A Note on a priori Estimations of Classification Circuit Complexity

Authors:
Andreas A. Albrecht;Alexander V. Chashkin;Costas S. Iliopoulos;Oktay M. Kasim-Zade;Georgios Lappas;Kathleen K. Steinhöfel
Affiliations:
CCRCB, Queen's University Belfast, Belfast, Northern Ireland, UK;Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia;Department of Computer Science, King's College London, London, England, UK;Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia;T.E.I. of Western Macedonia, Kastoria, Greece;(Correspd.) Department of Computer Science, King's College London, London, England, UK
Venue:
Fundamenta Informaticae - Hardest Boolean Functions and O.B. Lupanov
Year:
2010

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Abstract

The paper aims at tight upper bounds on the size of pattern classification circuits that can be used for a priori parameter settings in a machine learning context. The upper bounds relate the circuit size S(C) to nL := [log2mL], where mL is the number of training samples. In particular, we show that there exist unbounded fan-in threshold circuits with less than (a) SRcc := 2·√2nL + 3 gates for unbounded depth, (b) SLcc := 34.8 · √2nL + 14 · nL − 11 · log2nL + 2 gates for small bounded depth, where in both cases all mL samples are classified correctly. We note that the upper bounds do not depend on the length n of input (sample) vectors. Since nL Rcc or [0.07 · SLcc] gates are sufficient to achieve a high generalization rate of bounded-depth classification circuits.