## Abstract

In this paper we consider the problem of estimating a Bernoulli parameter using finite memory. Let X_{1}, X_{2}, . . . be a sequence of independent identically distributed Bernoulli random variables with expectation θ, where θ ∈ [0, 1]. Consider a finite-memory deterministic machine with S states, that updates its state M_{n} ∈ {1, 2, . . ., S} at each time according to the rule M_{n} = f(M_{n-1}, X_{n}), where f is a deterministic time-invariant function. Assume that the machine outputs an estimate at each time point according to some fixed mapping from the state space to the unit interval. The quality of the estimation procedure is measured by the asymptotic risk, which is the long-term average of the instantaneous quadratic risk. The main contribution of this paper is an upper bound on the smallest worst-case asymptotic risk any such machine can attain. This bound coincides with a lower bound derived by Leighton and Rivest, to imply that Θ(1/S) is the minimax asymptotic risk for deterministic S-state machines. In particular, our result disproves a longstanding Θ(log S/S) conjecture for this quantity, also posed by Leighton and Rivest.

Original language | American English |
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Pages (from-to) | 566-585 |

Number of pages | 20 |

Journal | Proceedings of Machine Learning Research |

Volume | 134 |

State | Published - 2021 |

Event | 34th Conference on Learning Theory, COLT 2021 - Boulder, United States Duration: 15 Aug 2021 → 19 Aug 2021 |

### Bibliographical note

Funding Information:This work was supported by the ISF under Grants 1791/17 and 1495/18.

Publisher Copyright:

© 2021 T. Berg, O. Ordentlich & O. Shayevitz.

## Keywords

- Learning with Memory Constraints
- Minimax Estimation
- Parametric Estimation