Mixing Implies Strong Lower Bounds for Space Bounded Learning

With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses graph. We prove that any hypothesis class whose hypotheses graph is mixing cannot be learned using less than 2 2 |H|) memory states unless the learner uses at least a large number of |H| labeled examples. In contrast, there is a learner that uses 2Θ(log|X | log |H|) memory states and only Θ(log |H|) labeled examples, and there is a learner that uses only |H| memory states but a large number Θ(|H| log |H|) of labeled examples. Our work builds on a combinatorial framework we suggested in a previous work for proving lower bounds on space bounded learning. The strong lower bound is obtained by considering a new notion of pseudorandomness for a sequence of graphs that represents the learner. ∗[email protected]. Department of Computer Science, UT Austin. This material is based upon work supported by the National Science Foundation under grants number 1218547 and 1648712. †[email protected]. The Edmond and Lily Safra Center for Brain Sciences, Hebrew University. This work is partially supported by the Gatsby Charitable Foundation, The Israel Science Foundation, and Intel ICRI-CI center. M.M. is grateful to the Harry and Sylvia Hoffman Leadership and Responsibility Program. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 116 (2017)