Paper ID: | 1030 |
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Title: | On the Curved Geometry of Accelerated Optimization |

Somewhere in Section 2, clarify what $\mathcal{M}$ is. I suppose it is simply R^n, with its usual manifold structure. Eq. between lines 54 and 55: Perhaps use a different notation to disambiguate between the acceleration of (the components of) \gamma (which is with respect to the affine connection), and the acceleration of the components of x, which seems to be in the classical sense. For example, in the book by Absil, Mahony and Sepulchre, they use D/dt for covariant derivatives, and d/dt for classical derivatives. So, for acceleration induced by the affine connection, we would have D^2/dt^2. Section 3: For smooth functions, strict convexity does not necessarily imply that the Hessian is everywhere positive definite (though the reverse is true). Isn't this an issue when using the Hessian of $\phi$ to define a Riemannian metric? Around equation 1: Can you explain more how the two discussions under "biorthogonal coordinate systems" are related? My understanding is: if we perturb x along the direction v, then y is perturbed along the direction D(\nabla \phi)(x)[v] = Hess \phi(x)[v] = H(x)v, and we call this u. If this is the right way to think about it, I would suggest to include it. When thinking about these two structures, I am wondering: would it help to think of two manifolds, M and N, and to think of \nabla \phi and \nabla \phi* as a diffeomorphism between these two manifolds? Then, the relationship between tangent spaces is clear through this map, and each manifold can have its own flat connection. I am more familiar with the definition of connection as a map which takes two smooth vector fields and outputs one smooth vector field, corresponding to a directional derivative of one along the other. If available, can you give a reference for a textbook that shows it is equivalent to specify the geodesics? Line 116: the velocity should be a vector; -f(x)/\rho seems to be missing \nabla. Also, it should be specified that this is the velocity in dual coordinates (on a similar note: the sentence goes on to say "... of the connection", but it would be good to specify which connection explicitly. Section 5: Here, phi was set to a special form (spell it out?) so the Bregman divergence is just the squared norm. But then, duality is used here by applying to f the concepts that were exposed for phi, regardless of Bregman divergence concerns. Specifically, for any smooth, strictly convex function q, we have a well defined q*, and the gradients of q and q* are inverse maps of one another, giving a diffeomorphism for R^n. This is used in two ways in the paper: it's used with phi in the context of Bregman divergences; and it's used with f in the context of primal-dual proximal point methods. Is this interpretation correct? Perhaps this could be clarified? Lines 138-140: Is the point that you would like an expression for g^k that the distance between $g$ and $g^{k-1}$ rather than $x^{k-1}$? This could be stated more directly. Line 154, "due to the flatness property, a simple closed-form solution can be derived by equating the derivative to 0" -- why is flatness necessary here? The definition of g^k at the bottom of page 5 is a the unique solution to a striclty convex problem; there's no need for flatness to claim that the optimum is attained when the gradient wrt g is zero? How do we initialize the algorithm, equation (5)? Do we need to be able to compute at least the very first "gradient of the conjugate of f"? Is that an issue in practice? Line 211: wouldn't \dot \gamma(t) be in the tangent space to the tangent bundle of M, rather than in the tangent space to the manifold M itself? Line 20: simpliest -> simplest Line 45: is type -> is a type Line 51: $x$on -> $x$ on Line 52: mention the name "Christoffel symbols"? Line 78: missing comma "coordinate, with" Line 130: missing reference after "Equation" (also, "i.e." should not start a sentence) Line 215: it's -> its

Starting from the well-know paper [2] about the explanation of acceleration, this area has been developed for several years. For this area, I have two criteria to measure the importance: 1) A good explanation should explain a large class of acceleration phenomenons. Nesterov's acceleration can be used for many settings, such as convex/strongly convex, composite/non-composite, first-order/high-order, etc. If we can explain various settings, then our explanation is less possible to be "overfitting". Inspired by estimation sequence, [1] proposed a unified theory for first-order algorithms to explain nearly all the settings of first-order algorithms, and showed substantial importance and elegance of the estimation sequence technique. 2) A good explanation should be able to give some new results and insights. Although the explanation itself is important, its practical value is its impact of developing new algorithms. As examples, [2] and [3] propose some restart heuristics according to the explanation. Unfortunately, the reviewer found that this paper can not satisfy the two criteria properly. As a unified theory has been given, this paper can only address the strongly convex setting. Meanwhile, I can not find something new induced by this explanation. The differential geometry perspective is novel. However, the dual linearity of Bregman iteration is a common sense in this community. The concept of flat connection is just another description of dual linearity. When the concept of differential geometry is used, we always expect something different from Euclidean space. If the author can show that acceleration can still work for "nontrivial" connections, this paper will have impacts. However, the authors seem only using the concept in differential geometry and not exploring it in depth. The introduction section is not state of the art and seems misleading. As [1] showed, the modern formulation of estimation sequence has strong power to explain acceleration. [1] Diakonikolas, Jelena, and Lorenzo Orecchia. "The approximate duality gap technique: A unified theory of first-order methods." SIAM Journal on Optimization 29.1 (2019): 660-689. [2] Su, Weijie, Stephen Boyd, and Emmanuel Candes. "A differential equation for modeling Nesterovâ€™s accelerated gradient method: Theory and insights." Advances in Neural Information Processing Systems. 2014. [3] Krichene, Walid, Alexandre Bayen, and Peter L. Bartlett. "Accelerated mirror descent in continuous and discrete time." Advances in neural information processing systems. 2015.

The paper explains well its connections to related work. I would like to see the authors' view of how the geometric descent method of [1] fits into the picture. The authors should also say more about the difference between their method ond that of Allen-Zhu and Orecchia 2017; isn't the distinction not the geometry, but motivating via the proximal method? It seems the authors have squeezed the space from above certain display equations. I consider this a grave sin. Certainly other authors also had plenty to say, and figured out how to say it in 8 pages, following the template. There are sections you could remove without much loss. For example, the heavy ball section, 6.3, adds little. I still found the definitions of biorthogonality and of flatness of a connection confusing. I am not sure whether you can explain these better, or whether these ideas are just not suited to an 8 page format. I hope you can explain them better. Why is the relation above line 87 called "biorthogonal", and why is it important? It seems like a simple change of variables. As for flat connections, can you say why the connection coefficients vanish for the dual connection? A reference back to the equation below line 54 might also help. I struggled in particular to extract meaning from lines 102-105. [1] Sebastian Bubeck, Yin Tat Lee, Mohit Singh. "A geometric alternative to Nesterovâ€™s accelerated gradient descent" https://arxiv.org/pdf/1506.08187.pdf Detailed comments: (I'm picky with you on grammar here because it's so close to perfect. Also an expository paper should have perfect grammar.) * capitalization errors in abstract, also line 30 * spacing error line 51 * the spacing above the eqn below line 68 is absurdly compressed. * line 116: have you left a \nabla off here? (to match the display equation below) * line 120: sentence fragment. * line 134: comma splice. * line 163: why do you think your method has three parameters? shouldn't two be enough? * below line 197: should be a nabla, not prime * line 215: its, not it's * line 216: write out wrt * line 218: sentence fragment * line 245: generally one avoids contractions in formal writing unless it serves a higher purpose. * line 248: why does momentum yield the accelerated method for Su 2014, but not for you? * Bibliography needs cleaning. You have two references to the same article by Su, Boyd, and Candes!