Expected Pinball Loss For Quantile Regression And Inverse CDF Estimation

TMLR Paper623 Authors

22 Nov 2022 (modified: 13 Mar 2023)Rejected by TMLREveryoneRevisionsBibTeX
Abstract: We analyze and improve a recent strategy to train a quantile regression model by minimizing an expected pinball loss over all quantiles. We give an asymptotic convergence rate that shows that minimizing the expected pinball loss can be more efficient at estimating single quantiles than training with the standard pinball loss for that quantile, an insight that generalizes the known deficiencies of the sample quantile in the unconditioned setting. Then, to guarantee a legitimate inverse CDF, we propose using flexible deep lattice networks with a monotonicity constraint on the quantile input to guarantee non-crossing quantiles, and show lattice models can be regularized to the same location-scale family. Our analysis and experiments on simulated and real datasets show that the proposed method produces state-of-the-art legitimate inverse CDF estimates that are likely to be as good or better for specific target quantiles.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: We have made a number of revisions to the paper to address these comments, which we summarize below. We have also attached a diff between our revised and original submission to the end of the revised PDF to make it easier to review our changes. **Theory:** - To elaborate on the theoretical implications for conditional quantile regression as requested by Reviewer VW4n and SsMT, we’ve added an additional Corollary 1 to Section 3 that formalizes a basic extension of Theorem 1 to conditional quantile regression with finite categorical features $X$ and a function class which is fully flexible and separable over the values of $X$. General asymptotic theory for conditional inverse CDF estimation is still an open problem. - We’ve elaborated on several assumptions in Theorem 1 in Section 3 and in the Appendix and made the theorem statement more precise, as requested by Reviewer Rz26. **Experiments:** - We have added Table 1, which cross-validates over the training data to perform keypoint selection in our unconditional quantile estimation simulations, as requested by Reviewer VW4n. - To build on Reviewer SsMT’s suggestion about exploring alternative $\tau$-sampling strategies that are concentrated around the desired quantile, we have augmented Figure 1, Table 7, and Table 8 to add results that train with a beta distribution over the quantiles. - We measured and reported results on training time for the different function classes in Table 4, as requested by Reviewer VW4n. **Writing:** - We expanded the introduction with related work on other uncertainty estimation problems and more motivation for non-crossing. - We greatly expanded Open Questions discussion in the Conclusions. - We added some discussion of the beta distribution, in line with the experiments described above and Reviewer SsMT’s comments. - We augmented our explanation of DLNs in Section 4.1 with new figures showing example 2D lattice functions and how their monotonicity-preserving layers can be incorporated into more general architectures.
Assigned Action Editor: ~Daniel_M_Roy1
Submission Number: 623
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