Foundations of Top-$k$ Decoding for Language Models

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 introduce *Bregman decoders* obtained by minimizing a separable Bregman divergence (for both the *primal* and *dual* cases) with a sparsity-inducing $\ell_0$-regularization; in particular, these decoders are *adaptive* in the sense that the sparsity parameter $k$ is chosen depending on the underlying token distribution. Despite the combinatorial nature of the sparse Bregman objective, we show how to optimize it efficiently for a large class of divergences. We prove that (i) the optimal decoding strategies are greedy, and further that (ii) the objective is discretely convex in $k$, such that the optimal $k$ can be identified in logarithmic time. We note that standard top-$k$ decoding arises as a special case for the KL divergence, and construct new decoding strategies with substantially different behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).