AdaBoost Does Not Always Cycle: A Computer-Assisted Counterexample
This resolves a long-standing theoretical problem in machine learning, specifically for researchers studying boosting algorithms and convergence properties.
The paper tackles the open question of whether exhaustive AdaBoost always converges to a finite cycle by providing a computer-assisted counterexample, demonstrating that it can exhibit non-periodic behavior with an irrational asymptotic frequency in the burst-winner sequence.
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.