The Rate of Convergence of AdaBoost

The AdaBoost algorithm was designed to combine many "weak" hypotheses that perform slightly better than random guessing into a "strong" hypothesis that has very low error. We study the rate at which AdaBoost iteratively converges to the minimum of the "exponential loss." Unlike previous work, our proofs do not require a weak-learning assumption, nor do they require that minimizers of the exponential loss are finite. Our first result shows that at iteration $t$, the exponential loss of AdaBoost's computed parameter vector will be at most $ε$ more than that of any parameter vector of $\ell_1$-norm bounded by $B$ in a number of rounds that is at most a polynomial in $B$ and $1/ε$. We also provide lower bounds showing that a polynomial dependence on these parameters is necessary. Our second result is that within $C/ε$ iterations, AdaBoost achieves a value of the exponential loss that is at most $ε$ more than the best possible value, where $C$ depends on the dataset. We show that this dependence of the rate on $ε$ is optimal up to constant factors, i.e., at least $Ω(1/ε)$ rounds are necessary to achieve within $ε$ of the optimal exponential loss.

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