Pareto-optimal Non-uniform Language Generation

TL;DR

The paper advances the theory of language generation in the limit by introducing Pareto-optimal non-uniform generation, proving that a canonical Pareto-optimal sequence of generation times exists for any countable language collection. It provides an insertion-sort-like construction to compute this sequence and an almost Pareto-optimal algorithm that matches it on large prefixes; it also extends the framework to practical settings with noise and representative generation. Additionally, it establishes sufficient conditions for exact Pareto-optimality and presents impossibility results illustrating limits of the approach. Altogether, the work offers a principled, geometrically motivated framework for balancing per-language generation effort against overall efficiency, with concrete algorithms for noisy and representative variants and guidance on when exact Pareto-optimality can be achieved.

Abstract

Kleinberg and Mullainathan (2024) recently proposed an interesting model for language generation in the limit: Given a countable collection of languages, and an adversary enumerating the strings of some language $L$ from the collection, the objective is to generate new strings from the target language, such that all strings generated beyond some finite time are valid. Li, Raman and Tewari (2024) and Charikar and Pabbaraju (2024) showed strong non-uniform generation guarantees in this model, giving algorithms that generate new valid strings from $L$ after seeing a number of distinct input strings $t(L)$ that depends only on $L$ (and the collection), but not the enumeration order. However, for both these works, the language-wise generation times $t(L)$ of the algorithm can be strictly sub-optimal. In this work, we study Pareto-optimality of non-uniform language generation in the limit. We propose an algorithm, whose generation times $t^\star(L)$ are (almost) Pareto-optimal: any other algorithm whose generation time for some language $L$ is strictly smaller than $t^\star(L)$, must satisfy that its generation time for some other language $L'$ is strictly worse than $t^\star(L')$. Pareto-optimality is essentially the best that one can achieve for non-uniform generation. Our algorithmic framework conveniently adapts to further give Pareto-optimal non-uniform generation algorithms in the practically motivated settings of noisy as well as representative generation.

Figures (4)

• Figure 1: Insertion Sort for Non-uniform Generation
• Figure 1: Diagonal traversal over languages arranged in a grid. The $i^{th}$ column entirely consists of $L_i$. When we arrive at a copy of $L_i$ in the $n^{\text{th}}$ row (i.e., at $L_{n,i}$), we compute $m^\star_n(L_i)$.
• Figure 2: Insertion Sort for Noisy Non-uniform Generation
• Figure 3: Insertion Sort for Representative Non-uniform Generation

Theorems & Definitions (47)

• Theorem 1: Almost Pareto-optimal Non-Uniform Language Generation
• Definition 1: Generation in the Limit kleinberg2024language
• Definition 2: Non-uniform Generation li2024generation
• Definition 3: Pareto-optimality
• Example 1
• proof
• Claim 3.1: $m^\star(\cdot)$ forms a Pareto-optimal sequence
• proof
• Claim 3.2: Arg max Maintained
• proof