We consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under certain condition on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ǫ-approximate second-order optimality which have been shown to be tight. Our Hessian approximation condition offers a range of advantages as compared with the prior works and allows for direct construction of the approximate Hessian with a priori guarantees through various techniques, including randomized sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding subsampled trust-region and adaptive cubic regularization methods.

In this paper, we consider variants of Newton-MR algorithm, initially proposed in [62], for solving unconstrained, smooth, but non-convex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient algorithm as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. Recently, [51] establishes certain useful monotonicity properties of MINRES as well as its inherent ability to detect non-positive curvature directions as soon as they arise. We leverage these recent results and show that our algorithms come with desirable properties including competitive first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.

The downside risk of crop production affects the entire supply chain of the agricultural industry nationally and globally. This also has a profound impact on food security, and thus livelihoods, in many parts of the world. The advent of high temporal, spatial and spectral resolution remote sensing platforms, specifically during the last five years, and the advancement in software pipelines and cloud computing have resulted in the collating, analysing and application of “BIG DATA” systems, especially in agriculture. Furthermore, the application of traditional and novel computational and machine learning approaches is assisting in resolving complex interactions, to reveal components of eco-physiological systems that were previously deemed either “too difficult” to solve or “unseen”. In this review, digital technologies encompass mathematical, computational, proximal- and remote sensing technologies. Here, we review the current state of digital technologies and their application in broad acre cropping systems globally and in Australia. More specifically, we discuss the advances in (i) remote sensing platforms, (ii) machine learning approaches to discriminate between crops, and (iii) the prediction of crop phenological stages from both sensing and crop simulation systems for major Australian winter crops. An integrated solution is proposed to allow accurate development, validation and scalability of predictive tools for crop phenology mapping at within-field scales, across extensive cropping areas.

We consider extensions of the Newton-MR algorithm for nonconvex optimization, proposed in [43], to the settings where Hessian information is approximated. Under additive noise model on the Hessian matrix, we investigate the iteration and operation complexities of these variants to achieve first and second-order sub-optimality criteria. We show that, under certain conditions, the algorithms achieve iteration and operation complexities that match those of the exact variant. Focusing on the particular nonconvex problems satisfying Polyak-Lojasiewicz condition, we show that our algorithm achieves a linear convergence rate. We finally compare the performance of our algorithms with several alternatives on a few machine learning problems.