421 research outputs found
Efficiency versus Convergence of Boolean Kernels for On-Line Learning Algorithms
The paper studies machine learning problems where each example is described
using a set of Boolean features and where hypotheses are represented by linear
threshold elements. One method of increasing the expressiveness of learned
hypotheses in this context is to expand the feature set to include conjunctions
of basic features. This can be done explicitly or where possible by using a
kernel function. Focusing on the well known Perceptron and Winnow algorithms,
the paper demonstrates a tradeoff between the computational efficiency with
which the algorithm can be run over the expanded feature space and the
generalization ability of the corresponding learning algorithm. We first
describe several kernel functions which capture either limited forms of
conjunctions or all conjunctions. We show that these kernels can be used to
efficiently run the Perceptron algorithm over a feature space of exponentially
many conjunctions; however we also show that using such kernels, the Perceptron
algorithm can provably make an exponential number of mistakes even when
learning simple functions. We then consider the question of whether kernel
functions can analogously be used to run the multiplicative-update Winnow
algorithm over an expanded feature space of exponentially many conjunctions.
Known upper bounds imply that the Winnow algorithm can learn Disjunctive Normal
Form (DNF) formulae with a polynomial mistake bound in this setting. However,
we prove that it is computationally hard to simulate Winnows behavior for
learning DNF over such a feature set. This implies that the kernel functions
which correspond to running Winnow for this problem are not efficiently
computable, and that there is no general construction that can run Winnow with
kernels
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
Complexity Results on Learning by Neural Nets
We consider the computational complexity of learning by neural nets. We are inter-
ested in how hard it is to design appropriate neural net architectures and to train
neural nets for general and specialized learning tasks. Our main result shows that
the training problem for 2-cascade neural nets (which have only two non-input nodes,
one of which is hidden) is NP-complete, which implies that nding an optimal net
(in terms of the number of non-input units) that is consistent with a set of exam-
ples is also NP-complete. This result also demonstrates a surprising gap between the
computational complexities of one-node (perceptron) and two-node neural net training
problems, since the perceptron training problem can be solved in polynomial time by
linear programming techniques. We conjecture that training a k-cascade neural net,
which is a classical threshold network training problem, is also NP-complete, for each
xed k 2. We also show that the problem of nding an optimal perceptron (in
terms of the number of non-zero weights) consistent with a set of training examples is
NP-hard.
Our neural net learning model encapsulates the idea of modular neural nets, which
is a popular approach to overcoming the scaling problem in training neural nets. We
investigate how much easier the training problem becomes if the class of concepts to
be learned is known a priori and the net architecture is allowed to be su ciently
non-optimal. Finally, we classify several neural net optimization problems within the
polynomial-time hierarchy
Neural Relax
We present an algorithm for data preprocessing of an associative memory
inspired to an electrostatic problem that turns out to have intimate relations
with information maximization
Communication Complexity Lower Bounds by Polynomials
The quantum version of communication complexity allows the two communicating
parties to exchange qubits and/or to make use of prior entanglement (shared
EPR-pairs). Some lower bound techniques are available for qubit communication
complexity, but except for the inner product function, no bounds are known for
the model with unlimited prior entanglement. We show that the log-rank lower
bound extends to the strongest model (qubit communication + unlimited prior
entanglement). By relating the rank of the communication matrix to properties
of polynomials, we are able to derive some strong bounds for exact protocols.
In particular, we prove both the "log-rank conjecture" and the polynomial
equivalence of quantum and classical communication complexity for various
classes of functions. We also derive some weaker bounds for bounded-error
quantum protocols.Comment: 16 pages LaTeX, no figures. 2nd version: rewritten and some results
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