The nonparametric learning of positive-valued functions appears widely in machine learning, especially in the context of estimating intensity functions of point processes. Yet, existing approaches either require computing expensive projections or semidefinite relaxations, or lack convexity and theoretical guarantees after introducing nonlinear link functions. In this paper, we propose a novel algorithm, pseudo mirror descent, that performs efficient estimation of positive functions within a Hilbert space without expensive projections. The algorithm guarantees positivity by performing mirror descent with an appropriately selected Bregman divergence, and a pseudo-gradient is adopted to speed up the gradient evaluation procedure in practice. We analyze both asymptotic and nonasymptotic convergence of the algorithm. Through simulations, we show that pseudo mirror descent outperforms the state-of-the-art benchmarks for learning intensities of Poisson and multivariate Hawkes processes, in terms of both computational efficiency and accuracy.
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机译：积极值函数的非参数学习在机器学习中广泛出现，尤其是在估算点流程的强度函数的上下文中。然而，在引入非线性链接功能之后，现有方法需要计算昂贵的预测或半纤维放松，或缺乏凸起和理论保证。在本文中，我们提出了一种新颖的算法，伪镜血迹，其在没有昂贵的投影的情况下执行高效估计Hilbert空间内的正函数。 The algorithm guarantees positivity by performing mirror descent with an appropriately selected Bregman divergence, and a pseudo-gradient is adopted to speed up the gradient evaluation procedure in practice.我们分析了算法的渐近和非对症融合。通过仿真，我们表明伪镜血清突出了用于学习泊松和多变量霍克斯流程的最先进的基准，这就是计算效率和准确性。
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