431 research outputs found

    Tight Bounds on Proper Equivalence Query Learning of DNF

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    We prove a new structural lemma for partial Boolean functions ff, which we call the seed lemma for DNF. Using the lemma, we give the first subexponential algorithm for proper learning of DNF in Angluin's Equivalence Query (EQ) model. The algorithm has time and query complexity 2(O~n)2^{(\tilde{O}{\sqrt{n}})}, which is optimal. We also give a new result on certificates for DNF-size, a simple algorithm for properly PAC-learning DNF, and new results on EQ-learning logn\log n-term DNF and decision trees

    Quantum machine learning: a classical perspective

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    Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde

    A Quantum Computational Learning Algorithm

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    An interesting classical result due to Jackson allows polynomial-time learning of the function class DNF using membership queries. Since in most practical learning situations access to a membership oracle is unrealistic, this paper explores the possibility that quantum computation might allow a learning algorithm for DNF that relies only on example queries. A natural extension of Fourier-based learning into the quantum domain is presented. The algorithm requires only an example oracle, and it runs in O(sqrt(2^n)) time, a result that appears to be classically impossible. The algorithm is unique among quantum algorithms in that it does not assume a priori knowledge of a function and does not operate on a superposition that includes all possible states.Comment: This is a reworked and improved version of a paper originally entitled "Quantum Harmonic Sieve: Learning DNF Using a Classical Example Oracle

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

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    Consider the following heuristic for building a decision tree for a function f:{0,1}n{±1}f : \{0,1\}^n \to \{\pm 1\}. Place the most influential variable xix_i of ff at the root, and recurse on the subfunctions fxi=0f_{x_i=0} and fxi=1f_{x_i=1} on the left and right subtrees respectively; terminate once the tree is an ε\varepsilon-approximation of ff. We analyze the quality of this heuristic, obtaining near-matching upper and lower bounds: \circ Upper bound: For every ff with decision tree size ss and every ε(0,12)\varepsilon \in (0,\frac1{2}), this heuristic builds a decision tree of size at most sO(log(s/ε)log(1/ε))s^{O(\log(s/\varepsilon)\log(1/\varepsilon))}. \circ Lower bound: For every ε(0,12)\varepsilon \in (0,\frac1{2}) and s2O~(n)s \le 2^{\tilde{O}(\sqrt{n})}, there is an ff with decision tree size ss such that this heuristic builds a decision tree of size sΩ~(logs)s^{\tilde{\Omega}(\log s)}. We also obtain upper and lower bounds for monotone functions: sO(logs/ε)s^{O(\sqrt{\log s}/\varepsilon)} and sΩ~(logs4)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
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