RatePreserving Reductions for Blackwell Approachability
Abstract
Abernethy et al. (2011) showed that Blackwell approachability and noregret learning are equivalent, in the sense that any algorithm that solves a specific Blackwell approachability instance can be converted to a sublinear regret algorithm for a specific noregret learning instance, and vice versa. In this paper, we study a more finegrained form of such reductions, and ask when this translation between problems preserves not only a sublinear rate of convergence, but also preserves the optimal rate of convergence. That is, in which cases does it suffice to find the optimal regret bound for a noregret learning instance in order to find the optimal rate of convergence for a corresponding approachability instance? We show that the reduction of Abernethy et al. (2011) does not preserve rates: their reduction may reduce a $d$dimensional approachability instance $I_1$ with optimal convergence rate $R_1$ to a noregret learning instance $I_2$ with optimal regretperround of $R_2$, with $R_{2}/R_{1}$ arbitrarily large (in particular, it is possible that $R_1 = 0$ and $R_{2} > 0$). On the other hand, we show that it is possible to tightly reduce any approachability instance to an instance of a generalized form of regret minimization we call improper $\phi$regret minimization (a variant of the $\phi$regret minimization of Gordon et al. (2008) where the transformation functions may map actions outside of the action set). Finally, we characterize when linear transformations suffice to reduce improper $\phi$regret minimization problems to standard classes of regret minimization problems in a rate preserving manner. We prove that some improper $\phi$regret minimization instances cannot be reduced to either subclass of instance in this way, suggesting that approachability can capture some problems that cannot be phrased in the language of online learning.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.07585
 arXiv:
 arXiv:2406.07585
 Bibcode:
 2024arXiv240607585D
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning