Accelerated Gradient Flow for Probability Distributions
This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in wibisono2016 from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton*s equations for the optimal gradient flow. The Hamilton*s equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton*s equations as a system of N interacting particles. The continuous limit of the Nesterov*s algorithm is shown to be a special case with N=1. The algorithm is illustrated with numerical examples.
 The articles adapt the framework developed in Wibisono & al to the (infinite dimensional) setting consisting in carrying out gradient descent in the space of probability distributions. PROS: - the text is well written, with clear references to the literature and a high-level description of the current state-of-the-art. - there is a good balance between mathematical details and high-level descriptions of the methods - although I have not been able to check all the details of the proofs, the results appear to be correct. CONS: - while I think that this type of article is interesting, I was really frustrated to discover at the end that the proposed methods either rely on strong Gaussian assumptions, or "density estimations". In other words, no "practical" method is really proposed. - no comparison with other existing method is provided.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Accelerated Gradient Flow for Probability Distributions
This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in wibisono2016 from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton*s equations for the optimal gradient flow. The Hamilton*s equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton*s equations as a system of N interacting particles. The continuous limit of the Nesterov*s algorithm is shown to be a special case with N=1. The algorithm is illustrated with numerical examples.
user
<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 articles adapt the framework developed in Wibisono & al to the (infinite dimensional) setting consisting in carrying out gradient descent in the space of probability distributions. PROS: - the text is well written, with clear references to the literature and a high-level description of the current state-of-the-art. - there is a good balance between mathematical details and high-level descriptions of the methods - although I have not been able to check all the details of the proofs, the results appear to be correct. CONS: - while I think that this type of article is interesting, I was really frustrated to discover at the end that the proposed methods either rely on strong Gaussian assumptions, or "density estimations". In other words, no "practical" method is really proposed. - no comparison with other existing method is provided.
assistant
 The articles adapt the framework developed in Wibisono & al to the (infinite dimensional) setting consisting in carrying out gradient descent in the space of probability distributions. PROS: - the text is well written, with clear references to the literature and a high-level description of the current state-of-the-art. - there is a good balance between mathematical details and high-level descriptions of the methods - although I have not been able to check all the details of the proofs, the results appear to be correct. CONS: - while I think that this type of article is interesting, I was really frustrated to discover at the end that the proposed methods either rely on strong Gaussian assumptions, or "density estimations". In other words, no "practical" method is really proposed. - no comparison with other existing method is provided.