Paper:
Robust Portfolio Optimization
We propose a robust portfolio optimization approach based on quantile statistics. The proposed method is robust to extreme events in asset returns, and accommodates large portfolios under limited historical data. Specifically, we show that the risk of the estimated portfolio converges to the oracle optimal risk with parametric rate under weakly dependent asset returns. The theory does not rely on higher order moment assumptions, thus allowing for heavy-tailed asset returns. Moreover, the rate of convergence quantifies that the size of the portfolio under management is allowed to scale exponentially with the sample size of the historical data. The empirical effectiveness of the proposed method is demonstrated under both synthetic and real stock data. Our work extends existing ones by achieving robustness in high dimensions, and by allowing serial dependence.

Review:
The paper presents a method to derive a portfolio based on minimizing the variance of the return estimator, where the covariance is estimated by a quantile-based scatter matrix. The authors discuss how to estimate this matrix R^Q from data. An important property of their estimator is that no assumptions on the tails of the distribution are made. In Section 4, the authors bound the error of the quantile-based scatter matrix and their empirical estimate. Moreover, they show that R^Q is proportional to the covariance matrix if the asset returns follow an elliptical distribution (including Gaussians).I'm inclined to accept the paper because especially the latter finding is of interest to the ML community. However, the authors should address the following issues.Minimizing the overall variance of the financial return estimator in Eq. 2.1 is a "proxy" to solve the actual problem of maximizing return. Even if the problem statement and terms such as long/short positions are well known in the finance community, they may be not in the ML community. Please provide a clear problem setting. In contrast, there is no need to introduce matrix or vector norms.Replacing the covariance by any matrix R^Q seems arbitrary and unmotivated in the first place. If I understand correctly, the actual problem is that the sample variance is a "bad" estimate of the real covariance (for several reasons including a lack of observations). The proposed estimator R-tilde^Q is supposed to be a more accurate estimate w.r.t. metric E[X^T,w], where w is defined by solving (3.2) given the covariance estimate and X is any stochastic process of asset returns. This should be made clear in the beginning of Section 3. Also I would like to see an empirical comparison of different covariance estimators for various matrix distances; not just for E[X^T,w] as in Figure 1.Is estimator Q-hat in (3.1) a consistent/unbiased estimator of Q? To me it seems that Q-hat overestimates the quantiles since pairs (z_s,z_t) are "drawn" from sample {z_i} without replacement.In terms of notation, there is no real distinction between random variables and their realization which is sometimes confusing.typos: line 70 and 71, M_jk should not be bold face line 72: ^q missing for the q-norm definition 4.2: P(A) - P(A|B) without |.| Eq. 8 contains equalities; the text thereafter refers to inequalities