Boosting with Tempered Exponential Measures

Richard Nock, Ehsan Amid, Manfred K Warmuth

Keywords: Boosting, optimization, exponential families

TL;DR: The recently introduced Tempered Exponential Measures allow to generalise AdaBoost "above" its classical entropic optimisation framework -- while keeping exponential convergence

Abstract:

One of the most popular ML algorithms, AdaBoost, can be
derived from the dual of a relative entropy
minimization problem subject to the fact that the positive weights
on the examples sum to one. Essentially, harder examples receive higher probabilities. We generalize this setup to the recently introduced *tempered
exponential measure*s (TEMs) where normalization is enforced on a specific power of the measure and not the measure itself.
TEMs are indexed by a parameter $t$ and generalize exponential families ($t=1$). Our algorithm, $t$-AdaBoost, recovers AdaBoost as a special case ($t=1$). We show that $t$-AdaBoost retains AdaBoost's celebrated exponential convergence rate when $t\in [0,1)$ while allowing a slight improvement of the rate's hidden constant compared to $t=1$. $t$-AdaBoost partially computes on a generalization of classical arithmetic over the reals and brings notable properties like guaranteed bounded leveraging coefficients for $t\in [0,1)$. From the loss that $t$-AdaBoost minimizes (a generalization of the exponential loss), we show how to derive a new family of *tempered* losses for the induction of domain-partitioning classifiers like decision trees. Crucially, strict properness is ensured for all while their boosting rates span the full known spectrum. Experiments using $t$-AdaBoost+trees display that significant leverage can be achieved by tuning $t$.

Submission Number: 8766