No-go Theorem for Acceleration in the Hyperbolic Plane

Hamilton, Linus; Moitra, Ankur

doi:10.48550/arxiv.2101.05657

Cited by 6 publications

(11 citation statements)

References 21 publications

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“…To explain the issue, let d(A, v, r) be the point r units from the given point A in direction v. Suppose that, due to the numerical precision issues, we represent the direction v as v , where |v − v | ≤ . However, a circle of radius r has circumference of 2π sinh(r), which is exponential in r, and thus the distance between d(A, v, r) and d(A, v , r) can be of order × e r [11]. In general, the number of tiles in r steps from the center is exponential in r, so any representation using a fixed number of bits will not be able to discern between the coordinates of two tiles if r is large enough, on the order of the number of bits.…”

Section: Previous Methodsmentioning

confidence: 99%

Generating Tree Structures for Hyperbolic Tessellations

Celińska-Kopczyńska¹,

Kopczyński²

2021

Preprint

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Section: Previous Methodsmentioning

confidence: 99%

Generating Tree Structures for Hyperbolic Tessellations

Celińska-Kopczyńska¹,

Kopczyński²

2021

Preprint

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“…Cutting plane methods, such as ellipsoid, require an exponential bound on the volume of a known region containing an approximate optimizer. This is the case for Rusciano's non-constructive query upper bound for cutting plane methods on manifolds of non-positive curvature [Rus20], which is essentially tight [HM21] . The volume of a ball in the manifold we consider grows exponentially in the radius (see Section 4.5), so this query bound will be exponential.…”

Section: Theorem 14 (Noncommutative Diameter Lower Bound)mentioning

confidence: 99%

Barriers for recent methods in geodesic optimization

Franks,

Reichenbach

2021

Preprint

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We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton's method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017, and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, Bürgisser et. al. FOCS 2019. One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling.We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.

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“…When studying the global complexity of Riemannian optimization algorithms, it is common to assume that the sectional curvature of M is bounded below by K min and bounded above by K max to prevent the manifold from being overly curved. Unfortunately, [13,14] show that even when sectional curvature is bounded, achieving global acceleration is impossible in general. Thus, one might need another common assumption, an upper bound D of diam(N ).…”

Section: Introductionmentioning

confidence: 99%

Nesterov Acceleration for Riemannian Optimization

Kim¹,

Yang²

2022

Preprint

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In this paper, we generalize the Nesterov accelerated gradient (NAG) method to solve Riemannian optimization problems in a computationally tractable manner. The iteration complexity of our algorithm matches that of the NAG method on the Euclidean space when the objective functions are geodesically convex or geodesically strongly convex. To the best of our knowledge, the proposed algorithm is the first fully accelerated method for geodesically convex optimization problems without requiring strong convexity. Our convergence rate analysis exploits novel metric distortion lemmas as well as carefully designed potential functions. We also identify a connection with the continuous-time dynamics for modeling Riemannian acceleration in Alimisis et al. [1] to understand the accelerated convergence of our scheme through the lens of continuous-time flows.

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No-go Theorem for Acceleration in the Hyperbolic Plane

Cited by 6 publications

References 21 publications

Generating Tree Structures for Hyperbolic Tessellations

Generating Tree Structures for Hyperbolic Tessellations

Barriers for recent methods in geodesic optimization

Nesterov Acceleration for Riemannian Optimization

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