论文标题
熵屏障的热带化
The tropicalization of the entropic barrier
论文作者
论文摘要
由Bubeck和Eldan（Proc。Mach。Learn。1015）研究的熵障碍是具有渐近最佳自我符合参数的自我屏障。在本文中，我们研究了与熵屏障相关的中心路径的热带化，即该中心路径的对数极限，用于在PUISEUX系列领域定义的线性程序的参数系列。我们的主要结果是，熵中心路径的热带化是一条分段线性曲线，它与Allamigeon等人研究的对数中心路径的热带化相吻合。 （Siam J. AppliedAlg。Geom。，2018）。结果是，热带熵中心路径中的线性碎片数量可以在定义线性程序的维度和不平等数量中指数。
The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. (SIAM J. Applied Alg. Geom., 2018). One consequence is that the number of linear pieces in the tropical entropic central path can be exponential in the dimension and the number of inequalities defining the linear program.
