Nesterov*s method is the discretization of a differential equation with Hessian damping
Su-Boyd-Candes (2014) made a connection between Nesterov*s method and an ordinary differential equation (ODE). We show if a Hessian damping term is added to the ODE from Su-Boyd-Candes (2014), then Nesterov*s method arises as a straightforward discretization of the modified ODE. Analogously, in the strongly convex case, a Hessian damping term is added to Polyak*s ODE, which is then discretized to yield Nesterov*s method for strongly convex functions. Despite the Hessian term, both second order ODEs can be represented as first order systems. Established Liapunov analysis is used to recover the accelerated rates of convergence in both continuous and discrete time. Moreover, the Liapunov analysis can be extended to the case of stochastic gradients which allows the full gradient case to be considered as a special case of the stochastic case. The result is a unified approach to convex acceleration in both continuous and discrete time and in both the stochastic and full gradient cases.
 The paper introduces an ordinary differential equation (ODE) with a term involving the Hessian and shows that Nesterov*s method can be derived from a direct discretization of the proposed ODE. The perspective of connecting Nesterov*s method and ODE is not new, but to the best of my knowledge, this is the first work where Nesterov*s method can be derived directly from the given ODE. The proposed approach is interesting but I find the contribution a bit weak. A large part of the paper is devoted to deriving Nesterov*s method from the proposed ODE, which is basically applying Euler method. However, the intuition behind the ODE is missing. In particular, it will be interesting to explain why acceleration is attained from discretizing the given ODE. I agree that the acceleration is followed by the convergence analysis, but the convergence analysis in the paper is not new, which follows the standard proof of Nesterov*s method using Lyapunov functions. The inexact version is not new neither, which can be viewed as a direct application of inexact accelerated gradient method in [1]. As a result, it will be important to discuss why the proposed framework intuitively provides acceleration, which is missing in the current version. Overall, I find the approach interesting, but a large portion of the paper is spent to reformulate existing result in accelerated gradient descent literature instead of explaining why the proposed ODE helps understanding acceleration. [1] M. Schmidt, N. Roux, and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Nesterov*s method is the discretization of a differential equation with Hessian damping
Su-Boyd-Candes (2014) made a connection between Nesterov*s method and an ordinary differential equation (ODE). We show if a Hessian damping term is added to the ODE from Su-Boyd-Candes (2014), then Nesterov*s method arises as a straightforward discretization of the modified ODE. Analogously, in the strongly convex case, a Hessian damping term is added to Polyak*s ODE, which is then discretized to yield Nesterov*s method for strongly convex functions. Despite the Hessian term, both second order ODEs can be represented as first order systems. Established Liapunov analysis is used to recover the accelerated rates of convergence in both continuous and discrete time. Moreover, the Liapunov analysis can be extended to the case of stochastic gradients which allows the full gradient case to be considered as a special case of the stochastic case. The result is a unified approach to convex acceleration in both continuous and discrete time and in both the stochastic and full gradient cases.
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<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



 The paper introduces an ordinary differential equation (ODE) with a term involving the Hessian and shows that Nesterov*s method can be derived from a direct discretization of the proposed ODE. The perspective of connecting Nesterov*s method and ODE is not new, but to the best of my knowledge, this is the first work where Nesterov*s method can be derived directly from the given ODE. The proposed approach is interesting but I find the contribution a bit weak. A large part of the paper is devoted to deriving Nesterov*s method from the proposed ODE, which is basically applying Euler method. However, the intuition behind the ODE is missing. In particular, it will be interesting to explain why acceleration is attained from discretizing the given ODE. I agree that the acceleration is followed by the convergence analysis, but the convergence analysis in the paper is not new, which follows the standard proof of Nesterov*s method using Lyapunov functions. The inexact version is not new neither, which can be viewed as a direct application of inexact accelerated gradient method in [1]. As a result, it will be important to discuss why the proposed framework intuitively provides acceleration, which is missing in the current version. Overall, I find the approach interesting, but a large portion of the paper is spent to reformulate existing result in accelerated gradient descent literature instead of explaining why the proposed ODE helps understanding acceleration. [1] M. Schmidt, N. Roux, and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization
assistant
 The paper introduces an ordinary differential equation (ODE) with a term involving the Hessian and shows that Nesterov*s method can be derived from a direct discretization of the proposed ODE. The perspective of connecting Nesterov*s method and ODE is not new, but to the best of my knowledge, this is the first work where Nesterov*s method can be derived directly from the given ODE. The proposed approach is interesting but I find the contribution a bit weak. A large part of the paper is devoted to deriving Nesterov*s method from the proposed ODE, which is basically applying Euler method. However, the intuition behind the ODE is missing. In particular, it will be interesting to explain why acceleration is attained from discretizing the given ODE. I agree that the acceleration is followed by the convergence analysis, but the convergence analysis in the paper is not new, which follows the standard proof of Nesterov*s method using Lyapunov functions. The inexact version is not new neither, which can be viewed as a direct application of inexact accelerated gradient method in [1]. As a result, it will be important to discuss why the proposed framework intuitively provides acceleration, which is missing in the current version. Overall, I find the approach interesting, but a large portion of the paper is spent to reformulate existing result in accelerated gradient descent literature instead of explaining why the proposed ODE helps understanding acceleration. [1] M. Schmidt, N. Roux, and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization