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Prediction with eventual almost sure guarantees

Changlong Wu, Narayana Santhanam

TL;DR

A general framework that models such predication problems as well as some general properties of the framework are introduced, and some concrete examples with the application of such properties are shown.

Abstract

We study the problem of sequentially predicting properties of a probabilistic model and its next outcome over an infinite horizon, with the goal of ensuring that the predictions incur only finitely many errors with probability 1. We introduce a general framework that models such prediction problems, provide general characterizations for the existence of successful prediction rules, and demonstrate the application of these characterizations through several concrete problem setups, including hypothesis testing, online learning, and risk domination. In particular, our characterizations allow us to recover the findings of Dembo and Peres (1994) with simple and elementary proofs, and offer a partial resolution to an open problem posed therein.

Prediction with eventual almost sure guarantees

TL;DR

A general framework that models such predication problems as well as some general properties of the framework are introduced, and some concrete examples with the application of such properties are shown.

Abstract

We study the problem of sequentially predicting properties of a probabilistic model and its next outcome over an infinite horizon, with the goal of ensuring that the predictions incur only finitely many errors with probability 1. We introduce a general framework that models such prediction problems, provide general characterizations for the existence of successful prediction rules, and demonstrate the application of these characterizations through several concrete problem setups, including hypothesis testing, online learning, and risk domination. In particular, our characterizations allow us to recover the findings of Dembo and Peres (1994) with simple and elementary proofs, and offer a partial resolution to an open problem posed therein.

Paper Structure

This paper contains 28 sections, 35 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Related work
  4. Problem setup
  5. Characterization of $e.a.s.$-predictability
  6. Specialized settings
  7. Supervised setting
  8. Unsupervised setting
  9. Capturing the final error
  10. Unions of learnable classes
  11. Characterization of $e.a.s.$-learnability
  12. Variations
  13. Weakly $e.a.s.$-predictable
  14. Prediction with finite expected loss
  15. Applications
  16. ...and 13 more sections

Key Result

lemma 1

Let $\mathcal{P}$ be a collection of models, $\{\mathcal{P}_i,i\ge 1\}$ be a nesting of $\mathcal{P}$. If for all $\eta>0$ and $i\in \mathbb{N}^+$, $(\mathcal{P}_i,\ell)$ is $\eta$-predictable. Then $(\mathcal{P},\ell)$ is $e.a.s.$-predictable.

Theorems & Definitions (87)

  • definition 1: $\eta$-predictability
  • definition 2: $e.a.s.$-predictable
  • definition 3
  • lemma 1
  • proof
  • definition 4: Universal nestings
  • definition 5: $\eta$-nestings
  • lemma 2
  • proof
  • theorem 1
  • ...and 77 more