{'fraction_non_alphanumeric': 0.02680067001675042, 'fraction_numerical': 0.006700167504187605, 'mean_word_length': 5.191709844559585, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Generalized drift analysis in continuous domain: linear convergence of (1 + 1)-ES on strongly convex functions with Lipschitz continuous gradients

We prove the linear convergence of the (1 + 1)-Evolution Strategy (ES) with a success based step-size adaptation on a broad class of functions, including strongly convex functions with Lipschitz continuous gradients, which is often assumed to analyze gradient based methods. Our proof is based on the methodology recently developed to analyze the same algorithm on the spherical function, namely the additive drift analysis on unbounded continuous domain. An upper bound of the expected first hitting time is derived, from which we can conclude that our algorithm converges linearly. We investigate the class of functions that satisfy the assumptions of our main theorem, revealing that strongly convex functions with Lipschitz continuous gradients and their strictly increasing transformation satisfy the assumptions. To the best of our knowledge, this is the first paper showing the linear convergence of the (1+1)-ES on such a broad class of functions. This opens the possibility to compare the (1 + 1)-ES and gradient based methods in theory.