Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: - This paper proposes to use the student-t distribution for data generation in financial markets, which can capture the effect of outliers better than the commonly used Gaussian distribution. 

- In particular, the data is modeled using a Laplacian Markov random field with student-t distribution, and a practical ADMM based algorithm to learn the underlying graph structure is proposed.

- The algorithms are designed under two settings namely, connected graphs and k-component graphs, and convergence is proven. 

- The proposed approach is shown to outperform the state-of-the-art Gaussian assumption-based approaches with real-world data from stock markets, foreign exchanges, and cryptocurrencies. 


main_review: - The paper addresses an important problem of designing graph learning algorithms under heavy-tailed data distributions which is relevant in the financial domain but challenging due to its non-convex nature. 

- The convergence of proposed algorithms is analytically proven and simulations suggest better performance compared to existing algorithms. 

- The paper is written well and easy to understand. The mathematical details are clear. 
limitations_and_societal_impact:  A section introducing the preliminary concepts such as MRF and the motivation to use them in financial data would be helpful.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a new estimator for learning graphical models under the t-distribution assumption for data generation process. For the new estimator, an ADMM-based algorithm is designed to solve the associated optimization problem in order to find the graph structures. The new estimator is evaluated extensively on real-world datasets to demonstrate its effectiveness. The paper also establishes the convergence of the proposed algorithm.

main_review: Originality:

The proposed estimator is not entirely new in the literature, given that t-distribution has been used in graphical models before (see [1]). Also, it is a direct result of well-known maximum-likelihood principle, so deriving such estimator should not be difficult. For the ADMM-type learning algorithm, I acknowledge that there are numbers of technical details to be figured out in order to have the exact algorithm, but at the high level this seems to be a standard application of ADMM recipe. So it seems to me that the novelty of this paper is somewhat limited.


Quality:

In terms of technical quality, though the ADMM algorithm seems straightforward, it is nice to see that the paper proves its convergence to stationary points, which requires some efforts and skills. Regarding the algorithm, one thing that is not clear to me is the reason why the trace regularization term is needed in Eq. (16). Conceptually if we drop the regularization term, the new formulation is still equivalent to problem (16) (in the sense that they would end up with the same set of minimizers if globally optimality could be achieved), as the constraints $rank(\Theta) = p - k$ and $\Theta = Lw$ are imposed. I understand that due to the nonconvexity, ADMM algorithm could end up with different solutions for the two formulations (with or without the regularization term), but it is not obvious to me that the regularized one could work better or there is a need to introduce it. This regularization seems to add some novelty to this paper, so I hope that either theoretical or empirical comparison could be provided to support it (e.g., faster convergence or better quality of solutions), otherwise the additional computational cost seems not well-justified.  

For the practical perspective, I like the extensive evaluation of the proposed algorithm on different datasets, showing its usefulness and advantage, but it would be better to extend the comparison beyond the algorithms that just work for Gaussian distributions. Below are a few questions I have for the experiments.

1. For S&P500 stocks, is there a particular reason for choosing the three GICS sectors (Communication Services, Utilities and Real Estate), as opposed to other sectors?
2. Have you considered using some type of residual returns from econometrics literature instead of log returns? If you use GICS sectors as the ground truth for clustering, residual returns may be preferred, as sometimes the co-movement of stock returns is mainly driven by some econometrics factors, which may connect stocks across different sectors.  


Clarity:

Overall the presentation of the paper is clear. The description of the problem setup and the derivation of the algorithm are easy to follow. The figures shown in the experiments give an nice qualitatively demonstration of the superiority of the algorithm.

Significance:

The proposed algorithm seems to be readily applicable to a few real-world problems, as demonstrated in the experiment section. So it should be a good addition to this area for handling heavy-tailed distributions.

[1] Finegold, Michael A., and Mathias Drton. "Robust graphical modeling with t-distributions." In UAI 09, pp. 169-176. 2009. 



********** post-rebuttal changes **********

After reading author response (especially on the concern about algorithmic novelty), I would like to increase my rating from 6 to 7.
limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a framework for learning a graphical model when the data generating distribution is heavy-tailed. More specifically, the author(s) propose a graph learning estimator under Markov Random Field framework when the data is generated from a Student's t-distribution. A scalable algorithm is proposed to handle connected and K-component graphs. The methodology is particularly applicable to financial time series data. The algorithm is shown to be superior in performance compared to existing methods in both simulated and real data applications.
main_review: The paper proposes a novel algorithm based on the alternating direction method of multipliers (ADMM) to a pertinent problem in the graphical model viz. estimating graph structure in the heavy-tailed distribution setting. The problem has a number of applications in financial time series especially clustering the financial time series. The algorithm looks technically correct and the numerical performances are good. However, I have following queries for the author(s) mainly centred on the framework and the methodology

-I understand that the lack of constraints on the degree of the nodes may lead to graphs with isolated nodes however, I am not sure whether in real world applications one will always have the number d available. The information on the graph whether it is sparse or dense may be known but the exact number d may not be known. Any other soft constraints to consider here when d is not known?

-Both theorem 3 and theorem 4 discusses the convergence of the respective algorithms for large $\rho$. But, can author(s) shed some light about any connection with the number of iterations required to achieve the convergence?

-I am not sure whether in the K-connected components case the number k would be always known. One may has to do an estimation of K at the first stage. Am I missing something?

-Do you need any further regularization on Lw in equation (4) to achieve sparsity?

-How does one choose the rank hyperparameter $\eta$?



******** Change after author rebuttal ************

I am happy with the technical justification author(s) provided about the degree of the network and the convergence of the algorithm. Further minor technical details about the regularization and the connected components have been appropriately answered. I have now changed my score to "acceptance".

limitations_and_societal_impact: I didn't see any negative societal impact of their work.
review_title: null

Meta-review: This paper formulates a graphical model learning problem by maximizing the likelihood of a multivariate student-t distribution subject to constraints on the Laplacian structured parameter matrix. The formulation is non convex. An explicit ADMM iteration is derived to arrive at a stationary point. 

The primary strength of the paper is in its multiple experiments demonstrating the utility in the context of financial time-series of the modeling assumptions over the more commonly studied Gaussian graphical model setup. 

The main weakness of the paper is that it does not contain a great deal of conceptual depth or technical novelty: the ideas, and their implementation, appear to be relatively straightforward. 

Overall, the paper seems likely to have impact by spurring further work on estimation of graphical models of various types. 
Decision: Accept (Poster)