The IMP game: Learnability, approximability and adversarial learning beyond $\Sigma^0_1$

TL;DR

This paper introduces the IMP game framework to analyze learnability and approximability of complex language classes, revealing that certain classes are learnable or not in adversarial and non-adversarial settings, with implications for computational learning theory.

Contribution

It presents a novel game-based framework for studying learnability and approximability beyond basic classes, providing new theorems on what can be learned or approximated in adversarial contexts.

Findings

All of $oldsymbol{ ext{Sigma}^0_1 ext{ and } ext{Pi}^0_1}$ can be learned by example.

In adversarial learning, $oldsymbol{ ext{Sigma}^0_1}$ can be learned, but $oldsymbol{ ext{Pi}^0_1}$ cannot.

Some $oldsymbol{ ext{Pi}^0_1}$ languages cannot be approximated by any $oldsymbol{ ext{Sigma}^0_1}$ language.

Abstract

We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e., non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (1) It is possible to learn by example all of $\Sigma^0_1 \cup \Pi^0_1$ as well as some supersets; (2) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, $\Sigma^0_1$ can be learned adversarially, but $\Pi^0_1$ not); (3) some languages in $\Pi^0_1$ cannot be approximated by any language in $\Sigma^0_1$. We show corresponding results also for $\Sigma^0_i$ and $\Pi^0_i$ for arbitrary $i$.