保速率归约用于Blackwell可逼近性
Rate-Preserving Reductions for Blackwell Approachability
AI总结：
本文研究Blackwell可逼近性与无悔学习之间归约的保速率性质，证明现有归约不保持最优速率，并提出不当φ-遗憾最小化作为紧密归约框架，揭示可逼近性可表达超出在线学习范畴的问题。
摘要：
Abernethy等人（2011）表明，Blackwell可逼近性与无悔学习是等价的，即任何解决特定Blackwell可逼近性实例的算法都可以转化为解决特定无悔学习实例的亚线性遗憾算法，反之亦然。本文研究这种归约的更细粒度形式，并探讨问题之间的这种转换何时不仅保持亚线性收敛速率，还保持最优收敛速率。也就是说，在哪些情况下，只需找到无悔学习实例的最优遗憾界，就能找到相应可逼近性实例的最优收敛速率？我们证明Abernethy等人（2011）的归约不保持速率：他们的归约可能将一个具有最优收敛速率R1的d维可逼近性实例I1归约为一个具有最优每轮遗憾R2的无悔学习实例I2，且R2/R1可以任意大（特别是，可能R1=0而R2>0）。另一方面，我们证明可以将任何可逼近性实例紧密归约为一种广义形式的遗憾最小化实例，我们称之为不当φ-遗憾最小化（Gordon等人（2008）提出的φ-遗憾最小化的一种变体，其中变换函数可以将动作映射到动作集之外）。最后，我们刻画了线性变换何时足以将不当φ-遗憾最小化问题以保速率的方式归约为标准类别的遗憾最小化问题。我们证明某些不当φ-遗憾最小化实例无法以这种方式归约为任一子类实例，这表明可逼近性可以捕捉一些无法用在线学习语言表述的问题。
英文摘要：
Abernethy et al. (2011) showed that Blackwell approachability and no-regret 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 no-regret learning instance, and vice versa. In this paper, we study a more fine-grained 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 no-regret 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 no-regret learning instance $I_2$ with optimal regret-per-round 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 $ϕ$-regret minimization (a variant of the $ϕ$-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 $ϕ$-regret minimization problems to standard classes of regret minimization problems in a rate preserving manner. We prove that some improper $ϕ$-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.
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