用于稀疏线性模型的GPU加速非线性分支定界框架
A GPU-accelerated Nonlinear Branch-and-Bound Framework for Sparse Linear Models
AI总结：
本文提出一种基于GPU的非线性分支定界框架，用于加速稀疏线性模型的精确回归，通过高效并行计算提升运行效率。
摘要：
我们研究了带有ℓ0-ℓ2惩罚的精确稀疏线性回归，并开发了一个专门设计用于GPU执行的分支定界（BnB）算法。从视角重新表述出发，我们推导出一个可以由ADMM以闭合形式、坐标-wise更新解决的区间松弛。我们将这些更新结构化，使得每个BnB节点的主要工作仅限于使用共享数据矩阵的批量矩阵-向量运算，从而在单个GPU上实现细粒度的坐标并行和粗粒度的BnB节点并行。可行解（上界）通过在活跃支持上实施的投影梯度法生成，以批量方式实现，使得许多候选支持在GPU上并行更新。我们讨论了内存布局、批量策略和节点之间负载平衡等实际设计选择，这些对于在现代GPU上获得良好的利用率至关重要。在合成和真实高维数据集上，我们的基于GPU的方法在我们的方法的CPU实现、现有的专门BnB方法和商业MIP求解器上均实现了显著的运行时间改进。
英文摘要：
We study exact sparse linear regression with an $\ell_0-\ell_2$ penalty and develop a branch-and-bound (BnB) algorithm explicitly designed for GPU execution. Starting from a perspective reformulation, we derive an interval relaxation that can be solved by ADMM with closed-form, coordinate-wise updates. We structure these updates so that the main work at each BnB node reduces to batched matrix-vector operations with a shared data matrix, enabling fine-grained parallelism across coordinates and coarse-grained parallelism across many BnB nodes on a single GPU. Feasible solutions (upper bounds) are generated by a projected gradient method on the active support, implemented in a batched fashion so that many candidate supports are updated in parallel on the GPU. We discuss practical design choices such as memory layout, batching strategies, and load balancing across nodes that are crucial for obtaining good utilization on modern GPUs. On synthetic and real high-dimensional datasets, our GPU-based approach achieves clear runtime improvements over a CPU implementation of our method, an existing specialized BnB method, and commercial MIP solvers.
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