Accelerated Information Gradient flow
We present a systematic framework for the Nesterov*s accelerated gradient flows in the spaces of probabilities embedded with information metrics. Here two metrics are considered, including both the Fisher-Rao metric and the Wasserstein- metric. For the Wasserstein- metric case, we prove the convergence properties of the accelerated gradient flows, and introduce their formulations in Gaussian families. Furthermore, we propose a practical discrete-time algorithm in particle implementations with an adaptive restart technique. We formulate a novel bandwidth selection method, which learns the Wasserstein- gradient direction from Brownian-motion samples. Experimental results including Bayesian inference show the strength of the current method compared with the state-of-the-art.
The paper attempts to develop a counterpart of the well-known Nesterov accelerated gradient method for gradient flows on the space of probability measures equipped with an information metric. This is an important problem which is useful for optimization on probability spaces. The accelerated gradient flow is developed by leveraging a damping Hamiltonian flow. The paper focuses mainly on the case with the Wasserstein metric and provides a convergence analysis. Practical considerations such as discretizing the accelerated flow and bandwidth selection are developed for the use of the method in practical problems. Although the paper has some important merit, I find the paper extremely hard to follow, partly because of its writing style. There is not enough motivation and explanation for the ideas presented. Some discussions and sentences either do not make much sense to me or read badly. For example, this sentence *For the Wasserstein gradient, many classical methods such as Markov Chain Monte Carlo .... are based on this framework...* doesn*t make sense, as the development of MCMC is never based on Wasserstein gradient. Or the sentence right before it *For the Fisher-Rao gradient, classical results including Adam ... and K-FAC .. demonstrate its effectiveness in ...*: it*s not clear what the authors are trying to say here. Adam is not relevant to the Fisher-Rao natural gradient while K-FAC is just an approximation method and isn*t a good reference for demonstrating the effectiveness of the natural gradient. Also, there are many English typos and grammar errors. I didn*t read the proof carefully due to the time constraint, so I cannot judge on the theoretical part of the paper. The numerical experiment is quite limited as it considers very simple problems (a toy example, a single Gaussian distribution and a logistic regression problem). As such, I think there isn*t enough evidence to judge the usefulness of the proposed method in practice. Having said that, I believe this paper can be an important contribution if the authors invest more time on refining its presentation and if more thorough experimental studies are conducted.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Accelerated Information Gradient flow
We present a systematic framework for the Nesterov*s accelerated gradient flows in the spaces of probabilities embedded with information metrics. Here two metrics are considered, including both the Fisher-Rao metric and the Wasserstein- metric. For the Wasserstein- metric case, we prove the convergence properties of the accelerated gradient flows, and introduce their formulations in Gaussian families. Furthermore, we propose a practical discrete-time algorithm in particle implementations with an adaptive restart technique. We formulate a novel bandwidth selection method, which learns the Wasserstein- gradient direction from Brownian-motion samples. Experimental results including Bayesian inference show the strength of the current method compared with the state-of-the-art.
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 paper attempts to develop a counterpart of the well-known Nesterov accelerated gradient method for gradient flows on the space of probability measures equipped with an information metric. This is an important problem which is useful for optimization on probability spaces. The accelerated gradient flow is developed by leveraging a damping Hamiltonian flow. The paper focuses mainly on the case with the Wasserstein metric and provides a convergence analysis. Practical considerations such as discretizing the accelerated flow and bandwidth selection are developed for the use of the method in practical problems. Although the paper has some important merit, I find the paper extremely hard to follow, partly because of its writing style. There is not enough motivation and explanation for the ideas presented. Some discussions and sentences either do not make much sense to me or read badly. For example, this sentence *For the Wasserstein gradient, many classical methods such as Markov Chain Monte Carlo .... are based on this framework...* doesn*t make sense, as the development of MCMC is never based on Wasserstein gradient. Or the sentence right before it *For the Fisher-Rao gradient, classical results including Adam ... and K-FAC .. demonstrate its effectiveness in ...*: it*s not clear what the authors are trying to say here. Adam is not relevant to the Fisher-Rao natural gradient while K-FAC is just an approximation method and isn*t a good reference for demonstrating the effectiveness of the natural gradient. Also, there are many English typos and grammar errors. I didn*t read the proof carefully due to the time constraint, so I cannot judge on the theoretical part of the paper. The numerical experiment is quite limited as it considers very simple problems (a toy example, a single Gaussian distribution and a logistic regression problem). As such, I think there isn*t enough evidence to judge the usefulness of the proposed method in practice. Having said that, I believe this paper can be an important contribution if the authors invest more time on refining its presentation and if more thorough experimental studies are conducted.
assistant
The paper attempts to develop a counterpart of the well-known Nesterov accelerated gradient method for gradient flows on the space of probability measures equipped with an information metric. This is an important problem which is useful for optimization on probability spaces. The accelerated gradient flow is developed by leveraging a damping Hamiltonian flow. The paper focuses mainly on the case with the Wasserstein metric and provides a convergence analysis. Practical considerations such as discretizing the accelerated flow and bandwidth selection are developed for the use of the method in practical problems. Although the paper has some important merit, I find the paper extremely hard to follow, partly because of its writing style. There is not enough motivation and explanation for the ideas presented. Some discussions and sentences either do not make much sense to me or read badly. For example, this sentence *For the Wasserstein gradient, many classical methods such as Markov Chain Monte Carlo .... are based on this framework...* doesn*t make sense, as the development of MCMC is never based on Wasserstein gradient. Or the sentence right before it *For the Fisher-Rao gradient, classical results including Adam ... and K-FAC .. demonstrate its effectiveness in ...*: it*s not clear what the authors are trying to say here. Adam is not relevant to the Fisher-Rao natural gradient while K-FAC is just an approximation method and isn*t a good reference for demonstrating the effectiveness of the natural gradient. Also, there are many English typos and grammar errors. I didn*t read the proof carefully due to the time constraint, so I cannot judge on the theoretical part of the paper. The numerical experiment is quite limited as it considers very simple problems (a toy example, a single Gaussian distribution and a logistic regression problem). As such, I think there isn*t enough evidence to judge the usefulness of the proposed method in practice. Having said that, I believe this paper can be an important contribution if the authors invest more time on refining its presentation and if more thorough experimental studies are conducted.