In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order $1/T$ up to logarithmic factors on the objective function scale, where $T$ denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository \texttt{LogConcComp} (Log-Concave Computation).

翻译：在统计中,在形状限制下,对coccocle 密度估计是非参数推断领域的一个中心问题。尽管近年来在Canonical 估测器的统计理论(即对数计算的最大概率估测器)方面取得了很大进展,但这种方法的采用却由于作为计算基础的非吸附的 convex优化问题的复杂性而受阻。我们加深了对这一优化问题结构特性的理解,这种结构特性促使人们提出基于随机和Nesterov 平稳的新的算法,同时对不断提高的准确性进行适当的整体分解。我们证明,在客观功能尺度上,这些方法在1/T美元到对数因素的趋同率方面都具有很高的可能性和预期的趋同率,其中,$T表示的重复数。我们的新的计算框架的好处既表现在合成数据上,也表现在真实数据上,我们的实施可以在一个 github 仓库 \ textt{LogConcomp}(Log-concomput)中找到。</s>