Current methods for document key information extraction (DKIE) rely heavily on labeled data with high annotation costs. To mitigate this issue, the semi-supervised learning (SSL) paradigm, which utilizes unlabeled document samples, has gained broad attention in DKIE. However, existing SSL methods require labeled and unlabeled data to share an identical label space, which is impractical in many DKIE tasks (i.e., some unlabeled samples do not belong to any known classes in the labeled set). In this paper, we formulate this problem as Class-Missing Semi-supervised (CMSS) DKIE. In DKIE, unknown classes usually belong to minority and fine-grained categories, intensifying the misconnections between known and unknown classes and making CMSS more challenging. To address this issue, we propose Synergistic Refinement Estimation (SRE), a progressive prototype estimation scheme that alleviates the unknown classes bias to the majority known classes on long-tailed unlabeled data. Furthermore, dynamic threshold hash rectification and structural calibration mechanisms are proposed to correct connections between fine-grained classes. Extensive experimental results demonstrate that SRE surpasses existing state-of-the-art methods on several DKIE benchmarks. Code is available at https://github.com/anonymoulink/SRE_DKIE .

Decision tree, despite its unmatched interpretability and lightweight structure, faces two key issues that limit its broader applicability: non-differentiability and low testing accuracy. This study addresses these issues by developing a differentiable oblique tree that optimizes the entire tree using gradient-based optimization. We propose an exact reformulation of hard-split trees based on "ReLU+Argmin" mechanism, and then cast the reformulated tree training as an unconstrained optimization task. The ReLU-based sample branching, expressed as exact-zero or non-zero values, preserve a unique decision path, in contrast to soft decision trees with probabilistic routing. The subsequent Argmin operation identifies the unique zero-violation path, enabling deterministic predictions. For effective gradient flow, we approximate Argmin behaviors by scaling softmin function. To ameliorate numerical instability, we propose a warm-start annealing scheme that solves multiple optimization tasks with increasingly accurate approximations. This reformulation alongside distributed GPU parallelism offers strong scalability, supporting 12-depth tree even on million-scale datasets where most baselines fail. Extensive experiments demonstrate that our optimized tree achieves a superior testing accuracy against 14 baselines, including an average improvement of 7.54\% over CART.

In recent years, researchers have proposed the approximation of model predictive control (MPC) using deep neural networks (DNNs). However, a limitation arises as DNNs inherently offer one-to-one mappings, posing a challenge when multiple optimal control inputs correspond to each system state, thereby leading to one-to-many mappings. Therefore, we propose an alternative scheme using mixture networks (MNs) with components of probability (density) distributions in the output layer. This method uses conditional probabilities offered by combining several estimated probability distributions, enabling generating multiple control inputs with the highest probabilities. Notably, this approach is applicable to several problems by choosing a suitable probability distribution, such as using a Gaussian distribution for nonlinear problems and a Bernoulli distribution for mixed-integer linear programming (MILP) problems. We investigate two case studies illustrating that the mixture network-based approximation outperforms the DNN-based approximation.Note to Practitioners—This article proposes using mixture (density) networks, a type of machine learning technique, to enable the online implementation of model predictive control. The prohibitive computation cost associated with model predictive control poses a significant challenge when implementing it in complex nonlinear and MILP problems. While approximation methods of control laws using deep neural networks have been studied to address this issue, it is unsuitable for problems where each state has multiple optimal control inputs. In contrast, our proposed approach can accurately approximate the control laws for these problems. This notable feature can facilitate the online implementation of model predictive control in a wide range of nonlinear and MILP problems.