Random Search under Additive Noise

From the early days in his career, Sid Yakowitz showed interest in noisy function optimization. He realized the universality of random search as an optimization paradigm, and was particularly interested in the minimization of functions Q without making assumptions on the form of Q. Especially the noisy optimization problem appealed to him, as exact computations of Q come often at a tremendous cost, while rough or noisy evaluations are computationally cheaper. His early contributions were with Fisher (Fisher and Yakowitz, 1976; Yakowitz and Fisher, 1973). The present paper builds on these fundamental papers and provides further results along the same lines. It is also intended to situate Sid’s contributions in the growing random search literature. Always motivated by the balance between accurate estimation or optimization and efficient computations, Sid then turned to so-called bandit problems, in which noisy optimization must be performed within a given total computational effort (Yakowitz and Lowe, 1991). The computational aspects of optimization brought him closer to learning and his work there included studies of game playing strategies (Yakowitz, 1989; Yakowitz and Kollier, 1992), epidemiology (Yakowitz, 1992; Yakowitz, Hayes and Gani, 1992) and communication theory (Yakowitz and Vesterdahl, 1993). Sid formulated machine learning invariably as a noisy optimization problem, both over finite and infinite sample spaces: Yakowitz (1992), Yakowitz and Lugosi (1990), and Yakowitz, Jayawardena and Li (1992) summarize his main views and results in this respect. Another thread he liked to follow was stochastic approximation, and in particular the KieferWolfowitz method (1952) for the local optimization in the presence of noise. In a couple of technical reports in 1989 and in his 1993 siam paper, Sid presented globally convergent extensions of this method by combining ideas of random search and stochastic approximation. We have learned from his insights and shared his passion for nonparametric estimation, machine learning and algorithmic statistics. Thank you, Sid.