Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Minimax Regret of Sequential Probability Assignment via Square-Root Entropy

Zeyu Jia, Yury Polyanskiy, Alexander Rakhlin

TL;DR

The objective is to analyze the minimax regret in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions, and shows that the minimax regret can be upper bounded in terms of sequential square-root entropy.

Abstract

We study the problem of sequential probability assignment under logarithmic loss, both with and without side information. Our objective is to analyze the minimax regret -- a notion extensively studied in the literature -- in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions. We show that the minimax regret for the case of no side information (equivalently, the Shtarkov sum) can be upper bounded in terms of sequential square-root entropy, a notion closely related to Hellinger distance. For the problem of sequential probability assignment with side information, we develop both upper and lower bounds based on the aforementioned entropy. The lower bound matches the upper bound, up to log factors, for classes in the Donsker regime (according to our definition of entropy).

On the Minimax Regret of Sequential Probability Assignment via Square-Root Entropy

TL;DR

The objective is to analyze the minimax regret in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions, and shows that the minimax regret can be upper bounded in terms of sequential square-root entropy.

Abstract

We study the problem of sequential probability assignment under logarithmic loss, both with and without side information. Our objective is to analyze the minimax regret -- a notion extensively studied in the literature -- in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions. We show that the minimax regret for the case of no side information (equivalently, the Shtarkov sum) can be upper bounded in terms of sequential square-root entropy, a notion closely related to Hellinger distance. For the problem of sequential probability assignment with side information, we develop both upper and lower bounds based on the aforementioned entropy. The lower bound matches the upper bound, up to log factors, for classes in the Donsker regime (according to our definition of entropy).

Paper Structure

This paper contains 44 sections, 31 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Towards a General Result
  3. Summary of Main Results
  4. An upper bound on minimax regret for the non-contextual case:
  5. Tight characterization of contextual sequential probability assignment:
  6. Notation
  7. Organization
  8. Upper Bound for Shtarkov Sum through Sequential Square-Root Covering
  9. Comparison with Previous Results
  10. Binary Contextual Sequential Probability Assignment
  11. Upper Bound with Sequential Square-Root Entropy
  12. Comparison with Previous Upper Bounds
  13. Lower Bound with Sequential Square-Root Entropy
  14. Examples
  15. Hilbert Ball
  16. ...and 29 more sections

Key Result

Theorem 1

For any $n\ge 7$ and class $\mathcal{Q}\subseteq \Delta(\mathcal{Y}^n)$, we have where $\tilde{\mathcal{O}}$ hides constants and logarithmic factors of $n$ and $|\mathcal{Y}|$.

Theorems & Definitions (68)

  • Definition 1: sequential square-root cover and entropy
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Definition 2: sequential square-root cover and entropy
  • Theorem 2
  • Corollary 2
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • ...and 58 more