An Unsupervised Learning Framework Combined with Heuristics for the Maximum Minimal Cut Problem

The Maximum Minimal Cut Problem (MMCP), a NP-hard combinatorial optimization (CO) problem, has not received much attention due to the demanding and challenging bi-connectivity constraint. Moreover, as a CO problem, it is also a daunting task for machine learning, especially without labeled instances. To deal with these problems, this work proposes an unsupervised learning framework combined with heuristics for MMCP that can provide valid and high-quality solutions. As far as we know, this is the first work that explores machine learning and heuristics to solve MMCP. The unsupervised solver is inspired by a relaxation-plus-rounding approach, the relaxed solution is parameterized by graph neural networks, and the cost and penalty of MMCP are explicitly written out, which can train the model end-to-end. A crucial observation is that each solution corresponds to at least one spanning tree. Based on this finding, a heuristic solver that implements tree transformations by adding vertices is utilized to repair and improve the solution quality of the unsupervised solver. Alternatively, the graph is simplified while guaranteeing solution consistency, which reduces the running time. We conduct extensive experiments to evaluate our framework and give a specific application. The results demonstrate the superiority of our method against two techniques designed.

Data Debugging is NP-hard for Classifiers Trained with SGD

Data debugging is to find a subset of the training data such that the model obtained by retraining on the subset has a better accuracy. A bunch of heuristic approaches are proposed, however, none of them are guaranteed to solve this problem effectively. This leaves an open issue whether there exists an efficient algorithm to find the subset such that the model obtained by retraining on it has a better accuracy. To answer this open question and provide theoretical basis for further study on developing better algorithms for data debugging, we investigate the computational complexity of the problem named Debuggable. Given a machine learning model $\mathcal{M}$ obtained by training on dataset $D$ and a test instance $(\mathbf{x}_\text{test},y_\text{test})$ where $\mathcal{M}(\mathbf{x}_\text{test})\neq y_\text{test}$, Debuggable is to determine whether there exists a subset $D^\prime$ of $D$ such that the model $\mathcal{M}^\prime$ obtained by retraining on $D^\prime$ satisfies $\mathcal{M}^\prime(\mathbf{x}_\text{test})=y_\text{test}$. To cover a wide range of commonly used models, we take SGD-trained linear classifier as the model and derive the following main results. (1) If the loss function and the dimension of the model are not fixed, Debuggable is NP-complete regardless of the training order in which all the training samples are processed during SGD. (2) For hinge-like loss functions, a comprehensive analysis on the computational complexity of Debuggable is provided; (3) If the loss function is a linear function, Debuggable can be solved in linear time, that is, data debugging can be solved easily in this case. These results not only highlight the limitations of current approaches but also offer new insights into data debugging.