Foundations of Top-$k$ Decoding For Language Models

TL;DR

This paper provides a theoretical foundation for top-$k$ decoding in language models, framing it as sparse probability recovery via Bregman divergence minimization, and introduces new decoding strategies beyond traditional top-$k$ methods.

Contribution

It develops a unified theoretical framework for top-$k$ decoding using Bregman divergence minimization, generalizes the method, and proposes novel decoding strategies.

Findings

Top-$k$ decoding is a special case of Bregman divergence-based recovery.

Optimal decoding strategies are greedy and can be efficiently found via binary search.

New decoding methods can non-linearly re-weight probabilities, offering alternative sampling behaviors.

Abstract

Top-$k$ decoding is a widely used method for sampling from LLMs: at each token, only the largest $k$ next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-$k$ and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-$k$ decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-$k$ decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We consider \emph{Bregman decoders} obtained by minimizing a separable Bregman divergence (for both the \emph{primal} and \emph{dual} cases) with a sparsity-inducing $\ell_0$ regularization. Despite the combinatorial nature of the objective, we show how to optimize it efficiently for a large class of divergences. We show that the optimal decoding strategies are greedy, and further that the loss function is discretely convex in $k$, so that binary search provably and efficiently finds the optimal $k$. We show that top-$k$ decoding arises as a special case for the KL divergence, and identify new decoding strategies that have distinct behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).