This paper is concerned by solving supervised machine learning problem as an inverse problem. Recently, many works have focused on defining a relationship between supervised learning and the well-known inverse problems. However, this connection between the learning problem and the inverse one has been done in the particular case where the inverse problem is reformulated as a minimization problem with a quadratic cost functional (\(L^2\) cost functional). Although, it is well known that the cost functional can be \(L^1\), \(L^2\) or any positive function that measures the gap between the predicted data and the observed one. Indeed, the use of \(L^1\) loss function for supervised learning problem gives more consistent results (see Rosasco et al. in Neural Comput 16:1063–1076, 2004). This strengthens the idea of reformulating the inverse problem, associated to machine learning problem, into a minimization problem using \( L^{1}\) functional. However, the \(L^{1}\) loss function is non-differentiable, which precludes the use of standard optimization tools. To overcome this difficulty, we propose in this paper a new technique of approximation based on the reformulation of the associated inverse problem into a minimizing one of a slanting cost functional Chen et al. (MIS Q Manag Inf Syst 36:1165–1188, 2012), which is solved using Tikhonov regularization and Newton’s method. This approach leads to an efficient numerical algorithm allowing us to solve supervised learning problem in the most general framework. To confirm this, we present some numerical results showing the efficiency of the proposed approach. Furthermore, the numerical experiment validation is made through academic and real-life data. Thus, the comparison with existing methods and numerical stability of the algorithm is presented in order to show that our approach is better in terms of convergence speed and quality of predicted models.

中文翻译：

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基于非光滑损失函数的有监督学习作为逆问题

本文关注通过将有监督的机器学习问题作为逆问题来解决。近来，许多工作集中在定义监督学习与众所周知的逆问题之间的关系。但是，在学习问题和逆问题之间的这种联系是在特定情况下完成的，在该特殊情况下，将逆问题重新构造为具有二次成本函数（\（L ^ 2 \）成本函数）的最小化问题。虽然，众所周知，成本函数可以是\（L ^ 1 \），\（L ^ 2 \）或任何测量预测数据与观测数据之间的差距的正函数。确实，\（L ^ 1 \）的使用监督学习问题的损失函数给出了更加一致的结果（参见Rosasco等人，Neural Comput 16：1063-1076，2004）。这强化了使用\（L ^ {1} \）函数将与机器学习问题相关的逆问题重新构造为最小化问题的思想。但是，\（L ^ {1} \）损失函数不可微，因此无法使用标准的优化工具。为了克服这个困难，我们在本文中提出了一种新的逼近技术，该技术基于将相关的反问题重构为最小化倾斜成本函数Chen等人的方法。（MIS Q Manag Inf Syst 36：1165-1188，2012），使用Tikhonov正则化和牛顿方法求解。这种方法导致一种有效的数值算法，使我们能够在最通用的框架中解决监督学习问题。为了证实这一点，我们提出了一些数值结果，表明了该方法的有效性。此外，数值实验验证是通过学术和现实数据进行的。从而，