On Exact Learning Monotone DNF from Membership Queries

Abstract

In this paper, we study the problem of learning a monotone DNF with at most s terms of size (number of variables in each term) at most r (s term r-MDNF) from membership queries. This problem is equivalent to the problem of learning a general hypergraph using hyperedge-detecting queries, a problem motivated by applications arising in chemical reactions and genome sequencing.We first present new lower bounds for this problem and then present deterministic and randomized adaptive algorithms with query complexities that are almost optimal. All the algorithms we present in this paper run in time linear in the query complexity and the number of variables n. In addition, all of the algorithms we present in this paper are asymptotically tight for fixed r and/or s.

Exact learning of juntas from membership queries

“…Therefore, applying the lower bound from [11,12] gives result (1) in Table 2. A lower bound for the randomized case follows from a general lower bound presented in [16]. See result (2).…”

Exact learning from an honest teacher that answers membership queries

“…See [6,11,12,14,15,42,55,87,91,94,100,114,115,122,139,144,211,212,235,262] for more details on the problem, learnability of subclasses of s-term r-MDNF and other applications. This problem is also called, "sets of positive subsets" [262] "complex group testing" [114,211] and "group testing in hypergraph" [139].…”

“…Adaptive algorithms for learning s-term r-MDNF is studied in [14,15] and [12]. The information theoretic lower bound for this class is rs log n. Angluin and Chen gave in [15] the lower bound Ω((2s/r) r/2 + rs log n) when s > r and Abasi et al gave in [12] the lower bound Ω((r/s) s−1 + rs log n) when s ≤ r. Angluin and Chen gave a polynomial time adaptive algorithm for learning s-term 2-MDNF that asks O(s log n) queries. Therefore, the class s-term 2-MDNF is adaptively optimally learnable.…”

“…Therefore, the class s-term 2-MDNF is adaptively optimally learnable. In [12] Abasi et al gave a polynomial time learning algorithm for s-term r-MDNF that asks rs log n + (r/s) s+o(s) queries when r > s and rs log n + s r/2+o(r) queries when r ≤ s. They also gave some randomized algorithms.…”

“…stand for deterministic algorithm and randomized algorithm, respectively. [12,15] Det. rs log n + s r/2+o(r) [12,15] Rand.…”