Generalized Hierarchical Conformal Prediction

Many prediction problems arise with data collected in groups. In this setting, hierarchical conformal prediction (HCP) (Lee et al., 2026) provides distribution-free prediction sets for a new observation from a previously unseen group under hierarchical exchangeability. In many applications, however, prediction is conducted only after a few observations from the group of interest have already been collected. Standard HCP cannot leverage these observations, as its required symmetry conditions do not hold in this setting. At the same time, the initial sample may still be too small for standard conformal prediction applied within the test group to be informative. We develop predictive inference methods for this setting. Our proposed method, Generalized HCP (GHCP), restores the relevant symmetry needed for conformal inference by assigning the test group a randomly "donated" reference group size. GHCP further leverages the initial test group observations to improve the quality of the nonconformity scores for prediction within that group. To improve efficiency, we introduce a variant that restricts the set of eligible donors. We demonstrate the performance of the proposed method through simulations and an illustration on the American Community Survey dataset.

Statistical Early Stopping for Reasoning Models

While LLMs have seen substantial improvement in reasoning capabilities, they also sometimes overthink, generating unnecessary reasoning steps, particularly under uncertainty, given ill-posed or ambiguous queries. We introduce statistically principled early stopping methods that monitor uncertainty signals during generation to mitigate this issue. Our first approach is parametric: it models inter-arrival times of uncertainty keywords as a renewal process and applies sequential testing for stopping. Our second approach is nonparametric and provides finite-sample guarantees on the probability of halting too early on well-posed queries. We conduct empirical evaluations on reasoning tasks across several domains and models. Our results indicate that uncertainty-aware early stopping can improve both efficiency and reliability in LLM reasoning, and we observe especially significant gains for math reasoning.

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 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).