In this work, we propose a novel Parameter-Efficient Fine-Tuning (PEFT) approach based on Gaussian Graphical Models (GGMs), marking the first application of GGMs to PEFT tasks, to the best of our knowledge. The proposed method utilizes the $\ell_{2,g}$-norm to effectively select critical parameters and capture global dependencies. The resulting non-convex optimization problem is efficiently solved using a Block Coordinate Descent (BCD) algorithm. Experimental results on the GLUE benchmark [24] for fine-tuning RoBERTa-Base [18] demonstrate the effectiveness of the proposed approach, achieving competitive performance with significantly fewer trainable parameters. The code for this work is available at: https://github.com/jzheng20/Course projects.git.

The prevailing mindset is that a single decision tree underperforms classic random forests in testing accuracy, despite its advantages in interpretability and lightweight structure. This study challenges such a mindset by significantly improving the testing accuracy of an oblique regression tree through our gradient-based entire tree optimization framework, making its performance comparable to the classic random forest. Our approach reformulates tree training as a differentiable unconstrained optimization task, employing a scaled sigmoid approximation strategy. To ameliorate numerical instability, we propose an algorithmic scheme that solves a sequence of increasingly accurate approximations. Additionally, a subtree polish strategy is implemented to reduce approximation errors accumulated across the tree. Extensive experiments on 16 datasets demonstrate that our optimized tree outperforms the classic random forest by an average of $2.03\%$ improvements in testing accuracy.

Recently, numerous tensor SVD (t-SVD)-based tensor recovery methods have emerged, showing promise in processing visual data. However, these methods often suffer from performance degradation when confronted with high-order tensor data exhibiting non-smooth changes, commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. Our objective in this study is to provide an effective tensor recovery technique for handling non-smooth changes in tensor data and efficiently explore the correlations of high-order tensor data across its various dimensions without introducing numerous variables and weights. To this end, we introduce a new tensor decomposition and a new tensor norm called the Tensor $U_1$ norm. We utilize these novel techniques in solving the problem of high-order tensor completion problem and provide theoretical guarantees for the exact recovery of the resulting tensor completion models. An optimization algorithm is proposed to solve the resulting tensor completion model iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers. Theoretical analysis showed the convergence of the algorithm to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. Numerical experiments demonstrated the effectiveness of the proposed method in high-order tensor completion, especially for tensor data with non-smooth changes.

Decision trees are essential yet NP-complete to train, prompting the widespread use of heuristic methods such as CART, which suffers from sub-optimal performance due to its greedy nature. Recently, breakthroughs in finding optimal decision trees have emerged; however, these methods still face significant computational costs and struggle with continuous features in large-scale datasets and deep trees. To address these limitations, we introduce a moving-horizon differential evolution algorithm for classification trees with continuous features (MH-DEOCT). Our approach consists of a discrete tree decoding method that eliminates duplicated searches between adjacent samples, a GPU-accelerated implementation that significantly reduces running time, and a moving-horizon strategy that iteratively trains shallow subtrees at each node to balance the vision and optimizer capability. Comprehensive studies on 68 UCI datasets demonstrate that our approach outperforms the heuristic method CART on training and testing accuracy by an average of 3.44% and 1.71%, respectively. Moreover, these numerical studies empirically demonstrate that MH-DEOCT achieves near-optimal performance (only 0.38% and 0.06% worse than the global optimal method on training and testing, respectively), while it offers remarkable scalability for deep trees (e.g., depth=8) and large-scale datasets (e.g., ten million samples).

This paper presents a practical global optimization algorithm for the K-center clustering problem, which aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. This algorithm is based on a reduced-space branch and bound scheme and guarantees convergence to the global optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency, we have designed a two-stage decomposable lower bound, the solution of which can be derived in a closed form. In addition, we also propose several acceleration techniques to narrow down the region of centers, including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-world datasets have demonstrated that our algorithm can solve the K-center problems to global optimal within 4 hours for ten million samples in the serial mode and one billion samples in the parallel mode. Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our algorithm can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.