Calibrated Multiple-Output Quantile Regression with Representation Learning

Abstract

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller (sometimes by a factor of 100) compared to existing techniques.

Multiple Representation Approach in Elementary School Science Learning: A Systematic Literature Review

“…Based on the discussion above, this research differs from other recent studies which did not use and study multiple representations in elementary schools. These include research on 3D multiple representations with M3DETR (Guan et al, 2022), representation learning on multiple family quantile regression material (Feldman et al, 2023), and multimodal representation and correlation learning (Mai et al, 2023). Our findings provide a more comprehensive concept because it discusses the approach, implementation, and impact of multiple representations in elementary school science learning.…”

Nonparametric Multiple-Output Center-Outward Quantile Regression

“…As for the depth-based quantile regions, moreover, they are necessarily convex, even for distributions with highly non-convex shapes as in Figure 1. To circumvent this convexity problem, Feldman, Bates and Romano (2021) propose a clever transformation of the data turning its distribution into a latent one with convex level sets by fitting a conditional variational auto-encoder (Sohn, Lee and Yan (2015)). The probability contents of the quantile regions resulting from this machine-learning type of "convexity reparation," however, remain out of control (they still depend on P).…”

Conformalized Online Learning: Online Calibration Without a Holdout Set

“…The time-series data sets we use in the experiments in Section 5 include the day of the week as an element in the feature vector. Therefore, we assess the violation of day-stratified coverage [22,37], as a proxy for conditional coverage. That is, we measure the average deviation of the coverage in each day of the week from the nominal coverage level; see Supplementary Section F.2 for a formal definition.…”

“…where E, D are the encoder and decoder networks of the STDQR model, m is the dimension of z, Q χ 2 m is the inverse cumulative distribution function (ICDF) of the chi-squared distribution with m degrees of freedom, and 0 m is a vectors of zeros of size m. We fit the model with a batch size of 64 with a loss that penalizes deviation of the output of the encoder E from the multivariate normal distribution using the KL-divergence loss [46,47]. We also added a mean squared error loss to encourage the decoder's D output to faithfully reconstruct Y , as explained in [37]. The model's hyper-parameters are summarized in Table 2.…”

“…We follow the experimental protocol described in Section 5.1, except for training an STDQR [37] as a base model, and using the first 12000 samples of the data stream for normalization and the next 3000 for evaluating performance. Additionally, we start applying our calibration scheme at time step 10000, when the raw model is stabilized.…”