## Abstract

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 `_{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 `_{∞}-metric. But, the time and space complexity of the algorithms designed for `_{∞}-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 `_{∞}-approachability problems to low-dimensional pseudonorm approachability problems, thereby resolving such issues. We first show that the `_{∞}-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 `_{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 `_{∞}-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 `_{∞}-distance can be computed efficiently. We also give an `_{∞}-approachability algorithm whose convergence is logarithmic in that dimension using an FTRL algorithm with a maximum-entropy regularizer. Finally, we illustrate the benefits of our framework by applying it to several problems in regret minimization.

Original language | English (US) |
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Pages (from-to) | 471-509 |

Number of pages | 39 |

Journal | Proceedings of Machine Learning Research |

Volume | 201 |

State | Published - 2023 |

Event | 34th International Conference onAlgorithmic Learning Theory, ALT 2023 - Singapore, Singapore Duration: Feb 20 2023 → Feb 23 2023 |

## Keywords

- Blackwell’s approachability
- regret minimization
- swap regret

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability