arXiv:2406.07585v1 Announce Kind: new
Summary: Abernethy et al. (2011) confirmed that Blackwell approachability and no-regret studying are equal, within the sense that any algorithm that solves a particular Blackwell approachability occasion will be transformed to a sublinear remorse algorithm for a particular no-regret studying occasion, and vice versa. On this paper, we research a extra fine-grained type of such reductions, and ask when this translation between issues preserves not solely a sublinear charge of convergence, but additionally preserves the optimum charge of convergence. That’s, during which instances does it suffice to search out the optimum remorse certain for a no-regret studying occasion with a view to discover the optimum charge of convergence for a corresponding approachability occasion?
We present that the discount of Abernethy et al. (2011) doesn’t protect charges: their discount could cut back a $d$-dimensional approachability occasion $I_1$ with optimum convergence charge $R_1$ to a no-regret studying occasion $I_2$ with optimum regret-per-round of $R_2$, with $R_{2}/R_{1}$ arbitrarily massive (particularly, it’s doable that $R_1 = 0$ and $R_{2} > 0$). Then again, we present that it’s doable to tightly cut back any approachability occasion to an occasion of a generalized type of remorse minimization we name improper $phi$-regret minimization (a variant of the $phi$-regret minimization of Gordon et al. (2008) the place the transformation capabilities could map actions exterior of the motion set).
Lastly, we characterize when linear transformations suffice to scale back improper $phi$-regret minimization issues to plain lessons of remorse minimization issues in a charge preserving method. We show that some improper $phi$-regret minimization situations can’t be diminished to both subclass of occasion on this manner, suggesting that approachability can seize some issues that can’t be phrased within the language of on-line studying.
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