## Abstract

Consider the following abstract coin tossing problem: Given a set of n coins with unknown biases, find the most biased coin using a minimal number of coin tosses. This is a common abstraction of various exploration problems in theoretical computer science and machine learning and has been studied extensively over the years. In particular, algorithms with optimal sample complexity (number of coin tosses) have been known for this problem for quite some time. Motivated by applications to processing massive datasets, we study the space complexity of solving this problem with optimal number of coin tosses in the streaming model. In this model, the coins are arriving one by one and the algorithm is only allowed to store a limited number of coins at any point - any coin not present in the memory is lost and can no longer be tossed or compared to arriving coins. Prior algorithms for the coin tossing problem with optimal sample complexity are based on iterative elimination of coins which inherently require storing all the coins, leading to memory-inefficient streaming algorithms. We remedy this state-of-affairs by presenting a series of improved streaming algorithms for this problem: we start with a simple algorithm which require storing only O(logn) coins and then iteratively refine it further and further, leading to algorithms with O(loglog(n)) memory, O((n)) memory, and finally a one that only stores a single extra coin in memory - the same exact space needed to just store the best coin throughout the stream. Furthermore, we extend our algorithms to the problem of finding the k most biased coins as well as other exploration problems such as finding top-k elements using noisy comparisons or finding an -best arm in stochastic multi-armed bandits, and obtain efficient streaming algorithms for these problems.

Original language | English (US) |
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Title of host publication | STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy |

Publisher | Association for Computing Machinery |

Pages | 1237-1250 |

Number of pages | 14 |

ISBN (Electronic) | 9781450369794 |

DOIs | |

State | Published - Jun 8 2020 |

Event | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States Duration: Jun 22 2020 → Jun 26 2020 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 |
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Country/Territory | United States |

City | Chicago |

Period | 6/22/20 → 6/26/20 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- Memory-efficient Algorithms
- Multi-Armed Bandits
- Noisy Comparison
- Pure Exploration
- Streaming Algorithms