Efficiency versus Convergence of Boolean Kernels for On-Line Learning Algorithms

TL;DR

This paper examines the tradeoff between computational efficiency and learning effectiveness when using Boolean kernels with Perceptron and Winnow algorithms, highlighting limitations in efficiency and mistake bounds.

Contribution

It introduces and analyzes Boolean kernels for Perceptron and Winnow, demonstrating their computational tradeoffs and proving hardness results for kernel-based Winnow learning of DNF.

Findings

Boolean kernels enable efficient Perceptron training over exponential feature spaces

Perceptron with Boolean kernels can make exponentially many mistakes on simple functions

Kernel-based Winnow cannot efficiently simulate its behavior for learning DNF due to computational hardness

Abstract

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.