arXiv subjects

Rocco A. Servedio

Publications and source records attributed to Rocco A. Servedio.

93 records · Page 6Linked to original sources

arXiv2005

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hatγ}^{C}}})$, where $\hatγ^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\log |C| \log \log |C|}{\sqrt{{\hatγ}^{C}}})$ quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for $(ε,δ)$-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain $\{0,1\}^n$ from $Ω(\frac{d}{n})$ to $Ω(\frac{1}ε\log \frac{1}δ+d+\frac{\sqrt{d}}ε)$. This new lower bound comes closer to matching known upper bounds for classical PAC learning.

quant-ph↗

arXiv2003

Toward Attribute Efficient Learning Algorithms

We make progress on two important problems regarding attribute efficient learnability. First, we give an algorithm for learning decision lists of length $k$ over $n$ variables using $2^{\tilde{O}(k^{1/3})} \log n$ examples and time $n^{\tilde{O}(k^{1/3})}$. This is the first algorithm for learning decision lists that has both subexponential sample complexity and subexponential running time in the relevant parameters. Our approach establishes a relationship between attribute efficient learning and polynomial threshold functions and is based on a new construction of low degree, low weight polynomial threshold functions for decision lists. For a wide range of parameters our construction matches a 1994 lower bound due to Beigel for the ODDMAXBIT predicate and gives an essentially optimal tradeoff between polynomial threshold function degree and weight. Second, we give an algorithm for learning an unknown parity function on $k$ out of $n$ variables using $O(n^{1-1/k})$ examples in time polynomial in $n$. For $k=o(\log n)$ this yields a polynomial time algorithm with sample complexity $o(n)$. This is the first polynomial time algorithm for learning parity on a superconstant number of variables with sublinear sample complexity.

cs.LG↗

arXiv2000

Quantum versus Classical Learnability

We consider quantum versions of two well-studied classical learning models: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Correct (PAC) model of learning from random examples. We give positive and negative results for quantum versus classical learnability. For each of the two learning models described above, we show that any concept class is information-theoretically learnable from polynomially many quantum examples if and only if it is information-theoretically learnable from polynomially many classical examples. In contrast to this information-theoretic equivalence betwen quantum and classical learnability, though, we observe that a separation does exist between efficient quantum and classical learnability. For both the model of exact learning from membership queries and the PAC model, we show that under a widely held computational hardness assumption for classical computation (the intractability of factoring), there is a concept class which is polynomial-time learnable in the quantum version but not in the classical version of the model.

quant-ph↗