This paper studies a simple data-driven approach to high-dimensional linear programs (LPs). Given data of past $n$-dimensional LPs, we learn an $n\times k$ \textit{projection matrix} ($n > k$), which reduces the dimensionality from $n$ to $k$. Then, we address future LP instances by solving $k$-dimensional LPs and recovering $n$-dimensional solutions by multiplying the projection matrix. This idea is compatible with any user-preferred LP solvers, hence a versatile approach to faster LP solving. One natural question is: how much data is sufficient to ensure the recovered solutions' quality? We address this question based on the idea of \textit{data-driven algorithm design}, which relates the amount of data sufficient for generalization guarantees to the \textit{pseudo-dimension} of performance metrics. We present an $\tilde{\mathrm{O}}(nk^2)$ upper bound on the pseudo-dimension ($\tilde{\mathrm{O}}$ compresses logarithmic factors) and complement it by an $\Omega(nk)$ lower bound, hence tight up to an $\tilde{\mathrm{O}}(k)$ factor. On the practical side, we study two natural methods for learning projection matrices: PCA- and gradient-based methods. While the former is simple and efficient, the latter sometimes leads to better solution quality. Experiments confirm that learned projection matrices are beneficial for reducing the time for solving LPs while maintaining high solution quality.

翻译：本文研究了一种针对高维线性规划（LP）的简单数据驱动方法。给定过去$n$维LP问题的数据，我们学习一个$n\times k$的投影矩阵（$n > k$），从而将维度从$n$降至$k$。随后，通过求解$k$维LP问题并乘以投影矩阵恢复$n$维解，即可处理未来的LP实例。该方法可与用户偏好的任何LP求解器兼容，因此是一种加速LP求解的通用方法。一个自然的问题是：需要多少数据才能确保恢复解的质量？我们基于数据驱动算法设计的思想回答该问题，该方法将确保泛化保证所需的数据量与性能度量的伪维度相关联。我们给出了伪维度的$\tilde{\mathrm{O}}(nk^2)$上界（$\tilde{\mathrm{O}}$隐含对数因子），并补充了$\Omega(nk)$下界，因此紧度至多相差$\tilde{\mathrm{O}}(k)$因子。在实践层面，我们研究了两种学习投影矩阵的自然方法：基于PCA的方法和基于梯度的方法。前者简单高效，而后者有时能获得更优的解质量。实验表明，学习得到的投影矩阵有助于在保持高求解质量的同时减少LP求解时间。