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：在统计学中，对数凹密度估计是形状约束下非参数推断领域的一个核心问题。尽管近年来对该经典估计量（即对数凹最大似然估计）的统计理论研究取得了重大进展，但该方法的应用因支撑其计算的非光滑凸优化问题的复杂性而受到阻碍。我们加深了对该优化问题结构性质的理解，并据此提出了基于随机平滑和内斯特罗夫平滑的新算法，同时结合了精度递增的适当积分离散化方法。我们证明了这些方法在目标函数尺度上以高概率和期望意义下均享有$1/T$阶的收敛速率（直至对数因子），其中$T$表示迭代次数。通过合成数据与真实数据的实验验证了该新计算框架的优势，且我们的实现代码已在GitHub仓库\texttt{LogConcComp}（对数凹计算）中公开。