Statistical Optimization in High Dimensions
针对参数仅能从含噪样本中近似获知的高维优化问题，提出三种结合统计、机器学习和鲁棒优化的算法，分别以最小化严重违反约束数、最大化精确满足约束数和算法运行时间随内在维度而非观测维度增长为目标。
We consider optimization problems whose parameters are known only approximately, based on noisy samples. In large-scale applications, the number of samples one can collect is typically of the same order of (or even less than) the dimensionality of the problem. This so-called high-dimensional statistical regime has been the object of intense recent research in machine learning and statistics, primarily due to phenomena inherent to this regime, such as the fact that the noise one sees here often dwarfs the magnitude of the signal itself. While relevant in numerous important operations research and engineering optimization applications, this setup falls far outside the traditional scope of robust and stochastic optimization. We propose three algorithms to address this setting, combining ideas from statistics, machine learning, and robust optimization. Our algorithms are motivated by three natural optimization objectives: minimizing the number of grossly violated constraints; maximizing the number of exactly satisfied constraints; and, finally, developing algorithms whose running time scales with the intrinsic dimension of a problem, as opposed to its observed dimension—a mismatch that, as we discuss in detail, can be dire in settings where constraints are meant to describe preferences of behaviors. The key ingredients of our algorithms are dimensionality reduction techniques from machine learning, robust optimization, and concentration of measure tools from statistics.