Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a universal sequence predictor in case of unknown mu. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Loef) random sequences remained open. Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of M itself. We show that there are universal semimeasures M which do not converge for all random sequences, i.e. we give a partial negative answer to the open problem. We also provide a positive answer for some non-universal semimeasures. We define the incomputable measure D as a mixture over all computable measures and the enumerable semimeasure W as a mixture over all enumerable nearly-measures. We show that W converges to D and D to mu on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role.
Broad stochastic configuration residual learning system for norm-convergent universal approximation
School of Control and Computer Engineering, North China Electric Power University（控制与计算机工程学院，华北电力大学）；School of Mathematics and Physics, North China Electric Power University（数学与物理学院，华北电力大学）；School of Intelligent Systems Engineering, Shenzhen Campus of Sun Yat-sen University（智能系统工程学院，中山大学深圳校区）
IDInit: A Universal and Stable Initialization Method for Neural Network Training
Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；The Chinese University of Hong Kong（香港中文大学）；The Hong Kong Polytechnic University（香港理工大学）；MetaAI（元宇宙人工智能实验室）；Fudan University（复旦大学）；Shanghai Academy of AI for Science（上海人工智能科学研究院）
