Learning using Local Membership Queries

We introduce a new model of membership query (MQ) learning, where the learning algorithm is restricted to query points that are \emph{close} to random examples drawn from the underlying distribution. The learning model is intermediate between the PAC model (Valiant, 1984) and the PAC+MQ model (where the queries are allowed to be arbitrary points). Membership query algorithms are not popular among machine learning practitioners. Apart from the obvious difficulty of adaptively querying labelers, it has also been observed that querying \emph{unnatural} points leads to increased noise from human labelers (Lang and Baum, 1992). This motivates our study of learning algorithms that make queries that are close to examples generated from the data distribution. We restrict our attention to functions defined on the $n$-dimensional Boolean hypercube and say that a membership query is local if its Hamming distance from some example in the (random) training data is at most $O(\log(n))$. We show the following results in this model: (i) The class of sparse polynomials (with coefficients in R) over $\{0,1\}^n$ is polynomial time learnable under a large class of \emph{locally smooth} distributions using $O(\log(n))$-local queries. This class also includes the class of $O(\log(n))$-depth decision trees. (ii) The class of polynomial-sized decision trees is polynomial time learnable under product distributions using $O(\log(n))$-local queries. (iii) The class of polynomial size DNF formulas is learnable under the uniform distribution using $O(\log(n))$-local queries in time $n^{O(\log(\log(n)))}$. (iv) In addition we prove a number of results relating the proposed model to the traditional PAC model and the PAC+MQ model

Top-Down Induction of Decision Trees: Rigorous Guarantees and Inherent Limitations

Consider the following heuristic for building a decision tree for a function $f : \{0,1\}^n \to \{\pm 1\}$. Place the most influential variable $x_i$ of $f$ at the root, and recurse on the subfunctions $f_{x_i=0}$ and $f_{x_i=1}$ on the left and right subtrees respectively; terminate once the tree is an $\varepsilon$-approximation of $f$. We analyze the quality of this heuristic, obtaining near-matching upper and lower bounds: $\circ$ Upper bound: For every $f$ with decision tree size $s$ and every $\varepsilon \in (0,\frac1{2})$, this heuristic builds a decision tree of size at most $s^{O(\log(s/\varepsilon)\log(1/\varepsilon))}$. $\circ$ Lower bound: For every $\varepsilon \in (0,\frac1{2})$ and $s \le 2^{\tilde{O}(\sqrt{n})}$, there is an $f$ with decision tree size $s$ such that this heuristic builds a decision tree of size $s^{\tilde{\Omega}(\log s)}$. We also obtain upper and lower bounds for monotone functions: $s^{O(\sqrt{\log s}/\varepsilon)}$ and $s^{\tilde{\Omega}(\sqrt[4]{\log s } )}$ respectively. The lower bound disproves conjectures of Fiat and Pechyony (2004) and Lee (2009). Our upper bounds yield new algorithms for properly learning decision trees under the uniform distribution. We show that these algorithms---which are motivated by widely employed and empirically successful top-down decision tree learning heuristics such as ID3, C4.5, and CART---achieve provable guarantees that compare favorably with those of the current fastest algorithm (Ehrenfeucht and Haussler, 1989). Our lower bounds shed new light on the limitations of these heuristics. Finally, we revisit the classic work of Ehrenfeucht and Haussler. We extend it to give the first uniform-distribution proper learning algorithm that achieves polynomial sample and memory complexity, while matching its state-of-the-art quasipolynomial runtime

Learning pseudo-Boolean k-DNF and Submodular Functions

We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f:{0,1}^n -> {0,1,...,k}. Our algorithm runs in time polynomial in n, k^{O(k \log k / \epsilon)}, 1/\epsilon and log(1/\delta) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi, Klivans, Kothari, Lee (SODA '12)] implies only n^{O(k)} query complexity for learning submodular functions in this setting, for fixed epsilon and delta. Our learning algorithm implies a property tester for submodularity of functions f:{0,1}^n -> {0, ..., k} with query complexity polynomial in n for k=O((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon