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 $\epsilon$ 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 $1/\epsilon$. We also provide lower bounds showing that a polynomial dependence on these parameters is necessary. Our second result is that within $C/\epsilon$ iterations, AdaBoost achieves a value of the exponential loss that is at most $\epsilon$ more than the best possible value, where $C$ depends on the dataset. We show that this dependence of the rate on $\epsilon$ is optimal up to constant factors, i.e., at least $\Omega(1/\epsilon)$ rounds are necessary to achieve within $\epsilon$ of the optimal exponential loss.Comment: A preliminary version will appear in COLT 201

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 the exponential loss of AdaBoost's computed parameter vector will be at most ε more than that of any parameter vector of ℓ[subscript 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 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 data set. We show that this dependence of the rate on ε is optimal up to constant factors, that is, at least Ω(1/ε) rounds are necessary to achieve within ε of the optimal exponential loss.National Science Foundation (U.S.) (Grant IIS-1016029)National Science Foundation (U.S.) (Grant IIS-1053407

Parallel coordinate descent for the Adaboost problem

We design a randomised parallel version of Adaboost based on previous studies on parallel coordinate descent. The algorithm uses the fact that the logarithm of the exponential loss is a function with coordinate-wise Lipschitz continuous gradient, in order to define the step lengths. We provide the proof of convergence for this randomised Adaboost algorithm and a theoretical parallelisation speedup factor. We finally provide numerical examples on learning problems of various sizes that show that the algorithm is competitive with concurrent approaches, especially for large scale problems.Comment: 7 pages, 3 figures, extended version of the paper presented to ICMLA'1