. However, with the rise of machine learning and data science, growing problem sizes have shifted the community’s focus to first-order methods such as gradient descent and stochastic gradient descent. Over the years, impressive theoretical progress has also been made here, helping elucidate problem characteristics and bringing insights that drive the discovery of provably faster algorithms, notably Nesterov’s accelerated gradient descent(Nesterov, 1983)
and variance reduced incremental gradient methods(e.g., Johnson and Zhang, 2013; Schmidt et al., 2013; Defazio et al., 2014).
Outside convex optimization, however, despite some recent progress on nonconvex optimization our theoretical understanding remains limited. Nonetheless, nonconvexity pervades machine learning applications and motivates identification and study of specialized structure that enables sharper theoretical analysis, e.g., optimality bounds, global complexity, or faster algorithms. Some examples include, problems with low-rank structure (Boumal et al., 2016b; Ge et al., 2017; Sun et al., 2017; Kawaguchi, 2016); local convergence rates (Ghadimi and Lan, 2013; Reddi et al., 2016; Agarwal et al., 2016; Carmon et al., 2016); growth conditions that enable fast convergence (Polyak, 1963; Zhang et al., 2016; Attouch et al., 2013; Shamir, 2015); and nonlinear constraints based on Riemannian manifolds (Boumal et al., 2016a; Zhang and Sra, 2016; Zhang et al., 2016; Mishra and Sepulchre, 2016), or more general metric spaces (Ambrosio et al., 2014; Bacák, 2014).
In this paper, we focus on nonconvexity from a Riemannian viewpoint and consider gradient based optimization. In particular, we are motivated by Nesterov’s accelerated gradient method (Nesterov, 1983), a landmark result in the theory of first-order optimization. By introducing an ingenious “estimate sequence” technique, Nesterov (1983) devised a first-order algorithm that provably outperforms gradient descent, and is optimal (in a first-order oracle model) up to constant factors. This result bridges the gap between the lower and upper complexity bounds in smooth first-order convex optimization (Nemirovsky and Yudin, 1983; Nesterov, 2004).
Following this seminal work, other researchers also developed different analyses to explain the phenomenon of acceleration. However, both the original proof of Nesterov and all other existing analyses rely heavily on the linear structure of vector spaces. Therefore, our central question is:
Is linear space structure necessary to achieve acceleration?
Given that the iteration complexity theory of gradient descent generalizes to Riemannian manifolds (Zhang and Sra, 2016), it is tempting to hypothesize that a Riemannian generalization of accelerated gradient methods also works. However, the nonlinear nature of Riemannian geometry poses significant obstructions to either verify or refute such a hypothesis. The aim of this paper is to study existence of accelerated gradient methods on Riemannian manifolds, while identifying and tackling key obstructions and obtaining new tools for global analysis of optimization on Riemannian manifolds as a byproduct.
It is important to note that in a recent work (Liu et al., 2017), the authors claimed to have developed Nesterov-style methods on Riemannian manifolds and analyzed their convergence rates. Unfortunately, this is not the case, since their algorithm requires the exact solution to a nonlinear equation (Liu et al., 2017, (4) and (5)) on the manifold at every iteration. In fact, solving this nonlinear equation itself can be as difficult as solving the original optimization problem.
1.1 Related work
The first accelerated gradient method in vector space along with the concept of estimate sequence is proposed by Nesterov (1983); (Nesterov, 2004, Chapter 2.2.1) contains an expository introduction. In recent years, there has been a surging interest to either develop new analysis for Nesterov’s algorithm or invent new accelerated gradient methods. In particular, Su et al. (2014); Flammarion and Bach (2015); Wibisono et al. (2016)
take a dynamical system viewpoint, modeling the continuous time limit of Nesterov’s algorithm as a second-order ordinary differential equation.Allen-Zhu and Orecchia (2014) reinterpret Nesterov’s algorithm as the linear coupling of a gradient step and a mirror descent step, which also leads to accelerated gradient methods for smoothness defined with non-Euclidean norms. Arjevani et al. (2015) reinvent Nesterov’s algorithm by considering optimal methods for optimizing polynomials. Bubeck et al. (2015) develop an alternative accelerated method with a geometric explanation. Lessard et al. (2016) use theory from robust control to derive convergence rates for Nesterov’s algorithm.
The design and analysis of Riemannian optimization algorithms as well as some historical perspectives were covered in details in (Absil et al., 2009), although the analysis only focused on local convergence. The first global convergence result was derived in (Udriste, 1994) under the assumption that the Riemannian Hessian is positive definite. Zhang and Sra (2016) established the globally convergence rate of Riemannian gradient descent algorithm for optimizing geodesically convex functions on Riemannian manifolds. Other nonlocal analyses of Riemannian optimization algorithms include stochastic gradient algorithm (Zhang and Sra, 2016), fast incremental algorithm (Zhang et al., 2016; Kasai et al., 2016), proximal point algorithm (Ferreira and Oliveira, 2002) and trust-region algorithm (Boumal et al., 2016a). Absil et al. (2009, Chapter 2) also surveyed some important applications of Riemannian optimization.
1.2 Summary of results
In this paper, we make the following contributions:
We propose the first computationally tractable accelerated gradient algorithm that, within a radius from the minimizer that depends on the condition number and sectional curvature bounds, is provably faster than gradient descent methods on Riemannian manifolds with bounded sectional curvatures. (Algorithm 2, Theorem 3)
We analyze the convergence of this algorithm using a new estimate sequence, which relaxes Nesterov’s original assumption and also generalizes to Riemannian optimization. (Lemma 3)
We develop a novel bound related to the bi-Lipschitz property of exponential maps on Riemannian manifolds. This fundamental geometric result is essential for our convergence analysis, but should also have other interesting applications. (Theorem 2)
We briefly review concepts in Riemannian geometry that are related to our analysis; for a thorough introduction one standard text is (e.g. Jost, 2011). A Riemannian manifold is a real smooth manifold equipped with a Riemannain metric . The metric induces an inner product structure on each tangent space associated with every . We denote the inner product of as ; and the norm of is defined as ; we omit the index for brevity wherever it is obvious from the context. The angle between is defined as . A geodesic is a constant speed curve that is locally distance minimizing. An exponential map maps in to on , such that there is a geodesic with and . If between any two points in there is a unique geodesic, the exponential map has an inverse and the geodesic is the unique shortest path with the geodesic distance between . Parallel transport is the Riemannian analogy of vector translation, induced by the Riemannian metric.
Let be linearly independent, so that they span a two dimensional subspace of . Under the exponential map, this subspace is mapped to a two dimensional submanifold of . The sectional curvature is defined as the Gauss curvature of at , and is a critical concept in the comparison theorems involving geodesic triangles (Burago et al., 2001).
The notion of geodesically convex sets, geodesically (strongly) convex functions and geodesically smooth functions are defined as straightforward generalizations of the corresponding vector space objects to Riemannian manifolds. In particular,
A set is called geodesically convex if for any , there is a geodesic with and for .
We call a function geodesically convex (g-convex) if for any and any geodesic such that , and for all , it holds that
It can be shown that if the inverse exponential map is well-defined, an equivalent definition is that for any , , where is the gradient of at (in this work we assume is differentiable). A function is called geodesically -strongly convex (-strongly g-convex) if for any and gradient , it holds that
We call a vector field geodesically -Lipschitz (-g-Lipschitz) if for any ,
where is the parallel transport from to . We call a differentiable function geodesically -smooth (-g-smooth) if its gradient is -g-Lipschitz, in which case we have
Throughout our analysis, for simplicity, we make the following standing assumptions:
is a geodesically convex set where the exponential map and its inverse are well defined.
The sectional curvature in is bounded, i.e. .
is geodesically -smooth, -strongly convex, and assumes its minimum inside .
All the iterates remain in .
With these assumptions, the problem being solved can be stated formally as .
3 Proposed algorithm: Ragd
Our proposed optimization procedure is shown in Algorithm 1. We assume the algorithm is granted access to oracles that can efficiently compute the exponential map and its inverse, as well as the Riemannian gradient of function . In comparison with Nesterov’s accelerated gradient method in vector space (Nesterov, 2004, p.76), we note two important differences: first, instead of linearly combining vectors, the update for iterates is computed via exponential maps; second, we introduce a paired sequence of parameters , for reasons that will become clear when we analyze the convergence of the algorithm.
Algorithm 1 provides a general scheme for Nesterov-style algorithms on Riemannian manifolds, leaving the choice of many parameters to users’ preference. To further simplify the parameter choice as well as the analysis, we note that the following specific choice of parameters
which leads to Algorithm 2, a constant step instantiation of the general scheme. We leave the proof of this claim as a lemma in the Appendix.
We move forward to analyzing the convergence properties of these two algorithms in the following two sections. In Section 4, we first provide a novel generalization of Nesterov’s estimate sequence to Riemannian manifolds, then show that if a specific tangent space distance comparison inequality (8) always holds, then Algorithm 1 converges similarly as its vector space counterpart. In Section 5, we establish sufficient conditions for this tangent space distance comparison inequality to hold, specifically for Algorithm 2, and show that under these conditions Algorithm 2 converges in iterations, a faster rate than the complexity of Riemannian gradient descent.
4 Analysis of a new estimate sequence
First introduced in (Nesterov, 1983), estimate sequences are central tools in establishing the acceleration of Nesterov’s method. We first note a weaker notion of estimate sequences for functions whose domain is not necessarily a vector space.
A pair of sequences and is called a (weak) estimate sequence of a function , if and for all we have:
This definition relaxes the original definition proposed by Nesterov (2004, def. 2.2.1), in that the latter requires to hold for all , whereas our definition only assumes it holds at the minimizer . We note that similar observations have been made, e.g., in (Carmon et al., 2017). This relaxation is essential for sparing us from fiddling with the global geometry of Riemannian manifolds.
However, there is one major obstacle in the analysis – Nesterov’s construction of quadratic function sequence critically relies on the linear metric and does not generalize to nonlinear space. An example is given in Figure 2, where we illustrate the distortion of distance (hence quadratic functions) in tangent spaces. The key novelty in our construction is inequality (4) which allows a broader family of estimate sequences, as well as inequality (8) which handles nonlinear metric distortion and fulfills inequality (4). Before delving into the analysis of our specific construction, we recall how to construct estimate sequences and note their use in the following two lemmas.
Let us assume that:
is geodesically -smooth and -strongly geodesically convex on domain .
is an arbitrary function on .
is an arbitrary sequence in .
: , .
Then the pair of sequences , which satisfy the following recursive rules:
is a (weak) estimate sequence.
If for a (weak) estimate sequence and we can find a sequence of iterates , such that
By Definition 1 we have . Hence . ∎
Lemma 2 immediately suggest the use of (weak) estimate sequences in establishing the convergence and analyzing the convergence rate of certain iterative algorithms. The following lemma shows that a weak estimate sequence exists for Algorithm 1. Later in Lemma 5, we prove that the sequence in Algorithm 1 satisfies the requirements in Lemma 2 for our estimate sequence.
Let . Assume for all , the sequences , , , and satisfy
then the pair of sequence and , defined by
is a (weak) estimate sequence.
The next lemma asserts that the iterates of Algorithm 1 satisfy the requirement that the function values are upper bounded by defined in our estimate sequence.
Theorem 1 (Convergence of Algorithm 1).
5 Local fast rate with a constant step scheme
By now we see that almost all the analysis of Nesterov’s generalizes, except that the assumption in (8) is not necessarily satisfied. In vector space, the two expressions both reduce to and hence (8) trivially holds with . On Riemannian manifolds, however, due to the nonlinear Riemannian metric and the associated exponential maps, and in general do not equal (illustrated in Figure 2). Bounding the difference between these two quantities points the way forward for our analysis, which is also our main contribution in this section. We start with two lemmas comparing a geodesic triangle and the triangle formed by the preimage of its vertices in the tangent space, in two constant curvature spaces: hyperbolic space and the hypersphere.
Lemma 6 (bi-Lipschitzness of the exponential map in hyperbolic space).
Let be the side lengths of a geodesic triangle in a hyperbolic space with constant sectional curvature , and is the angle between sides and . Furthermore, assume . Let be the comparison triangle in Euclidean space, with , then
We also state without proof that by the same techniques one can show the following result holds.
Lemma 7 (bi-Lipschitzness of the exponential map on hypersphere).
Let be the side lengths of a geodesic triangle in a hypersphere with constant sectional curvature , and is the angle between sides and . Furthermore, assume . Let be the comparison triangle in Euclidean space, with , then
Albeit very much simplified, spaces of constant curvature are important objects to study, because often their properties can be generalized to general Riemannian manifolds with bounded curvature, specifically via the use of powerful comparison theorems in metric geometry (Burago et al., 2001). In our case, we use these two lemmas to derive a tangent space distance comparison theorem for Riemannian manifolds with bounded sectional curvature.
Theorem 2 (Multiplicative distortion of squared distance on Riemannian manifold).
Let , , , be four points in a g-convex, uniquely geodesic set where the sectional curvature is bounded within , for some nonnegative number . Define . Assume for (otherwise ), then we have
The high level idea is to think of the tangent space distance distortion on Riemannian manifolds of bounded curvature as a consequence of bi-Lipschitzness of the exponential map. Specifically, note that and are two geodesic triangles in , whereas and are side lengths of two comparison triangles in vector space. Since is of bounded sectional curvature, we can apply comparison theorems.
First, we consider bound on the distortion of squared distance in a Riemannian manifold with constant curvature . Note that in this case, the hyperbolic law of cosines becomes
which corresponds to the geodesic triangle in hyperbolic space with side lengths , with the corresponding comparison triangle in Euclidean space having lengths . Apply Lemma 6 we have , i.e. . Now consider the geodesic triangle . Let , so that . By Toponogov’s comparison theorem (Burago et al., 2001), we have hence
Similarly, using the spherical law of cosines for a space of constant curvature
and Lemma 7 we can show , where is the side length in Euclidean space corresponding to . Hence by our uniquely geodesic assumption and (Meyer, 1989, Theorem 2.2, Remark 7), with similar reasoning for the geodesic triangle , we have , so that
Theorem 2 suggests that if , we could choose and to guarantee . It then follows that the analysis holds for -th step. Still, it is unknown that under what conditions can we guarantee hold for all , which would lead to a convergence proof. We resolve this question in the next theorem.
Theorem 3 (Local fast convergence).
Recall that in Theorem 1 we already establish that if the tangent space distance comparison inequality (8) holds, then the general Riemannian Nesterov iteration (Algorithm 1) and hence its constant step size special case (Algorithm 2) converge with a guaranteed rate. By the tangent space distance comparison theorem (Theorem 2), the comparison inequality should hold if and are close enough. Indeed, we use induction to assert that with a good initialization, (8) holds for each step. Specifically, for every , if is close to and the comparison inequality holds until the -th step, then is also close to and the comparison inequality holds until the -th step. We postpone the complete proof until Appendix F.
In this work, we proposed a Riemannian generalization of the accelerated gradient algorithm and developed its convergence and complexity analysis. For the first time (to the best of our knowledge), we show gradient based algorithms on Riemannian manifolds can be accelerated, at least in a neighborhood of the minimizer. Central to our analysis are the two main technical contributions of our work: a new estimate sequence (Lemma 3), which relaxes the assumption of Nesterov’s original construction and handles metric distortion on Riemannian manifolds; a tangent space distance comparison theorem (Theorem 2), which provides sufficient conditions for bounding the metric distortion and could be of interest for a broader range of problems on Riemannian manifolds.
Despite not matching the standard convex results, our result exposes the key difficulty of analyzing Nesterov-style algorithms on Riemannian manifolds, an aspect missing in previous work. Critically, the convergence analysis relies on bounding a new distortion term per each step. Furthermore, we observe that the side length sequence can grow much greater than , even if we reduce the “step size” in Algorithm 1, defeating any attempt to control the distortion globally by modifying the algorithm parameters. This is a benign feature in vector space analysis, since (8) trivially holds nonetheless; however it poses a great difficulty for analysis in nonlinear space. Note the stark contrast to (stochastic) gradient descent, where the step length can be effectively controlled by reducing the step size, hence bounding the distortion terms globally (Zhang and Sra, 2016).
A topic of future interest is to study whether assumption (8) can be further relaxed, while maintaining that overall the algorithm still converges. By bounding the squared distance distortion in every step, our analysis provides guarantee for the worst-case scenario, which seems unlikely to happen in practice. It would be interesting to conduct experiments to see how often (8) is violated versus how often it is loose. It would also be interesting to construct some adversarial problem case (if any) and study the complexity lower bound of gradient based Riemannian optimization, to see if geodesically convex optimization is strictly more difficult than convex optimization. Generalizing the current analysis to non-strongly g-convex functions is another interesting direction.
The authors thank the anonymous reviewers for helpful feedback. This work was supported in part by NSF-IIS-1409802 and the DARPA Lagrange grant.
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Appendix A Constant step scheme
Pick . If in Algorithm 1 we set
then we have
Suppose that , then from Algorithm 1 we have is the positive root of