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 $Ω(1/s)$ (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.

We propose the DPSM method, a density-based node clustering approach that automatically determines the number of clusters and can be applied in both data space and graph space. Unlike traditional density-based clustering methods, which necessitate calculating the distance between any two nodes, our proposed technique determines density through a propagation process, thereby making it suitable for a graph space. In DPSM, nodes are partitioned into small clusters based on propagated density. The partitioning technique has been proved to be sound and complete. We then extend the concept of spectral clustering from individual nodes to these small clusters, while introducing the CluCut measure to guide cluster merging. This measure is modified in various ways to account for cluster properties, thus provides guidance on when to terminate the merging process. Various experiments have validated the effectiveness of DOSM and the accuracy of these conclusions.

In Diffusion Probabilistic Models (DPMs), the task of modeling the score evolution via a single time-dependent neural network necessitates extended training periods and may potentially impede modeling flexibility and capacity. To counteract these challenges, we propose leveraging the independence of learning tasks at different time points inherent to DPMs. More specifically, we partition the learning task by utilizing independent networks, each dedicated to learning the evolution of scores within a specific time sub-interval. Further, inspired by residual flows, we extend this strategy to its logical conclusion by employing separate networks to independently model the score at each individual time point. As empirically demonstrated on synthetic and image datasets, our approach not only significantly accelerates the training process by introducing an additional layer of parallelization atop data parallelization, but it also enhances density estimation performance when compared to the conventional training methodology for DPMs.

The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimization. In this paper, we propose a distributed implementation of the stochastic subgradient method with a theoretical guarantee. Specifically, we show the global convergence of stoDPSM using the Moreau envelope stationarity measure. Furthermore, under a so-called sharpness condition, we show that deterministic DPSM (with a proper initialization) converges linearly to the sharp minima, using geometrically diminishing step-size. We provide numerical experiments to support our theoretical analysis.