Direct Prediction Set Minimization via Bilevel Conformal Classifier Training

Conformal prediction (CP) is a promising uncertainty quantification framework which works as a wrapper around a black-box classifier to construct prediction sets (i.e., subset of candidate classes) with provable guarantees. However, standard calibration methods for CP tend to produce large prediction sets which makes them less useful in practice. This paper considers the problem of integrating conformal principles into the training process of deep classifiers to directly minimize the size of prediction sets. We formulate conformal training as a bilevel optimization problem and propose the {\em Direct Prediction Set Minimization (DPSM)} algorithm to solve it. The key insight behind DPSM is to minimize a measure of the prediction set size (upper level) that is conditioned on the learned quantile of conformity scores (lower level). We analyze that DPSM has a learning bound of $O(1/\sqrt{n})$ (with $n$ training samples), while prior conformal training methods based on stochastic approximation for the quantile has a bound of $\Omega(1/s)$ (with batch size $s$ and typically $s \ll \sqrt{n}$). Experiments on various benchmark datasets and deep models show that DPSM significantly outperforms the best prior conformal training baseline with $20.46\%\downarrow$ in the prediction set size and validates our theory.

Consistent Estimation of Low-Dimensional Latent Structure in High-Dimensional Data

We consider the problem of extracting a low-dimensional, linear latent variable structure from high-dimensional random variables. Specifically, we show that under mild conditions and when this structure manifests itself as a linear space that spans the conditional means, it is possible to consistently recover the structure using only information up to the second moments of these random variables. This finding, specialized to one-parameter exponential families whose variance function is quadratic in their means, allows for the derivation of an explicit estimator of such latent structure. This approach serves as a latent variable model estimator and as a tool for dimension reduction for a high-dimensional matrix of data composed of many related variables. Our theoretical results are verified by simulation studies and an application to genomic data.