Heather Battey

Kean Ming Tan, Heather Battey, Wen-Xin Zhou

We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an $\ell_1$-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.

Nicholas Beale, Heather Battey, Anthony C. Davison et al.

If an artificial intelligence aims to maximise risk-adjusted return, then under mild conditions it is disproportionately likely to pick an unethical strategy unless the objective function allows sufficiently for this risk. Even if the proportion $η$ of available unethical strategies is small, the probability ${p_U}$ of picking an unethical strategy can become large; indeed unless returns are fat-tailed ${p_U}$ tends to unity as the strategy space becomes large. We define an Unethical Odds Ratio Upsilon ($Υ$) that allows us to calculate ${p_U}$ from $η$, and we derive a simple formula for the limit of $Υ$ as the strategy space becomes large. We give an algorithm for estimating $Υ$ and ${p_U}$ in finite cases and discuss how to deal with infinite strategy spaces. We show how this principle can be used to help detect unethical strategies and to estimate $η$. Finally we sketch some policy implications of this work.