A Space-bounded Learning of Classes with Vc-dim = 1 Proof: Combine Theorem 3 with Corollary 2. 5 Extensions
If the algorithm has in its memory two points x + ; x ? then the algorithm performs a binary search on the path connecting x + and x ? as follows: the algorithm picks the node y 2 which is the middle point on the path connecting x + and x ? ; the algorithm deenes an hypothesis h using the branch leading to y 2 and asks an equivalence query EQ(h). If the answer to the query is \YES" it stops with output h. Otherwise, if the counterexample z is a point on the path connecting x + and x ? satisfying f(z) 6 = h(z) (i.e., a point below y 2) the algorithm sets x + z; if the counterexample z is a point on the path connecting x + and x ? satisfying f(z) = h(z) (i.e., a point above y 2) the algorithm sets x ? z; if z is not at the path at all (in this case f(z) 6 = h(z)) the algorithm sets x + z and removes x ? (that is, in the next iteration it will perform Step 1). Since Step 2 performs a binary search on the path connecting x + and x ? then each sequence of iterations that perform Step 2 is of length at most the logarithm of the length of this path which is at most log jXj. Such a sequence ends with either identifying f or with a new x + outside the path. The number of times that Step 1 is performed is log jXj, because at each step we either nd a new x + which by the deenition of y 1 has in its subtree at most 1=2 of the number of nodes that we had before, or we start a sequence of iterations of Step 2 in which (unless we identify f before) we nd a new x + outside the path to x ? (and y 1) in which again, by the deenition of y 1 , leaves us with at most half the size of the tree. All together the number of iterations is bounded by (log jXj) 2 and the space is bounded by 2 points. Corollary 8 For every class C as above, the class C ? is learnable using number of queries which is polynomial in log X and m. 14 8f : f(x) = …