracle objective, argmin x ∈K 〈 g , Exp -1 x 0 x 〉 , remains a notable open problem. This paper does not discuss the membership oracle, primarily because related work (Mhammedi, 2022a; Lu et al., 2023) heavily relies on the convexity of the Minkowski functional in Euclidean space, a property not guaranteed to hold on Hadamard manifolds (Theorem 7). However, this does not rule out the potential for executing OCO or convex optimization on manifolds using a membership oracle. Thus, uncovering alternative strategies to tackle this issue remains a compelling research question.

Acknowledgments

We would like to thank five anonymous referees for constructive comments and suggestions. We gratefully thank the AI4OPT Institute for funding, as part of NSF Award 2112533. We also acknowledge the NSF for their support through Award IIS-1910077. GW would like to acknowledge an ARC-ACO fellowship provided by Georgia Tech. We would also like to thank Xi Wang from UCAS and Andre Wibisono from Yale for helpful discussions.

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A Omitted Proofs in Section 4

A.1 Proof of Lemma 1

Proof. Fix t , since ˜ y t +1 is an infeasible projection of y t +1 onto ˜ K , and y t +1 = Exp ˜ y t ( -η t ∇ t ) , by Lemma 29, we have for any x ∈ ˜ K ,

To verify that ζ represents the correct geometric distortion, we need to show d (˜ y t , x ) ≤ 2 R holds for any ˜ y t ∈ B p ( R ) and x ∈ ˜ K . Since ˜ K ⊆ K ⊆ B p ( R ) , we can demonstrate this by the triangle inequality:

Rearranging Equation (4), we have

Fix 1 ≤ s ≤ e ≤ T and sum over [ s, e ] , we have

Using gsc-convexity of f t ( · ) and set η t = η , we have

In case all losses are α -strongly gsc-convex, using f t ( y ) -f t ( x ) ≤ 〈 ∇ f t ( y ) , -Exp -1 y x 〉 -αd ( x , y ) 2 2 and set ( s, e ) = (1 , T ) , we have

Setting η t = 1 αt , we get the guarantee for the strongly gsc-convex case.

A.2 Proof of Lemma 2

where the first inequality relies on Lemma 29 and d ( y , z ) ≤ 2 R , the third inequality holds because -〈 Exp -1 y z , g 〉 ≥ Q , and for the last line, we plug in γ = Q ζC 2 .

B Omitted Proofs in Section 5

The technical difficulty lies in constructing an infeasible projection oracle using a separation oracle. The following lemma is instrumental to the later results.

B.1 Proof of Lemma 3

Proof. Let x = Exp y ( δ ¯ r g ∥ g ∥ + Exp -1 y z ) , if we can show x ∈ K , then by the definition of the separation oracle, we have

which in turn implies 〈 -Exp -1 y z , g 〉 > δ ¯ r ∥ g ∥ . Note that when ¯ r = 0 , x = z ∈ (1 -δ ) K ⊂ K , thus there exists positive ¯ r such that x ∈ K for any z ∈ (1 -δ ) K . Let ˜ r = √ -κ ( R + r ) sinh( √ -κ ( R + r )) · r . By Lemma 19, we have B z ( δ ˜ r ) ⊂ K . Thus we only need to ensure d ( x , z ) ≤ δ ˜ r because this implies x ∈ B z ( δ ˜ r ) ⊂ K . We define an admissible family of curves:

where v = g ∥ g ∥ . Then Γ(1 , 0) = z and Γ(1 , 1) = x . Let J s ( t ) := ∂ Γ( t,s ) ∂s , which is a variation field of the variation Γ , and is thus a Jacobi field. We can compute J s (0) = ∂ Γ(0 ,s ) ∂s = 0 and by Lemma 14,

To apply Lemma 15, we need to reparametrize Γ to make it unit speed. Let Γ( t, s ) = ˜ Γ ( t R ( s ) , s ) where

Then we have J s ( t ) = ˜ J s ( R ( s ) t ) and D t J s ( t ) = R ( s ) D t ˜ J s ( R ( s ) t ) . Now we can apply Lemma 15:

We would like to have d ( x , z ) ≤ δ ˜ r , and it suffices to choose ¯ r such that

We then find a valid ¯ r is

B.2 Proof of Lemma 4

Proof. We use k to denote the number of iterations in Algorithm 2, then for any i < k , we have y i / ∈ K . By Lemma 3, we have 〈 -Exp -1 y i z , g i 〉 ≥ δ ¯ r ∥ g i ∥ for any z ∈ (1 -δ ) K . We then invoke Lemma 2 with C = ∥ g i ∥ , Q = δ ¯ r ∥ g i ∥ to get

holds for any z ∈ (1 -δ ) K . To ensure the geometric distortion ζ is valid here, we need to prove d ( y i , z ) ≤ 2 R holds for any i ≥ 1 and z ∈ (1 -δ ) K , which can be guaranteed by induction. The case of i = 1 is straightforward. As y 1 ∈ B p ( R ) , z ∈ (1 -δ ) K ⊆ B p ( R ) , we have

Now assume d ( y i , z ) ≤ 2 R holds for some i ≥ 1 and any z ∈ (1 -δ ) K , then by Lemma 29 and Lemma 2,

Thus, we know d ( y i , z ) ≤ 2 R holds for any i ≥ 1 and z ∈ (1 -δ ) K .

By Equation (7), d ( y , z ) 2 ≤ d ( y 1 , z ) 2 for all z ∈ (1 -δ ) K . Since y 1 is the projection of y 0 onto B p ( R ) , we have d ( y 1 , z ) 2 ≤ d ( y 0 , z ) 2 holds for z ∈ B p ( R ) . Thus, we indeed have d ( y , z ) 2 ≤ d ( y 0 , z ) 2 for any z ∈ (1 -δ ) K .

It remains to bound the number of iterations in Algorithm 2. We must be careful because, on manifolds, (1 -δ ) K is not guaranteed to be gsc-convex (Theorem 6). Note that argmin x ∈ (1 -δ ) K d ( y i , x ) 2 is consistently non-empty because (1 -δ ) K is a closed set. Let x ∗ i ∈ argmin x ∈ (1 -δ ) K d ( y i , x ) 2 and invoke Equation (7), we have

The first inequality is due to the definition of x ∗ i +1 and the second one follows from Equation (7). We also need to show d ( y 1 , (1 -δ ) K ) ≤ d ( y 0 , (1 -δ ) K ) to finish the proof. We have

where the first inequality is again due to the definition of x ∗ 1 , while the second one comes from y 1 is the metric projection of y 0 onto B p ( R ) and x ∗ 0 ∈ (1 -δ ) K ⊆ B p ( R ) . Reminding y = y k and unrolling the recurrence, we have

Thus, in the worst case, we need

iterations to stop.

B.3 Proof of Theorem 1

Proof. Combining Algorithm 3 with Lemma 4, we know ˜ y t ∈ K thus Algorithm 3 plays a feasible point at each round. For a fixed interval I = [ s, e ] ⊆ [ T ] , let x I = argmin x ∈K ∑ e t = s f t ( x ) and ˜ x I = Exp p ((1 -δ ) Exp -1 p x I ) . Then

By Lemma 4, ˜ y t ∈ K is an infeasible projection of y t over (1 -δ ) K . Then by Lemma 1

On the other hand, by the gradient-Lipschitzness, we have

Combining the above two equations, we have

Setting η = 2 R G √ ζT and δ = 1 2 √ T < 1 , then we have

where we use ζ ≥ 1 . It remains to bound the number of calls to the separation oracle. Denote x ∗ t ∈ argmin x ∈ (1 -δ ) K d ( x , ˜ y t ) . Due to y t +1 = Exp ˜ y t ( -η ∇ f t (˜ y t )) ,

where the first inequality is because x ∗ t ∈ (1 -δ ) K , the second is due to the triangle inequality, while the third one follows from the definition of x ∗ t and y t +1 = Exp ˜ y t ( -η ∇ f t (˜ y t )) . Squaring both sides, we have

where the second inequality is because

Combining Equation (9) with Lemma 4, we can bound the number of separation oracle calls as:

where we use ˜ y 1 = p ∈ (1 -δ ) K and thus d (˜ y 1 , (1 -δ ) K ) = 0 to derive the third inequality.

B.4 Proof of Theorem 2

We first prove Lemma 7, which characterizes the relation between δ ′ and τ .

Lemma 7. For a gsc-convex subset K ⊆ M where M is Hadamard, any point y ∈ K and B p ( r ) ⊆ K ⊆ B p ( R ) , we have

Proof. The proof of this proposition takes inspiration from Lemma 19 (Wang et al., 2023), but with several significant adjustments. Notably, Lemma 19 hinges on the gsc-convexity of K to bound the radius of the geodesic ball B y (˜ r ) , where y ∈ (1 -τ ) K , which resides within K . However, the gsc-convexity of (1 + τ ) K is unknown. 5 Therefore, we must explore alternative strategies that leverage the gsc-convexity of K to meet our objectives.

Let v ∈ T y M be a unit vector such that ∥ v ∥ = 1 and z = Exp y ( θτr v ) . The goal is to find the maximum θ which ensures z ∈ (1 + τ ) K for any v . Let y ′ = Exp p ( Exp -1 p y 1+ τ ) and z ′ = Exp p ( Exp -1 p z 1+ τ ) . It is immediate to see z ′ ∈ K iff z ∈ (1 + τ ) K . We denote ξ τ ( s ) as the geodesic satisfying ξ τ (0) = y ′ and ξ τ (1) = z ′ .

We define an admissible family of curves:

We can verify that Γ(0 , 1) = y , Γ(1 , 0) = y ′ , Γ ( 1+ τ τ , 0 ) = Exp y ( 1+ τ τ · Exp -1 y y ′ ) = p , and Γ(1 , 1) = z ′ . We also denote w := Γ ( 1+ τ τ , 1 ) . The idea of the proof is to show d ( p , w ) ≤ r by Lemma 15, which implies w ∈ K since B p ( r ) ⊆ K . Combining with the fact that y ∈ K and z ′ is on the geodesic connecting y and w , we have z ′ ∈ K . Thus z ∈ (1 + τ ) K .

We notice that v ( t, s ) = ∂ Γ( s,t ) ∂s is a Jacobi field since it is a variation field of γ s ( t ) = Exp y ( t Exp -1 y ξ τ ( s )) . Let R ( s ) = d (Γ(0 , s ) , Γ(1 , s )) . To apply Lemma 15, we need to normalize the geodesic ˜ γ s ( t ) = γ s ( t R ( s ) ) . Since M is Hadamard, by Lemma 15

Remind that ∥ v (1 , s ) ∥ = ∥ ∥ ∥ ∂ξ τ ( s ) ∂s ∥ ∥ ∥ = ∥ ˙ ξ τ ( s ) ∥ = d ( y ′ , z ′ ) . Now we use Lemma 20 to bound d ( y ′ , z ′ ) . We construct ∆(¯ p , ¯ y , ¯ z ) in Euclidean space, a comparison triangle of ∆( p , y , z ) with comparison points ¯ y ′ ∈ [¯ p , ¯ y ] and ¯ z ′ ∈ [¯ p , ¯ z ] . We restrict d ( p , y ′ ) = d E (¯ p , ¯ y ′ ) and d ( p , z ′ ) = d E (¯ p , ¯ z ′ ) , then by Lemma 20, we have

On the other hand, it is immediate to verify ∆(¯ p , ¯ y , ¯ z ) is similar to ∆(¯ p , ¯ y ′ , ¯ z ′ ) by considering

Thus we have

where the first equation is due to the property of similar triangles, the second one follows from the definition of the comparison triangle, and the last one is due to the definition of z .

Combining Equations (11), (12) and (13), we have

We also need to apply Lemma 15 at t = 1+ τ τ . Since the sectional curvature of M is lower bounded by κ , we have

Putting Equations (14) and (15) together, we have

Now we can bound θ . Note that

Thus we can take

to ensure ∥ ∥ v ( 1+ τ τ , s )∥ ∥ ≤ r . The length of the curve v ( 1+ τ τ , s ) can be bounded by

This implies d ( p , w ) ≤ r and thus w ∈ K . Then we know z ′ ∈ K because z ′ lies on the geodesic connecting w and y , with both endpoints in K . This finally leads to the fact that z ∈ (1 + τ ) K .

Now we give the proof of Theorem 2.

Proof of Theorem 2. Algorithm 4 always plays feasible points because by

and Lemma 7, we have z t ∈ (1 + τ ) K and ˜ z t ∈ K .

By the first item of Lemma 23,

where ˆ f t ( x ) is a smoothed version of f t ( x ) . More specifically,

where ω is the volume form.

Applying Lemma 4 to Algorithm 4, we know ˜ y t is an infeasible projection of y t over (1 -δ ) K , which means

holds for any x ∈ (1 -δ ) K , where the second inequality is due to Lemma 29.

Equation (18) is equivalent to

By the third item of Lemma 23, we have

where ρ is a constant solely depends on K .

Combining Equations (17), (19), and (20), we have

holds for any x ∈ (1 -δ ) K , where we use the fact that d (˜ y s , x ) ≤ 2 R . We also need to bound E [ f t ( ˜ z t ) -ˆ f t (˜ y t ) ] and E [ ˆ f t (˜ x I ) -f t ( x I ) ] . For the former term,

where we use Lemma 22 and the gradient Lipschitzness. Similarly, for the latter term,

By the second term of Lemma 23, we have

Combining Equations (21), (22), (23) and (24), summing from t = s to e , we get

After plugging in η = T -1 2 , δ = T -1 4 , τ = T -1 4 and δ ′ = √ -κ ( R + r ) sinh ( √ -κ ( R + r ) ) T -1 4 r , we see δ ′ ≤ T -1 4 r and

Then as claimed.

Denote x ∗ t ∈ argmin x ∈ (1 -δ ) K d ( x , ˜ y t ) . We now bound the number of calls to the separation oracle. Due to y t +1 = Exp ˜ y t ( -η g t ) , d ( y t +1 , (1 -δ ) K ) ≤ d ( x ∗ t , y t +1 ) ≤ d ( x ∗ t , ˜ y t ) + d (˜ y t , y t +1 ) ≤ d (˜ y t , (1 -δ ) K ) + η ∥ g t ∥ . Squaring both sides, we have

where the second inequality is because d (˜ y t , (1 -δ ) K ) ≤ d (˜ y t , Exp p ((1 -δ ) Exp -1 p ˜ y t )) = d (˜ y t , Exp ˜ y t ( δ Exp -1 ˜ y t p )) = δd (˜ y t , p ) ≤ δR and ∥ g t ∥ = | f t ( z t ) | ≤ M . Combining Equation (26) with Lemma 4, we can bound the number of separation oracle calls as:

where the last inequality follows from ˜ y 1 = p ∈ (1 -δ ) K and thus d (˜ y 1 , (1 -δ ) K ) = 0 .

B.5 Proof of Theorem 3

Proof. We maintain the loop invariant x t ∈ β K . So by Lemma 19, B ( x t , δ ′ ) ⊆ K , which means z t and z ′ t are feasible points.

For the two-point feedback setting, by Lemma 23,

By Lemma 17 of Wang et al. (2023),

Thus,

where we use the ζ -smoothness of 1 2 d ( x , y ) 2 and the definition of δ ′ .

Since y t +1 is an infeasible projection of y ′ t +1 onto (1 -δ ) K ,

holds for any x ∈ (1 -δ ) K , which means

Taking expectation and using S δ ′ V δ ′ ≤ n δ ′ + n | κ | , we have

Combining Equations (28), (29), (30) and (31), and summing from t = s to e , we have

holds for any x ∈ (1 -δ ) K .

Following Equations (22) and (23), and taking x = Exp p ( (1 -δ ) Exp -1 p x ∗ ) , we have

and

In sum,

By choosing η = 1 , δ = 1 -β = T -1 2 , then δ ′ = O ( T -1 2 ) and we can get O ( √ T ) regret.

Denote x ∗ t ∈ argmin x ∈ (1 -δ ) K d ( x , y t ) and we again bound the number of calls to the separation oracle. Due to y ′ t +1 = Exp y t ( -η Γ y t x t g t ) ,

where we use ∥ g t ∥ = 1 2 | f t ( z t ) -f t (˜ z t ) | ≤ δ ′ G by Assumption 4. Squaring both sides, we have

where the second inequality is again due to y t ∈ K and d ( y t , (1 -δ ) K ) ≤ d ( y t , Exp p ((1 -δ ) Exp -1 p y t )) = d ( y t , Exp y t ( δ Exp -1 y t p )) = δd ( y t , p ) ≤ δR. Combining Equation (32) with Lemma 4, we can bound the number of separation oracle calls as:

where the last inequality follows from ˜ y 1 = p ∈ (1 -δ ) K and thus d (˜ y 1 , (1 -δ ) K ) = 0 . Since η = 1 , δ = 1 -β = T -1 2 and

Plugging these parameters into Equation (33), we have

C Omitted Proofs in Section 6

We have the following convergence guarantee for RFW.

C.1 Proof of Lemma 5

Proof. Since Algorithm 6 is indeed RFW with line-search (Algorithm 9) when f ( x ) = d ( x , y ) 2 / 2 , which is ζ gsc-smooth by Lemma 27 and ∇ f ( x ) = -Exp -1 x y , the upper bound on the number of iterations follows from Lemma 26 with L = ζ and D = 2 R . Item 1 follows from the line-search because f ( x i ) = d ( x i , y ) 2 / 2 does not increase as i increases.

For item 2 , Algorithm 6 stops when either d (˜ x , y ) 2 ≤ 3 ϵ or d (˜ x , y ) 2 > 3 ϵ and 〈 Exp -1 ˜ x y , Exp -1 ˜ x x 〉 ≤ ϵ . For the first case, item 2 obviously holds. Now we consider the second case. We first show 〈 Exp -1 y z , Exp -1 y ˜ x 〉 + 〈 Exp -1 ˜ x z , Exp -1 ˜ x y 〉 ≥ d (˜ x , y ) 2 as follows. By Lemma 30, we have

Adding the above two inequalities we have 〈 Exp -1 y z , Exp -1 y ˜ x 〉 + 〈 Exp -1 ˜ x z , Exp -1 ˜ x y 〉 ≥ d (˜ x , y ) 2 . Now item 2 follows from

For the third item, denote x ∗ = argmin x ∈K d ( x , y ) 2 , Suppose d ( y , K ) 2 = d ( x ∗ , y ) 2 < ϵ and d (˜ x , y ) 2 > 3 ϵ . Denote f ( x ) = d ( x , y ) 2 / 2 and ∇ f ( x ) = -Exp -1 x y . For the last iteration,

which implies

And thus

as claimed.

C.2 Proof of Lemma 6

Before proving Lemma 6, we need the following lemma.

Lemma 8. Consider Algorithm 7 and fix some ϵ such that 0 < 3 ϵ < d ( x 0 , y 0 ) 2 . Setting γ = 2 ϵ d ( x 0 , y 0 ) 2 , we have d ( x i , y i ) ≤ d ( x 0 , y 0 ) for any i .

Proof. For all i > 1 , we define the sequence y i by the relation y i = Exp x i -1 ((1 -γ ) Exp -1 x i -1 y i -1 ) , and γ ∈ [0 , 1) . From this, we can deduce the following sequence of inequalities:

The inequality follows since γ ∈ (0 , 1) .

By the first item of Lemma 5, for any i ≥ 1 , we have d ( x i , y i ) ≤ d ( x i -1 , y i ) . Combining this with the previous inequality, we get

where the last inequality follows from the fact that x 0 ∈ K and y 1 is the projection of y 0 onto B p ( R ) , while K ⊆ B p ( R ) .

Proof of Lemma 6. Given that y 1 is the projection of y 0 onto B p ( R ) and K ⊆ B p ( R ) , we have ∀ z ∈ K : d ( y 1 , z ) 2 ≤ d ( y 0 , z ) 2 . The lemma trivially holds when d ( x 0 , y 0 ) 2 ≤ 3 ϵ or d ( x 1 , y 1 ) 2 ≤ 3 ϵ .

Without loss of generality, we assume that d ( x 1 , y 1 ) > 3 ϵ . Let k > 1 be the number of iterations in Algorithm 7. This means that d ( x k , y k ) 2 ≤ 3 ϵ and d ( x i , y i ) 2 > 3 ϵ for any i < k . According to the second item of Lemma 5, we obtain ⟨ Exp -1 y i z , Exp -1 y i x i ⟩ ≥ 2 ϵ for all i < k and z ∈ K . Lemma 8 also gives us d ( x i , y i ) ≤ d ( x 0 , y 0 ) for all i < k .

Applying Lemma 2 with g = -Exp -1 y i x i , C = d ( x 0 , y 0 ) , and Q = 2 ϵ , we deduce that for every 1 ≤ i < k ,

Consequently, for any z ∈ K , the returned point y satisfies d ( y , z ) 2 ≤ d ( y 1 , z ) 2 ≤ d ( y 0 , z ) 2 . Because p ∈ K , we have d ( y , p ) ≤ d ( y 1 , p ) ≤ R , which implies y ∈ B p ( R ) . Note that x ∈ K as it is the output of Algorithm 6. Thus, we know Algorithm 7 returns ( x , y ) ∈ K × B p ( R ) .

Now we bound the number of iterations in Algorithm 7. Denote x ∗ i = argmin x ∈K d ( x , y i ) 2 for any i < k . Using Equation (34),

By applying the inequality d ( y 1 , K ) 2 ≤ d ( y 0 , K ) 2 and subsequently performing a telescoping sum, we can demonstrate that

After at most k -1 = ( ζd ( x 0 , y 0 ) 2 ( d ( x 0 , y 0 ) 2 -ϵ )) / 4 ϵ 2 iterations, we have d ( y k , K ) 2 ≤ ϵ . By the third item of Lemma 5, the next iteration will be the last one, and the returned points x , y satisfy d ( x , y ) 2 ≤ 3 ϵ , as claimed.

C.3 Proof of Theorem 4

We first prove Lemma 9, which bounds the regret of the infeasible projection for gsc-convex losses.

Lemma 9. With some fixed block size B , η i = η and ϵ i = ϵ for any i = 1 , . . . , T B . Assume all losses are gsc-convex. We use i ( t ) to denote the block index of round t . Then the regret of playing ˜ y i ( t ) -1 as defined in Algorithm 8 can be bounded by

Proof. For convenience, we assume B divides T . Let T i = { ( i -1) B +1 , . . . , iB } be the set of rounds in the i -th block. By Lemma 6, ˜ y i +1 is the infeasible projection of y iB +1 onto K :

holds for any x ∈ K . By Algorithm 8, we have

Thus

where the last inequality is due to Lemma 29 and ∥∇ f t ( x ) ∥ ≤ G . This is equivalent to

For a certain interval [ s, e ] ⊆ [ T ] , let i s and i e be the smallest and the largest block indices such that T i s and T i e are fully contained in [ s, e ] . Then we have

There are some undesirable terms in blocks i s -1 and i e +1 , and we can bound them by noticing

Combining Equations (35), (36) and (37), we have

where the last inequality is due to the number of blocks in [ s, e ] is upper bounded by T B .

Now we give the proof of Theorem 4.

Proof of Theorem 4. We still use i ( t ) to denote the block index of the t -th round. For an interval [ s, e ] ⊆ [ T ] , we can do the following decomposition

Algorithm 8 calls Algorithm 7 in each block to compute an infeasible projection. By Lemma 6 and Algorithm 8, ( x i , ˜ y i ) ∈ K × B p ( R ) , which satisfies d ( x i , ˜ y i ) 2 ≤ 3 ϵ i = 3 ϵ . Thus

Combining Equations (39), (40) and Lemma 9, we have

holds for any x ∈ K . By plugging in η , ϵ , and B , we get the claimed regret bound.

We also need to bound the number of calls to the linear optimization oracle. Note that d ( x 1 , ˜ y 1 ) = 0 due to the initialization, and for any i ≥ 2 , we have d ( x i , ˜ y i ) 2 ≤ 3 ϵ due to Lemma 6. By Algorithm 8, we have

Therefore

Squaring on both sides, we have

where ( a + b ) 2 ≤ 2( a 2 + b 2 ) is used.

By Lemma 6, on the i -th block, Algorithm 7 stops after at most

iterations. According to Lemma 5, each iteration calls the linear optimization oracle for at most ζ ⌈ 27 R 2 ϵ 2 -2 ⌉ times. Thus in the i -th block, the number of calls to the linear optimization oracle is bounded by

There are T B blocks in total, so the total number of calls to the linear optimization oracle is at most

Plugging in the values of B,η and ϵ , we find N calls ≤ T as claimed.

C.4 Proof of Theorem 5

Proof. Based on a similar argument as in Equation (35), we have

where we apply the strong gsc-convexity on the i -th block under the help of Lemma 28; the last inequality follows from the definition of η i . If we consider the sum of regret over each block as in Equation (43),

On the other hand, since f t ( · ) is G -Lipschitz, by Lemma 6,

where we use the fact that for the first block, i ( t ) = 1 and thus x i ( t ) -1 = ˜ y i ( t ) -1 . Now as we combine Equations (44) and (45), we have

By plugging in { ϵ i } T B i =1 , we get the stated regret guarantee.

Now it remains to bound the number of calls to the linear optimization oracle. Note that for each block i ∈ [ T B ] , by Lemma 6, d ( x i , ˜ y i ) 2 ≤ 3 ϵ i . In view of Algorithm 8, we also have

By the triangle inequality,

Squaring both sides and making use of ( a + b ) 2 ≤ 2( a 2 + b 2 ) , we have

By Lemma 6, the number of iterations in Algorithm 7 for the i +1 -th block is at most

Thus, the number of calls to the linear optimization oracle is bounded by

where the last line follows from ζ ≥ 1 , i +3 i +1 ≤ 5 3 and i +3 i -1 ≤ 5 holds for any i ≥ 2 . On the other hand, we have

Combining Equations (49), (50) and plugging in B = ( αR G ) 2 3 T 2 / 3 , we get the claimed N calls ≤ ζT .

D Some Basic Facts about Convex Sets on Hadamard Manifolds

In this section, we highlight two significant characteristics of convex sets in Euclidean space which do not apply to Hadamard manifolds, emphasizing the fundamental differences between these two spaces. Our counterexamples are constructed within the Poincaré half-plane, H 2 . Therefore, understanding some basic properties of this unique manifold will prove beneficial.

Lemma 10. (Udriste, 2013; Wang et al., 2016; Kristály et al., 2016) The Poincaré half plane H 2 can be described by

with metric

We have the following conclusions about geodesics and the distance on the Poincaré half-plane.

1) For the geodesic connecting x = ( t 1 , t 2 ) and y = ( s 1 , s 2 ) , let

then the geodesic γ xy := ( γ 1 xy , γ 2 xy ) connecting x and y is,

̸

̸

̸

Indeed, we have ( γ 1 xy ( s ) -b xy ) 2 +( γ 2 xy ( s )) 2 = r 2 xy for any s ∈ [0 , 1] when t 1 = s 1 .

2) The distance between two points ( u 1 , v 1 ) and ( u 2 , v 2 ) can be calculated by

D.1 Shrinking Gsc-convex Sets on Hadamard Manifolds Does Not Preserve Convexity

In the context of projection-free online learning on manifolds, the set c K = Exp p ( c Exp -1 p x ) | x ∈ K takes on a pivotal role, particularly when c < 1 . Consider a convex set K in Euclidean space, then the convexity of c K is guaranteed for any positive c . Nonetheless, this property does not necessarily hold on Hadamard manifolds. We can construct a counterexample within the Poincaré half-plane H 2 . To achieve this, we rely on the following lemma.

Lemma 11. On the Poincaré half-plane H 2 , the mid-point of the geodesic γ xy which connects x = ( -λ, µ ) and y = ( λ, µ ) is (0 , √ λ 2 + µ 2 ) .

Proof. By symmetry we know γ 1 xy ( 1 2 ) = 0 . To compute γ 2 xy ( 1 2 ) , note that b xy = 0 and r xy = √ λ 2 + µ 2 , then by Lemma 10,

where we use the fact that arctanh( · ) is an odd function and cosh(0) = 1 .

Now we prove the following theorem.

Theorem 6. There exists a counter-example on the Poincaré half plane H 2 such that the radially contracted set (1 -α ) K where α ∈ (0 , 1) is non-convex.

Proof. Let p = (0 , 1) , u = ( -1 , √ 2) , v = (1 , √ 2) , w = (0 , √ 3) . Observe that γ uv is an arc satisfying { ( λ, µ ) | λ 2 + µ 2 = 3 } and, by Lemma 11, w is its midpoint. Following this, we calculate γ pu , γ pw , and γ pv , and shrink these three geodesics by a factor of (1 -α ) . Let's denote the resulting geodesics as γ pu ′ , γ pw ′ , and γ pv ′ . Ultimately, we demonstrate that for some α , w ′ is positioned beneath γ u ′ v ′ , implying that the radial contraction of K does not encompass a geodesic, hence proving

its non-convexity. We first compute γ pu and γ pv to facilitate this. Note that b pu = -1 and r pu = √ 2 , by Lemma 10, we can write down

and

By symmetry, we have and

We can also compute γ pw ( s ) = ( 0 , e s ln √ 3 ) . Now we have

and w ′ = ( 0 , e (1 -α ) ln √ 3 ) . We just need to demonstrate that for some α ∈ (0 , 1) , w ′ is positioned below γ u ′ v ′ ( 1 2 ) , where

by Lemma 11. Specifically, we need to show ∃ α ∈ (0 , 1) ,

We set α = . 1 , then

Therefore, the set (1 -α ) K fails to contain the geodesic γ u ′ v ′ ( s ) and is thus non-convex.

D.2 Non-convexity of the Minkowski Functional on Hadamard Manifolds

In this part, we show the Minkowski functional (Gauge function), which is defined as

can be non-convex on Hadamard manifolds, where K is a gsc-convex set of a Hadamard manifold M and p is an interior point in K .

Theorem 7. The Minkowski functional defined in Equation (53) can be non-convex on Hadamard manifolds.

Proof. We again construct a counterexample on the Poincaré plane H 2 :

To verify the geodesic convexity, we can verify an equivalent statement: a function f is gsc-convex iff f is convex when restricted to any geodesic. That is f ( γ ( t )) ≤ (1 -t ) f ( γ (0)) + tf ( γ (1)) holds for any geodesic γ ( t ) : [0 , 1] →M . The counterexample is as follows.

Assume K is the gsc-convex set consists of all points ( u 1 , u 2 ) such that u 2 1 + u 2 2 ≤ 3 and u 2 > 0 . x = ( -1 , √ 3) , y = ( -2 , 2) and p = (0 , 1) . We show that γ K ( x ) is non-convex on the geodesic γ xy .

We first compute b xy = -2 and r xy = 2 . By Lemma 10, we have

where we let θ ( s ) := tanh ( (1 -s ) arctanh ( -1 2 )) to ease the elaboration. We denote z := γ xy ( s ) = ( -2 -2 cos θ ( s ) , 2 sin θ ( s )) . Then we compute the geodesic γ pz connecting z and p = (0 , 1) , and we have

and r pz ( s ) = √ b pz ( s ) 2 +1 . Now we can compute the intersection of γ pz ( s ) with K as the solution ( q 1 , q 2 ) of the following system of equations

The intersection point is ( 1 b pz ( s ) , √ 3 -1 b pz ( s ) 2 ) . For each s ∈ [0 , 1] , we can evaluate the Minkowski functional at γ xy ( s ) w.r.t. K as

where d H is the hyperbolic distance defined in the second item of Lemma 10. If the Minkowski functional is gsc-convex, then we should have h ( s ) ≤ (1 -s ) h (0) + s · h (1) for any s ∈ [0 , 1] . However, by numerical computation, we find

which refutes the assumption that h ( s ) is convex. Thus our conclusion is established.

E Technical Lemmas

E.1 Background on the Jacobi Field

The role of the Jacobi field as an essential tool for understanding the impact of curvature on the behavior of neighboring geodesics cannot be overstated. To make our discussion more comprehensive and self-contained, we will introduce related definitions and properties sourced from Lee (2006) and Lee (2018).

We define an admissible curve as a continuous map, γ ( t ) : [ a, b ] → M , representing a piecewise regular curve segment. An admissible family of curves, on the other hand, is a continuous map Γ( t, s ) : [ a, b ] × ( -ϵ, ϵ ) → M which is smooth on each rectangle [ a i -1 , a i ] × ( -ϵ, ϵ ) for a finite subdivision a = a 0 < · · · < a k = b . Here, Γ s ( t ) := Γ( t, s ) is recognized as an admissible curve for each s ∈ ( -ϵ, ϵ ) . When γ ( t ) is admissible and Γ( t, 0) = γ ( t ) , we refer to Γ as a variation of γ ( t ) .

Upon an admissible curve Γ , we can delineate two sets of curves: the principal curves Γ s ( t ) , which hold s as constant, and the transverse curves Γ ( t ) ( s ) := Γ( t, s ) , which consider t as constant. We

consider Γ as a variation through geodesics if each principal curve Γ s ( t ) = Γ( t, s ) is a geodesic segment. For an admissible family of curves Γ , we define a vector field along Γ as a map V : [ a, b ] × ( -ϵ, ϵ ) → T M such that V ( t, s ) ∈ T Γ( t,s ) M and for each s , V ( s, t ) is piecewise smooth. The Jacobi field is a vector field along a variation through geodesics and is formally defined as follows.

Lemma 12. (Lee, 2018, Theorem 10.1, The Jacobi Equation) Let ( M , g ) be a Riemannian manifold, γ be a geodesic in M , and J be a vector field along γ . If J is the variation field of a variation through geodesics, then J needs to satisfy the following Jacobi Equation :

where R is the Riemann curvature endomorphism. A smooth vector field J satisfying the Jacobi Equation is called a Jacobi field.

Given initial values of J and D t J at one point, the Jacobi Equation can be uniquely determined by the following proposition.

Lemma 13. (Lee, 2018, Prop. 10.2, Existence and Uniqueness of Jacobi Fields) Let ( M , g ) be a Riemannian manifold and γ : I →M is a geodesic where I is an interval. Fix a ∈ I , let p = γ ( a ) , u, v ∈ T p M , there is a unique Jacobi field J along γ which satisfies

The Jacobi field seems to be a complicated object to study, but the following lemma shows we can write down the initial conditions of the Jacobi field for certain variations through geodesics.

Lemma 14. (Lee, 2018, Lemma 10.9) Let ( M , g ) be a Riemannian manifold, I be an interval containing 0 , and γ : I →M be a geodesic. Suppose J : I →M be a Jacobi field with J (0) = 0 . If M is geodesically complete or I is compact, then J is the variation field of the following variation of γ :

where p = γ (0) , u = ˙ γ (0) , and v = D t J (0) .

The advantage of introducing the Jacobi field is the norm of J ( t ) can be bounded by the initial value ∥ D t J (0) ∥ , as shown in the following Lemma 15.

Definition 4. We define is a closed gsc-convex set.

Lemma 15. (Lee, 2018, Theorem. 11.9 ) Suppose ( M , g ) is a Riemannian manifold, γ : [0 , b ] →M is a unit-speed geodesic segment, and J is any Jacobi field along γ such that J (0) = 0 . For each c ∈ R , let s ( · , · ) be the function defined by Definition 4.

1) If all sectional curvatures of M are bounded above by a constant κ 2 , then

for all t ∈ [0 , b 1 ] , where b 1 = b if κ 2 ≤ 0 , and b 1 = min ( b, π √ κ 2 ) if κ 2 > 0 .

2) If all sectional curvatures of M are bounded below by a constant κ 1 , then

for all t ∈ [0 , b 2 ] , where b 2 is chosen so that γ ( b 2 ) is the first conjugate point to γ (0) along γ if there is one, and otherwise b 2 = b .

E.2 Miscellaneous Technical Lemmas

Lemma 16. (Bacák, 2014, Section 2.1) Let M be a Hadamard manifold, f : M→ ( -∞ , ∞ ) be a gsc-convex lower semicontinuous function. Then any β -sublevel set of f :

Lemma 17. (Bacák, 2014, Section 2.2) Let M be a Hadamard manifold, f i : M → ( -∞ , ∞ ) for i = 1 , . . . , m be a series of gsc-convex lower semicontinuous functions. Then sup i ∈ [ m ] f i ( x ) is gsc-convex.

Lemma 18. For a set K := { x | max 1 ≤ i ≤ m h i ( x ) ≤ 0 } ⊆ M where each h i ( x ) is gsc-convex, M is Hadamard and any y / ∈ K , there exists g ∈ T y M such that

Proof. By Lemma 17, max 1 ≤ i ≤ m h i ( x ) is gsc-convex. Further, by Lemma 16, K is a gsc-convex subset of M . For some y / ∈ K , there exists j ∈ [ m ] such that h j ( y ) > 0 . If not, we have h i ( y ) ≤ 0 holds for any i = 1 , . . . , m , which implies y ∈ K and leads to a contradictory. Applying the gsc-convexity for any x ∈ K , we have

where the second inequality is because h j ( y ) > 0 and h j ( x ) ≤ 0 . This is a valid separation oracle between y and K .

Remark 4. Given K = { x | max 1 ≤ i ≤ m h i ( x ) ≤ 0 } where each h i ( x ) is gsc-convex and y / ∈ K , we can check the sign of h i ( y ) for i = 1 , . . . , m until we find i ∗ such that h i ∗ ( y ) > 0 , then using the proof of Lemma 18, we obtain

for every x ∈ K . This gives us a separation oracle between y and K . The computation of this separation oracle requires no more than m function value evaluations and a single gradient evaluation. This is efficiently implementable when m is of moderate size.

Remark 5. Using Gauss's Lemma (Lee, 2006, Theorem 6.8), we can efficiently compute the projection onto a geodesic ball. Consider x / ∈ B p ( r ) for which we aim to compute its projection onto B p ( r ) . First, we determine the geodesic segment, denoted as γ ( t ) , connecting x and p . This segment intersects the geodesic ball at point z . Gauss's Lemma then tells us that the tangent vector of γ ( t ) at z is orthogonal to the geodesic sphere S p ( r ) . This indicates that z is the projection of x onto B p ( r ) . By performing O (log ( 1 ϵ ) ) rounds of binary search on γ ( t ) , we can approximate z to within an ϵ precision. In each round, it suffices to verify if the condition d ( x t , p ) ≥ r is met.

Lemma 19. (Wang et al., 2023, Lemma 45) Suppose the sectional curvature of M is in [ κ 1 , κ 2 ] where κ 1 ≤ 0 and κ 2 ≥ 0 . Assume a gsc-convex set K satisfies B p ( r ) ⊆ K ⊆ B p ( R ) where R ≤ π 2 √ κ 2

when κ 2 > 0 . Then for any y ∈ (1 -τ ) K , the geodesic ball B y ( s ( κ 2 ,R + r ) · τr s ( κ 1 ,R + r ) ) ∈ K , where s ( · , · ) is defined in Definition 4.

Lemma 20. (Bridson & Haefliger, 2013, Chapter II.1) Suppose ∆( p , q , r ) is a triangle on a Hadamard manifold M and ∆(¯ p , ¯ q , ¯ r ) is a comparison triangle in Euclidean space such corresponding side lengths are the same for both triangles. Then for any x ∈ [ p , q ] , y ∈ [ p , r ] , ¯ x ∈ [¯ p , ¯ q ] , ¯ y ∈ [¯ p , ¯ r ] satisfying d ( p , x ) = d E (¯ p , ¯ x ) and d ( p , y ) = d E (¯ p , ¯ y ) , we have d ( x , y ) ≤ d E (¯ x , ¯ y ) .

Lemma 21. The function sinh x x is increasing for any x ∈ [0 , ∞ ] .

Proof. We denote g ( x ) = sinh x x , then

It suffices to show h ( x ) := x cosh x -sinh x ≥ 0 holds for any x ∈ [0 , ∞ ) . This is obvious because h (0) = 0 and h ′ ( x ) = x sinh x ≥ 0 for any x ∈ [0 , ∞ ) . It remains to compute g ′ (0) . By L ´ Hôpital's rule,

Lemma 22. Let f : M→ R be gsc-convex and G -Lipschitz on a gsc-convex set K ⊆ M . Then

satisfies

Proof. We have

where v ∈ T x M . The first inequality is due to Jensen's inequality, and the second one is by the gradient Lipschitzness of f .

Definition 5. Given a homogeneous Riemannian manifold M , we use S δ and V δ to denote the surface area and the volume of a geodesic ball with radius δ .

Remark 6. We note that on homogeneous Riemannian manifolds, the volume and the area of a geodesic ball only depend on its radius (Kowalski & Vanhecke, 1982), so S δ and V δ are well-defined.

Lemma 23. (Wang et al., 2023, Lemmas 11 and 13) For a homogeneous Hadamard manifold ( M , g ) with sectional curvature lower bounded by κ ≤ 0 , f be a C 1 and gsc-convex function defined on M such that | f | ≤ C for any x ∈ M . We define

is a corresponding gradient estimator. Then we have

1) S δ V δ g ( u ) is an unbiased estimator of ∇ ˆ f ( x ) . More specifically, for any x ∈ M ,

The expected norm of S δ V δ g ( u ) can be bounded by

Suppose K ∈ M is a gsc-convex set and ∥∇ f ( x ) ∥ ≤ G , then ˆ f satisfies

where ρ solely depends on the set K .

Remark 7. From the proof of Lemma 23, we can figure out the formal definition of ρ is

By Wang et al. (2023), this quantity can be O ( e D ) even on a 2 -dimensional Poincaré disk.

Algorithm 9: Riemannian Frank-Wolfe with line-search

Data: feasible gsc-convex set K , initial point x 0 ∈ K , gsc-convex objective f ( · ) .

for i = 0 , . . . do

where ω is the volume element, and

Lemma 24. (Weber & Sra, 2022b, Theorem 1) Suppose the diameter of the gsc-convex set K ⊆ M is upper bounded by D and f is gsc-convex as well as L gsc-smooth on K . With a linear optimization oracle in hand, the primal convergence rate of Algorithm 9 can be described by

Lemma 25. (Weber & Sra, 2022b, Lemma 2) Under the same assumptions as in Lemma 24, for a step x k +1 = Exp x k ( η Exp -1 x k v ) with η ∈ [0 , 1] ,

Lemma 26. Under the same assumptions as in Lemma 24, denote

If Algorithm 9 is run for K ≥ 2 rounds, then there exists k ∗ such that 1 ≤ k ∗ ≤ K and

Proof. The proof is an adaptation of Jaggi (2013, Theorem 2). We first assume the duality gap is large in the last one third of K iterations, then get a contradictory to prove conclusion. For simplicity, we define h k := f ( x k ) -f ( x ∗ ) and g k := g ( x k ) .

To ease the elaboration, we denote C := 2 LD 2 , α := 2 3 , b := 27 8 and d := K + 2 . We assume g k > bC K +2 = bC d holds for the last one third of the K iterations. More specifically,

By Lemma 25 with η = 1 k +2 , we have

Plugging in the assumption that the duality gap is large, we arrive at

for all k = ⌈ αd ⌉ -2 , . . . , K , when the second inequality follows from ⌈ αd ⌉ -2 ≤ k ≤ K implies αd ≤ k +2 ≤ d .

Denote k min := ⌈ αd ⌉-2 and h min = h k min , then there are K -k min +1 = K -( ⌈ αd ⌉-2)+1 ≥ (1 -α ) d steps in the last one third iterations. Thus summing Equation (57) from k = ⌈ αd ⌉ -2 to K , we have

where h min ≤ C αd is due to Lemma 24, while for the last step, we plug in the values α = 2 3 and b = 27 8 . However, h K +1 < 0 is impossible because h K +1 = f ( x K +1 ) -f ( x ∗ ) . Thus, there always exists k ∗ in the last one third iterations of K such that

Lemma27. (Ahn & Sra, 2020, Prop. H.1) Let M be a Riemannian manifold with sectional curvatures lower bounded by κ ≤ 0 and the distance function d ( x ) = 1 2 d ( x , p ) 2 where p ∈ M . For D ≥ 0 , d ( · ) is ζ gsc-smooth within the domain { u ∈ M : d ( u , p ) ≤ D } where ζ = √ -κD coth( √ -κD ) is the geometric constant defined in Definition 1.

Lemma 28. Suppose f i ( x ) is α -strongly gsc-convex for any i = 1 , . . . , N , then ∑ N i =1 f i ( x ) is αN -strongly gsc-convex.

Proof. By the α -strongly gsc-convexity of each f i ( x ) ,

Summing from i = 1 to N , we have

This indeed implies ∑ N i =1 f i ( y ) is αN -strongly gsc-convex.

Lemma 29. (Zhang & Sra, 2016, Lemma 5). Let M be a Riemannian manifold with sectional curvature lower bounded by κ ≤ 0 . Consider a geodesic triangle fully lies within M with side lengths a, b, c , we have

Lemma 30. (Sakai, 1996, Prop. 4.5) Let M be a Riemannian manifold with sectional curvature upper bounded by κ ≤ 0 . Consider N , a gsc-convex subset of M with diameter D . For a geodesic triangle fully lies within N with side lengths a, b, c , we have

E.3 Extension to CAT ( κ ) Spaces

It is not too difficult to generalize the result in this work to CAT ( κ ) spaces, which are simply connected manifolds with sectional curvature upper bounded by κ . To achieve this extension, the following adjustments would be necessary in our manuscript:

We may assume that the sectional curvature lies in the range [ κ, K ] . In this context, we can replace Lemma 30 in our draft with Alimisis et al. (2020, Corollary 2.1). When K > 0 , we must further assume that the diameter of the decision set is upper-bounded by π √ K .

The separation theorem, specifically for Hadamard manifolds as outlined in Silva Louzeiro et al. (2022), would require generalization to CAT ( κ ) spaces. This change ensures that the separation oracle remains well-defined.

Lemma 3 requires modification to accommodate the recomputation of the Jacobi field, considering the manifold's sectional curvature. Lemmas 15 and 19 will be instrumental in achieving this.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present online algorithms for geodesically convex losses on Riemannian manifolds. Their algorithms do not call the expensive operation of projection onto a feasible set. Instead of projection, they rely on two oracles to provide a direction of descent: a separation oracle and a linear oracle. Both require an extension of the concept of hyperplanes from Euclidean space to the manifold, for which they use an inverse exponential map. This map transforms points on the manifold into tangent space vectors, enabling operations on the manifold without the need for explicit projection operations. In addition, they also consider the projection onto a geodesic ball, which is computable with high accuracy and thus effectively projection free. Their algorithms give adaptive regret guarantees which are sublinear in the horizon.\n\nSTRENGTHS:\n- The authors' contribution is significant as they address the challenge of projecting onto a Riemannian manifold, which is computationally expensive and can lead to geometric distortions. This area of research seems relatively less explored. \n\n- The writing style is clear and they provide a comprehensive background on Reimannian geometry, making it easy for a read to grasp the problem and motivation.\n\nWEAKNESSES:\nAs I am not familiar with this area, I am unable to identify any weaknesses or limitations.\n\nQUESTIONS:\nI am not sure why $\\tilde{\\mathcal{K}} = (1-\\delta) \\mathcal{K}$ is considered as the feasible set for the separation oracle.\n\nLIMITATIONS:\nThe authors have acknowledged the limitations of their work, specifically the absence of a membership oracle, and have discussed potential enhancements to their algorithms. These improvements include utilizing the separation oracle for strongly convex losses, reducing the reliance on the number of calls regarding the set's diameter and a faster method to optimise the objective in the linear oracle.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers Riemannian online optimization problems over sets of constraints and tries to tackle them by avoiding projections onto the constraint set. This is an already established line of research in Euclidean optimization and the results follow the structure of Garber and Kretzu (2022). The results of the latter are adapted to the Riemannian setting using well-known geometric bounds.\n\nSTRENGTHS:\nThe paper is concerned with problems of profound importance for the neurips community. It is well-written and can be followed easily by people with reasonable background in Riemannian optimization. The use of geometric bounds (law of cosines in negative curvature, spread of Jacobi fields) is clearly explained. The authors pay special attention in computability issues behind the separating and linear oracles used, building upon the contributions of Weber and Sra (2022b). The convergence guarantees much the ones of the Euclidean setting in Garber and Kretzu (2022) up to constants.\n\nWEAKNESSES:\nI am not thrilled by the level of originality since the paper follows closely the one by Garber and Kretzu (2022) and used geometric bounds that are well-known in the Riemannian optimization community for quite some time. Still, it is a clearly-written concrete contribution to a problem of interest, thus I think that publication in the conference is totally justifiable.\n\nThere are some points in the paper that I find peculiar as I specify in my questions' section. This is the main reason that I give a score only slightly above the acceptance threshold (which I will revisit in case the authors answer me concerns convincingly).\n\nI would advise the authors to revisit the title of the paper. When I see \"On ...\" I automatically form the impression that the paper features some vague discussion about a research topic without really useful contributions, which is not the case here.\n\nQUESTIONS:\n1. I see some slight discrepancy in the selection of parameters and worst-case regret bounds between this paper and Garber and Kretzu (2022) (I will refer to that as GK from now on). Could you explain why this is happening? To be more specific: \n\nTheorem 1 in this paper and Theorem 6 in GK do not have exactly the same constants. I think you choose $c=1/2$ and $c_1=4$, but I cannot see how the regret bound and the bound in the number of oracle calls become the same for $\\zeta=1$.\n\nThe parameters $\\eta$ and $\\delta$ in Theorem 2 of this paper are not chosen the same way as the ones in Theorem 7 of GK with respect to $T$. This is quite worrying.\n\nTheorem 3 in both papers have the same parameters (for $\\zeta=1$), but the regret bound features slightly different constants.\n\nDiscrepancies like that seem not to be the same among Lemmas 1 and Theorems 4 between the papers.\n\n\n2. I struggle to understand Algorithm 5, I think that after the if statement, you should have $< \\exp^{-1}_{x_i}(y),\\exp^{-1}_{x_i}(v_i) > \\leq \\epsilon$ or $d(x_i,v_i)^2 \\leq 3 \\epsilon$. Is that true?\n\n3. As I see the results, it should pretty straightforward to have similar ones for manifolds of positive curvature (with $\\zeta=1$ and $\\bar r=r$). Is there any specific reason that you didn't attempt that?\n\nLIMITATIONS:\nThe authors adequately discuss some limitations of their work in the conclusion of the paper.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper focuses on constrained Riemannian online optimization on the Hadamard manifold. Existing Riemannian online optimization methods often require projections, which present computational complexity challenges in high-dimensional settings. To address this issue, the authors have developed a projection-free Riemannian online optimization structure, implemented under two scenarios - the separation oracle and the linear optimization oracle.\n\nInitially, they explore the Infeasible R-OGD and Infeasible Riemannian bandit algorithm under a separation oracle, for which the regret bounds are proved to be $\\mathcal O(\\sqrt T)$ and $\\mathcal O(T^{\\frac{3}{4}})$ respectively for geodesically convex functions. Furthermore, they consider the Block R-OGD under a linear optimization oracle, and provide a proof for regret bounds of $\\mathcal O(T^{\\frac{3}{4}})$ and $\\mathcal O(T^\\frac{2}{3})$ for geodesically convex and geodesically strongly convex functions respectively. It's noteworthy that all these regret bounds match their respective Euclidean counterparts.\n\nSTRENGTHS:\n-  Originality & Significance: Althogh the paper has retricted novelty (discuss in the following), it provides, to my best of knowledge.  the first no-regret guarantee for projection-free Riemannian OCO. Also, it is nice to see that all regret bounds match their Euclidean counterparts.\n\n- Clarity & Significance: The paper is well-organized, techinally sound and easy to follow. The assumption is standard in the literature of Riemannian optimization and Riemannian OCO.\n\nWEAKNESSES:\n- Retricted novelty: The majority of the analysis leans heavily on the Euclidean Analogue and Jacobian/Hessian comparison. Although it is the standard structure in Riemannian optimization literature, it would have been enriching to see some novel conceptual ideas that engage more specifically with the geometry of the problem.\n\n- Positive curvature: The study focuses on the Hadamard manifold, known for its non-positive sectional curvature. Conversely, in practical scenarios, there's considerable work undertaken within spaces possessing positive sectional curvature like SO(3). The methodology and algorithms presented in this paper appear, at first glance, to be easily extendable to accommodate $cat(K)$ spaces. However, the authors have not provided any explanation or context for why such positive curvature spaces were not considered in their study.  A consideration of such spaces could have added more depth and wider applicability to the research.\n\n- Possible typo: formula about $\\bar r $ after l227: there should be a additional $r$ at RHS.  \n Algorithm 5: $y$ seems to be a point rather than a tangent vector.\n\nQUESTIONS:\n- I'd appreciate if the authors could expound on their primary insights in this paper. Specifically, it would be beneficial to understand the significant challenges encountered while extending the Euclidean projection-free algorithm to the Riemannian manifold. \n\n- Additionally, I'm interested in knowing whether the proposed Riemannian projection-free method could be easily extended to $cat(K)$ spaces. If this is not feasible, could the authors shed light on the primary hurdles that prevent such an extension?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study online learning on homogeneous Hadamard manifolds with geodesically-convex, Lipschitz losses. They focus on projection-free methods and in particular they develop algorithms that use either a separaton oracle or a linear optimization oracle. They study adaptive regret algorithms for the full information and the bandit feedback settings.\n\nSTRENGTHS:\nBesides from the strengths of having the theoretical results authors have, I want to emphasize the following.\n\nThis work usually states clearly the assumptions that is making, even though this could be improved, as I suggested. It was good to see the authors taking into account the non convexity of (1-\\delta)\\mathcal{K}. Some works ignore this fact, I wonder if they learn about this in the review of a paper. But the authors here are rigorous and work with the possible non-convex set.\n\nThe results with linear minimization oracle look fine to me\n\nWEAKNESSES:\n# Main points of my review\n\nSeveral results in this paper follow closely Garber and Kretzu (2022). This is not a criticism, of course there are several challenges in the Hadamard case that need to be surmounted. \n\nI have several points of criticism though:\n\n+ The paper contains some mistakes, and I am not sure if one could fix at least one of them while keeping the stated results (see below). \n+ Moreover algorithm 4 has a regret guarantee in Theorem 2 which is non standard (algorithm plays a point and feedback is given for another point but the regret is still measured with respect to the played point). This seems an artifact of the proof, as in it was the notion that made the proof work, but without any other justification for this made-up definition of regret, usefulness of it or an application of it, the result is very weak.\n+ Lemmas 3 and 4 yield that the number of iterations of algorithm 2 is exponential on R. This is not desirable. I do not want to criticize this point harshly, since Hadamard is hard and sometimes the geometry makes these exponential dependences appear, and this could be a result deserving its own paper. This dependence also appears in previous Riemannian online learning works. But rather, I just want to encourage the authors, possibly for future work, to try to improve this to a polynomial dependence (see below for a few comments on this).\n\nWhile I think this works has value, I also think this is not ready for publication. At the very least the errors should be corrected, and the bandit results should not be presented as results (e.g. in the table) unless the authors can justify that the modify notion or regret makes sense. And in any case this should be clearly stated for the readers that only want to firstly check the introduction out.\n\n# Proof Errors\n\nCorrect me if I am wrong but I think I found the following errors in your proofs. I can increase my score if the technical errors can indeed be fixed.\n\n+ The proof of Lemma 1 is wrong, although fixable. The points y_{t+1} are not guaranteed to be in B_p(R) and so you cannot use lemma 29 as you stated it. However, if you look closely at Lemma 5 of Zhang and Sra, you will realize that the lemma actually says that \\zeta depends on \\kappa and c, which a stronger inequality than what you stated (dependence on \\kappa and D). Using that stronger statement then you can show in your Lemma 1 that the \\zeta in the proof depends on d(\\tilde{y}_t, x) \\leq D, and so the proof follows.\n\n+ However, your lemma 2 has the same problem, and when you use it in the proof of lemma 4, you use it with the point y_i, which as you point out in L435, it is y_i \\not \\in \\mathcal{K}. Therefore, the geometric deformation is not \\zeta (which was defined as depending on D), and this does not seem to be so easily fixable.\n\n\n# Other comments / suggestions\n\nThe results with linear minimization oracle look fine to me.\n\nYou should specify in lemma 5, that you always use y\\in B_p(R), since you use f(x) = d(x y)^2/2 and you claim that it is \\zeta smooth, which you can have because when you use lemma 5 in alg 6 it is always y\\in B_p(R). But this is not stated and it was confusing for me.\n\nThis work usually states clearly the assumptions that is making, even though this could be improved, as I suggested. It was good to see the authors taking into account the non convexity of (1-\\delta)\\mathcal{K}. Some works ignore this fact, I wonder if they learn about this in the review of a paper. But the authors here are rigorous and work with the possible non-convex set.\n\n\"(pinpointing a compact set that contains the true optimum can improve time complexity) This makes metric projection onto a subset of Riemannian manifold seemingly indispensable, yet this operation is not only computationally taxing but can also lead to unwanted geometric distortion, which can undermine convergence. Certain works have depended on simplifying assumptions to prove results...\" I disagree with the message that is given here. For several reasons:\n    + Firstly, in many situations, one can \"pinpoint\" a compact set where the optimum is by just estimating the initial distance to minimizer and imposing ball constraints (and using a doubling trick or other tricks if necessary to avoid assuming knowledge of this distance ). In that case, the operation is \"projection-free\" according to your own definition in line 178, due to the simplicity of the operation, so not necessarily computationally taxing.\n    + Secondly, while it is true that using projections can lead to greater geometric distortion that can undermine convergence (e.g. see https://arxiv.org/pdf/2305.16186.pdf where the authors make projected gradient descent work for smooth functions while quantifying the distortion), several works make use of projections to obtain algorithms that previously had to make the assumption of iterates staying in some set, and at the same time the geometric dependence on the convergence rates does not worsen. See for instance https://arxiv.org/pdf/2211.14645.pdf and theorem 7 in https://arxiv.org/pdf/2305.16186.pdf, and https://arxiv.org/pdf/2111.13263v2.pdf section 6 \n    + Thirdly, the works that mandate the iterates to be in a feasible set by assumption do so because they do not know how to enforce or guarantee their algorithms are in some feasible set, even if they have access to projections. It is not like the assumption is made to avoid projections that are \"computationally taxing and with extra geometric distortions, which can undermine convergence\" by using projections. Their techniques do not allow them to use projections and as explained above and more clever algorithms do without extra distortions.\n\n\nYou probably cannot compute \\sigma_i in Algorithm 5 in close form. You probably would need a binary search and then have the algorithm be able to account for an error. Is that right?\n\n# Exponential dependence on the diameter\n\nRegarding the exponential nature of the constants in lemma 3 and running time in lemma 4, this phenomenon is very similar to the exponential constants that appear in the deformations of lemma 2 in https://arxiv.org/pdf/2012.03618.pdf However, in that paper they are interested on optimization, and they show that one can get away with this exponential complexity. The trick is a reduction from global optimization to approximately implementing a ball optimization oracle, so that the algorithms would only need to run in balls where R\\sqrt{-\\kappa} = O(1). The reduction goes as follows: optimize in one such constant-radius ball with linear rates (regularize as it is done in reductions if necessary, although you'd need smoothness of the losses) and then optimize in another ball with center equal to the previously computed point. In the online learning setting, it is harder but you could in principle try to run Riemannian OGC by minimizing the sort of FTRL objective you have by using this reduction (you'd have to optimize with linear rates in the intersection of constant curvature balls and your sets, but every time the infeasible projection would be easy). Or just implement the separation oracle by looking at making progress in constant-curvature balls. Anyway, maybe it doesn't work for you, but I hope it is useful to you to at least know that a problem of this kind has been sort of solved in Riemannian optimization.\n\n# Minor suggestions / typos\n\nL62 \"While a horosphere gsc-convex\" ->\"\"While a horosphere is gsc-convex\"\nL200 \"theLipschitz\" -> \"the Lipschitz\"\nL259 Alg 5. I would not say target vector y, since y is a point.\nL269 Alg 6. \"initial vector y\" -> \"initial vector y_i\" (although again, I would not say initial *vector* )\n\n# Edit after rebuttal\nThe authors provided fixes for the proof errors and a new algorithm two work with two point bandit feedback. I increased my score from reject to accept (from 3 to 7)\n\nQUESTIONS:\nsee above\n\nLIMITATIONS:\nsee above", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a generalization for online learning on the Riemannian manifolds via separation and linear optimization oracles. The core idea here is to use the oracles to construct an infeasible projection, which may not be the nearest point in the constrain/decision set $\\mathcal{K}$, but are in $\\mathcal{K}$ and is closer to a shrunk feasible set $(1-\\delta)\\mathcal{K}$ than the original point. The interesting point finding here is that because there is a buffer between $\\mathcal{K}$ and $(1-\\delta)\\mathcal{K}$, we have a constant distance decrease of $O(\\delta^2)$ (Lemma 4 eq (5) in the appendix) w.r.t. by the separating oracle direction to the shrunk set $(1-\\delta)\\mathcal{K}$, and hence the projection to $\\mathcal{K}$ can be done in constant time to $\\delta$. This means we can construct an infeasible/inaccurate projection by the separation oracle in $O(1/\\delta^2)$ time, and the algorithm follows from replacing the projection part with the infeasible projection and bound the errors. The authors also consider the linear optimization oracles, and it's done by constructing the separation oracles using the linear optimization oracles in Frank-Wolfe. Actually, this draft is a combination between Garber & Kretzu (2022) and Wang et al. (2021), where the authors generalize the online learning by SO/LOO framework by Garber & Kretzu (2022) with the tools and assumptions in Wang et al. (2021).\n\nSTRENGTHS:\nThe idea of constructing an infeasible projection (the solution is feasible to $\\mathcal{K}$) and providing guarantees to a shrunk set $(1-\\delta)\\mathcal{K}$ is very interesting, and it's good to see it works on the Riemannian manifold. The introduction is pretty well-written, and there is no major error in the proof, and generalizing the result from Euclidean space to the Riemannian space may be challenging in some corner cases.\n\nWEAKNESSES:\nThe major weakness of the paper is its complexity, the lack of experiments, and the limiting assumptions.\n\nFirst, the proof in the paper is pretty complex and is not self-contained. To verify any proof in the paper, the readers usually need to go to Garber & Kretzu (2022) and Wang et al. (2021), which have another set of symbols and may again point to other papers. And this may lead to confusion for both the authors and the readers. For example, the Lemma 19 in the draft points to Wang et al. 2023 Lemma 45, which requires the curvature to be also upper-bounded, but such assumption and coefficient disappears in the draft. Garber & Kretzu (2022) assume that the feasible set contains the origin. The paper generalized it to arbitrary point $p$, but in that case, the shrinking set $\\mathcal{K}$ should be redefined toward $p$ for $(1-\\delta)\\mathcal{K}\\subset\\mathcal{K}$, but it is also not done. The proof in the paper is mostly correct when self-contained, but it is very time-consuming to verify the mentioned errors from cited lemmas by chasing around the links, so I am not very sure about the correctness of the theoretical paper.\n\nSecond, I am not very sure about the practical aspect of the paper due to its limiting assumptions and lack of experiments. Surely, theoretical papers may not have been experimented with, but the paper is too complex to verify its correctness, so it would be good to show the reader that it works in a sense. I am concerned about the limitation in the assumptions because most of the examples mentioned in the draft's introduction don't fit the assumption, e.g., the spherical constraint in K-means clustering. The manifold must be bounded, containing a non-empty unit ball, and contained within a unit ball of radius $\\leq \\pi/\\sqrt{4\\kappa_2}$ (Lemma 19). And I am unsure when the separation oracle and linear optimization oracle are cheaper than the projection in a practical example. Please give the readers more examples to understand when the setting would be suitable.\n\nFinally, the paper is a direct combination between Garber & Kretzu (2022) and Wang et al. (2021). Most of the work here looks like a Riemannian generalization of Garber & Kretzu's (2022)'s framework with Wang et al. (2021,2023)'s tools, replacing the distance with exponential maps and bound them with Wang et al. (2021,2023)'s inequalities. And there are not many new surprising results, but double the complexity. It is not necessary to be novel to publish, but given that I'm not convinced that the results are totally correct, I would recommend a weak rejection. This draft should be more suited to a journal than a conference.\n\nQUESTIONS:\n1. L32: Computing the orthogonal projection of \"polytope\" to half-spaces is NP-hard, but point projection is not. So this is unrelated to the draft.\n2. L161: May you explain the existence of g in Lemma 18 implies an efficient implementation?\n3. Assumptions & Lemma 19 requires $R\\leq \\pi/\\sqrt{4\\kappa_2}$, but it's not in the assumptions and coefficients.\n4. Lemma 29 is different from the original paper. Please note that you use the monotonicity of $x coth(x)$ here.\n5. L179: Please cite the projection algorithm in $O(\\log(1/\\epsilon)$\n6. Algorithm 2 takes $\\delta$ as an input but the $\\delta$ doesn't appears in the algorithm. Please at least define $\\gamma$ as a function.\n7. Appendix L430 eq (3): Please annotate that the later line suffices the inequality.\n8. Algorithm 2: It's also using one membership oracle at each step.\n9. Appendix L456: You use the bounded gradient property, not the Lipschitzness of the gradient.\n10. Algorithm 4: This looks pretty much like the R-BAN. Why R-BAN sample from a sphere but the algorithm samples from a ball?\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-30"}