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Decoding Game: On Minimax Optimality of Heuristic Text Generation Strategies

Sijin Chen, Omar Hagrass, Jason M. Klusowski

TL;DR

Decoding strategies for open-ended text generation exhibit a gap between MAP-like likelihood maximization and practical, diverse outputs. The authors formulate Decoding Game, a two-player zero-sum framework where Strategist seeks high true-distribution log-likelihood and Nature adversarially perturbs the distribution within a TV budget around the model, yielding a minimax objective $\max_{\mathbb{Q}}\min_{\mathbb{P}\in N(\widehat{\mathbb{P}})} \mathbb{E}_{\mathbb{Q}}[\log\mathbb{P}(X_1,\dots,X_T|X_0)]$. They derive that Nature induces an $\ell_\infty$-type regularization on the log-likelihood and show tail truncation-normalization as a first-order optimal strategy (with precise per-step conditions and thresholds), while also generalizing to other objective functions that recover temperature-based methods. The framework unifies and justifies popular heuristic sampling strategies and introduces Game sampling, which demonstrates competitive or superior open-ended generation performance across several models. Overall, the work provides a rigorous, minimal-assumption theory for decoding strategy design with practical, testable implications for robust text generation.

Abstract

Decoding strategies play a pivotal role in text generation for modern language models, yet a puzzling gap divides theory and practice. Surprisingly, strategies that should intuitively be optimal, such as Maximum a Posteriori (MAP), often perform poorly in practice. Meanwhile, popular heuristic approaches like Top-$k$ and Nucleus sampling, which employ truncation and normalization of the conditional next-token probabilities, have achieved great empirical success but lack theoretical justifications. In this paper, we propose Decoding Game, a comprehensive theoretical framework which reimagines text generation as a two-player zero-sum game between Strategist, who seeks to produce text credible in the true distribution, and Nature, who distorts the true distribution adversarially. After discussing the decomposibility of multi-step generation, we derive the optimal strategy in closed form for one-step Decoding Game. It is shown that the adversarial Nature imposes an implicit regularization on likelihood maximization, and truncation-normalization methods are first-order approximations to the optimal strategy under this regularization. Additionally, by generalizing the objective and parameters of Decoding Game, near-optimal strategies encompass diverse methods such as greedy search, temperature scaling, and hybrids thereof. Numerical experiments are conducted to complement our theoretical analysis.

Decoding Game: On Minimax Optimality of Heuristic Text Generation Strategies

TL;DR

Decoding strategies for open-ended text generation exhibit a gap between MAP-like likelihood maximization and practical, diverse outputs. The authors formulate Decoding Game, a two-player zero-sum framework where Strategist seeks high true-distribution log-likelihood and Nature adversarially perturbs the distribution within a TV budget around the model, yielding a minimax objective . They derive that Nature induces an -type regularization on the log-likelihood and show tail truncation-normalization as a first-order optimal strategy (with precise per-step conditions and thresholds), while also generalizing to other objective functions that recover temperature-based methods. The framework unifies and justifies popular heuristic sampling strategies and introduces Game sampling, which demonstrates competitive or superior open-ended generation performance across several models. Overall, the work provides a rigorous, minimal-assumption theory for decoding strategy design with practical, testable implications for robust text generation.

Abstract

Decoding strategies play a pivotal role in text generation for modern language models, yet a puzzling gap divides theory and practice. Surprisingly, strategies that should intuitively be optimal, such as Maximum a Posteriori (MAP), often perform poorly in practice. Meanwhile, popular heuristic approaches like Top- and Nucleus sampling, which employ truncation and normalization of the conditional next-token probabilities, have achieved great empirical success but lack theoretical justifications. In this paper, we propose Decoding Game, a comprehensive theoretical framework which reimagines text generation as a two-player zero-sum game between Strategist, who seeks to produce text credible in the true distribution, and Nature, who distorts the true distribution adversarially. After discussing the decomposibility of multi-step generation, we derive the optimal strategy in closed form for one-step Decoding Game. It is shown that the adversarial Nature imposes an implicit regularization on likelihood maximization, and truncation-normalization methods are first-order approximations to the optimal strategy under this regularization. Additionally, by generalizing the objective and parameters of Decoding Game, near-optimal strategies encompass diverse methods such as greedy search, temperature scaling, and hybrids thereof. Numerical experiments are conducted to complement our theoretical analysis.
Paper Structure (21 sections, 6 theorems, 50 equations, 1 figure, 15 tables, 1 algorithm)

This paper contains 21 sections, 6 theorems, 50 equations, 1 figure, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and our framework
  3. Contribution
  4. Related works
  5. Existing theoretical interpretations
  6. Text generation as decision making
  7. Robust optimization and regularization
  8. Formulation
  9. Notations
  10. Decoding Game
  11. Reduction from multiple steps
  12. Theoretical analysis
  13. $p$-strategy: implicit regularization
  14. $q$-strategy: heuristic sampling methods
  15. Generalization from log-likelihood
  16. ...and 6 more sections

Key Result

Proposition 3.3

Given arbitrary $\widehat{\mathbb{P}}$ from the space of probability measures on $\mathcal{V}^{T}$, let $\mathbb{Q}=\mathbb{Q}(\widehat{\mathbb{P}})$ be any strategy with no foresight. Moreover, let $\mathbb{P}^*=\mathbb{P}^*(\widehat{\mathbb{P}},\mathbb{Q})$ be the optimal strategy of player N agai

Figures (1)

  • Figure 1: Next-token probability distribution in GPT-2 XL model and truncation threshold of Game sampling and Nucleus sampling.

Theorems & Definitions (7)

  • Definition 3.1
  • Proposition 3.3
  • Theorem 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Theorem 4.7
  • Corollary 4.8