Conditional validity and length efficiency are two crucial aspects of conformal prediction (CP). Achieving conditional validity ensures accurate uncertainty quantification for data subpopulations, while proper length efficiency ensures that the prediction sets remain informative and non-trivial. Despite significant efforts to address each of these issues individually, a principled framework that reconciles these two objectives has been missing in the CP literature. In this paper, we develop Conformal Prediction with Length-Optimization (CPL) - a novel framework that constructs prediction sets with (near-) optimal length while ensuring conditional validity under various classes of covariate shifts, including the key cases of marginal and group-conditional coverage. In the infinite sample regime, we provide strong duality results which indicate that CPL achieves conditional validity and length optimality. In the finite sample regime, we show that CPL constructs conditionally valid prediction sets. Our extensive empirical evaluations demonstrate the superior prediction set size performance of CPL compared to state-of-the-art methods across diverse real-world and synthetic datasets in classification, regression, and text-related settings.

Counting and classifying blood cells is an important diagnostic tool in medicine. Support Vector Machines are increasingly popular and efficient and could replace artificial neural network systems. Here a method to classify blood cells is proposed using SVM. A set of statistics on images are implemented in C++. The MPEG-7 descriptors Scalable Color Descriptor, Color Structure Descriptor, Color Layout Descriptor and Homogeneous Texture Descriptor are extended in size and combined with textural features corresponding to textural properties perceived visually by humans. From a set of images of human blood cells these statistics are collected. A SVM is implemented and trained to classify the cell images. The cell images come from a CellaVision DM-96 machine which classify cells from images from microscopy. The output images and classification of the CellaVision machine is taken as ground truth, a truth that is 90-95% correct. The problem is divided in two -- the primary and the simplified. The primary problem is to classify the same classes as the CellaVision machine. The simplified problem is to differ between the five most common types of white blood cells. An encouraging result is achieved in both cases -- error rates of 10.8% and 3.1% -- considering that the SVM is misled by the errors in ground truth. Conclusion is that further investigation of performance is worthwhile.