AdaBoost Does Not Always Cycle: A Computer-Assisted Counterexample

TL;DR

This paper presents a computer-assisted counterexample demonstrating that AdaBoost does not always converge to a finite cycle, challenging previous assumptions about its convergence behavior.

Contribution

It provides the first explicit counterexample to the conjecture that AdaBoost always cycles finitely, using a novel construction verified by exact rational arithmetic.

Findings

Counterexample shows AdaBoost can have non-periodic convergence behavior

The construction involves a block-product gadget with irrational eigenvalue ratios

All assertions are certified by exact rational arithmetic

Abstract

We give a computer-assisted counterexample to the open question, posed by Rudin, Schapire, and Daubechies in COLT 2012, of whether exhaustive AdaBoost always converges to a finite cycle. The construction is based on a block-product gadget whose two factors share an exact period-2 orbit for their 5-step branch maps, but whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio. This irrationality forces the burst-winner sequence to have an irrational asymptotic frequency, precluding eventual periodicity. All assertions are certified by exact rational arithmetic. This work was developed in collaboration with GPT-5.4 Pro and Claude Opus 4.6.