Large Language Models (LLMs) have demonstrated remarkable capabilities across various tasks but require substantial computational resources, limiting their deployment in resource-constrained environments. While one-shot pruning methods can reduce model size without expensive retraining, they typically optimize for single objectives, ignoring LLMs' multi-faceted applications. We introduce Multi-Objective One-Shot Pruning (MOSP), which formulates LLM pruning as a multi-objective optimization problem. MOSP efficiently generates a Pareto set of pruned models representing different capability trade-offs, allowing users to select solutions aligned with their preferences. The proposed approach identifies share core support while enabling specialized support. Experiments across various LLMs and sparsity levels demonstrate MOSP's superior performance in navigating multi-objective trade-offs compared to baseline methods.

In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated as low-rank semi-definite programming (SDP) problems. Traditional formulations use the square loss, which is notorious for being sensitive to outliers. We propose to replace this with more robust noise models, including the$\ell _1$-loss and other nonconvex losses. However, the resultant optimization problem becomes difficult as the objective is no longer convex or smooth. To alleviate this problem, we design an efficient algorithm based on majorization-minimization. The crux is on constructing a good optimization surrogate, and we show that this surrogate can be efficiently obtained by the alternating direction method of multipliers (ADMM). By properly monitoring ADMM's convergence, the proposed algorithm is empirically efficient and also theoretically guaranteed to converge to a critical point. Extensive experiments are performed on four machine learning applications using both synthetic and real-world data sets. Results show that the proposed algorithm is not only fast but also has better performance than the state-of-the-arts.

Robustness recently becomes one of the major concerns among machine learning community, since learning algorithms are usually vulnerable to outliers or corruptions. Motivated by such a trend and needs, we pursue robustness in semi-definite programming (SDP) in this paper. Specifically, this is done by replacing the commonly used squared loss with the more robust L1-loss in the low-rank SDP. However, the resulting objective becomes neither convex nor smooth. As no existing algorithms can be applied, we design an efficient algorithm, based on majorization-minimization, to optimize the objective. The proposed algorithm not only has cheap iterations and low space complexity but also theoretically converges to some critical points. Finally, empirical study shows that the new objective armed with proposed algorithm outperforms state-of-the-art in terms of both speed and accuracy.

Many machine learning problems involve learning a low-rank positive semidefinite matrix. However, existing solvers for this low-rank semidefinite program (SDP) are often expensive. In this paper, by factorizing the target matrix as a product of two matrices and using a Courant penalty to penalize for their difference, we reformulate the SDP as a biconvex optimization problem. This allows the use of multiconvex optimization techniques to define simple surrogates, which can be minimized easily by block coordinate descent. Moreover, while traditionally this biconvex problem approaches the original problem only when the penalty parameter is infinite, we show that the two problems are equivalent when the penalty parameter is sufficiently large. Experiments on a number of SDP applications in machine learning show that the proposed algorithm is as accurate as other state-of-the-art algorithms, but is much faster, especially on large data sets.

Nonparametric kernel learning (NPKL) is a flexible approach to learn the kernel matrix directly without assuming any parametric form. It can be naturally formulated as a semidefinite program (SDP), which, however, is not very scalable. To address this problem, we propose the combined use of low-rank approximation and block coordinate descent (BCD). Low-rank approximation avoids the expensive positive semidefinite constraint in the SDP by replacing the kernel matrix variable with V(T)V, where V is a low-rank matrix. The resultant nonlinear optimization problem is then solved by BCD, which optimizes each column of V sequentially. It can be shown that the proposed algorithm has nice convergence properties and low computational complexities. Experiments on a number of real-world data sets show that the proposed algorithm outperforms state-of-the-art NPKL solvers.

The goal of semisupervised kernel matrix learning (SS-KML) is to learn a kernel matrix on all the given samples on which just a little supervised information, such as class label or pairwise constraint, is provided. Despite extensive research, the performance of SS-KML still leaves some space for improvement in terms of effectiveness and efficiency. For example, a recent pairwise constraints propagation (PCP) algorithm has formulated SS-KML into a semidefinite programming (SDP) problem, but its computation is very expensive, which undoubtedly restricts PCPs scalability in practice. In this paper, a novel algorithm, called kernel propagation (KP), is proposed to improve the comprehensive performance in SS-KML. The main idea of KP is first to learn a small-sized sub-kernel matrix (named seed-kernel matrix) and then propagate it into a larger-sized full-kernel matrix. Specifically, the implementation of KP consists of three stages: 1) separate the supervised sample (sub)set X(l) from the full sample set X; 2) learn a seed-kernel matrix on X(l) through solving a small-scale SDP problem; and 3) propagate the learnt seed-kernel matrix into a full-kernel matrix on X . Furthermore, following the idea in KP, we naturally develop two conveniently realizable out-of-sample extensions for KML: one is batch-style extension, and the other is online-style extension. The experiments demonstrate that KP is encouraging in both effectiveness and efficiency compared with three state-of-the-art algorithms and its related out-of-sample extensions are promising too.