伪可逼近性及其在遗憾极小化中的应用
Pseudonorm Approachability and Applications to Regret Minimization
摘要：
布莱克威尔著名的可接近性理论为各种学习问题提供了一个通用框架，包括后悔最小化。然而，Blackwell的证明和隐式算法使用$\ell_2$（欧几里德）距离测量可接近性。我们认为，在许多应用中，如遗憾最小化，它是更有用的研究下的其他距离度量，最常见的$\ell_\infty$度量的可接近性。但是，为$\ell_\infty$可接近性设计的算法的时间和空间复杂度取决于向量支付空间的维数，这通常是非常大的。因此，我们提出了一个框架转换高维$\ell_\infty$的可接近性问题，低维的可接近性问题，从而解决这些问题。我们首先表明，平均支付和可接近集之间的$\ell_\infty$ -距离可以等价地定义为一个低维平均向量支付和一个新的凸集之间的伪距离，我们定义。接下来，我们开发了伪范可逼近性的算法理论，类似于之前关于$\ell_2$和其他规范的可逼近性的工作，表明它可以通过单位伪范球的Fenchel对偶给出的凸集上的在线线性优化（OLO）来实现。然后，我们用它来显示，模温和的规范化假设，存在一个$\ell_\infty$ -可逼近算法，其收敛性是独立的原始向量支付的维度。我们进一步表明，该算法承认多项式时间的复杂性，假设原来的$\ell_\infty$距离可以有效地计算。我们还给出了一个$\ell_\infty$ -逼近算法，其收敛性是对数在这方面使用的FTRL算法与最大熵正则化。
英文摘要：
Blackwell's celebrated approachability theory provides a general framework for a variety of learning problems, including regret minimization. However, Blackwell's proof and implicit algorithm measure approachability using the $\ell_2$ (Euclidean) distance. We argue that in many applications such as regret minimization, it is more useful to study approachability under other distance metrics, most commonly the $\ell_\infty$-metric. But, the time and space complexity of the algorithms designed for $\ell_\infty$-approachability depend on the dimension of the space of the vectorial payoffs, which is often prohibitively large. Thus, we present a framework for converting high-dimensional $\ell_\infty$-approachability problems to low-dimensional pseudonorm approachability problems, thereby resolving such issues. We first show that the $\ell_\infty$-distance between the average payoff and the approachability set can be equivalently defined as a pseudodistance between a lower-dimensional average vector payoff and a new convex set we define. Next, we develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $\ell_2$ and other norms, showing that it can be achieved via online linear optimization (OLO) over a convex set given by the Fenchel dual of the unit pseudonorm ball. We then use that to show, modulo mild normalization assumptions, that there exists an $\ell_\infty$-approachability algorithm whose convergence is independent of the dimension of the original vectorial payoff. We further show that that algorithm admits a polynomial-time complexity, assuming that the original $\ell_\infty$-distance can be computed efficiently. We also give an $\ell_\infty$-approachability algorithm whose convergence is logarithmic in that dimension using an FTRL algorithm with a maximum-entropy regularizer.
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