Stochastic Sparse Attention for Memory-Bound Inference

Autoregressive decoding becomes bandwidth-limited at long contexts, as generating each token requires reading all $n_k$ key and value vectors from KV cache. We present Stochastic Additive No-mulT Attention (SANTA), a method that sparsifies value-cache access by sampling $S \ll n_k$ indices from the post-softmax distribution and aggregates only those value rows. This yields an unbiased estimator of the post-softmax value aggregation while replacing value-stage multiply-accumulates with gather-and-add. We introduce stratified sampling to design variance-reduced, GPU-friendly variants, demonstrating $1.5\times$ decode-step attention kernel speedup over FlashInfer and FlashDecoding on an NVIDIA RTX 6000 Ada while matching baseline accuracy at 32k-token contexts. Finally, we propose Bernoulli $qK^\mathsf{T}$ sampling as a complementary technique to sparsify the score stage, reducing key-feature access through stochastic ternary queries. Both methods are orthogonal to upstream techniques such as ternary quantization, low-rank projections, and KV-cache compression. Together, they point toward sparse, multiplier-free, and energy-efficient inference. We open-source our kernels at: https://github.com/OPUSLab/SANTA.git

Learning to Correct: Calibrated Reinforcement Learning for Multi-Attempt Chain-of-Thought

State-of-the-art reasoning models utilize long chain-of-thought (CoT) to solve increasingly complex problems using more test-time computation. In this work, we explore a long CoT setting where the model makes up to K successive attempts at solving a problem, in which each attempt is allowed to build on earlier ones after the model receives a hard verifier feedback. This motivates RL methods that can harness per-attempt rewards by carefully weighting individual attempts. We study optimizing the Verification@K reward (the model succeeds by the K-th attempt) and show that naively weighing the attempts by their pass/fail results in biased gradients. We introduce Calibrated Attempt-Level (CAL) GRPO by devising a weighing strategy to obtain unbiased gradients while maintaining small variance. Our theory reveals how incorporating per-attempt rewards influence the training and the eventual Verification@K performance. Experiments, baselines, and ablations on synthetic and real data corroborate our theory and the benefits of CAL-GRPO over vanilla GRPO as well as naive weighting.

Enabling Intrinsic Reasoning over Dense Geospatial Embeddings with DFR-Gemma

Representation learning for geospatial and spatio-temporal data plays a critical role in enabling general-purpose geospatial intelligence. Recent geospatial foundation models, such as the Population Dynamics Foundation Model (PDFM), encode complex population and mobility dynamics into compact embeddings. However, their integration with Large Language Models (LLMs) remains limited. Existing approaches to LLM integration treat these embeddings as retrieval indices or convert them into textual descriptions for reasoning, introducing redundancy, token inefficiency, and numerical inaccuracies. We propose Direct Feature Reasoning-Gemma (DFR-Gemma), a novel framework that enables LLMs to reason directly over dense geospatial embeddings. DFR aligns high-dimensional embeddings with the latent space of an LLM via a lightweight projector, allowing embeddings to be injected as semantic tokens alongside natural language instructions. This design eliminates the need for intermediate textual representations and enables intrinsic reasoning over spatial features. To evaluate this paradigm, we introduce a multi-task geospatial benchmark that pairs embeddings with diverse question-answer tasks, including feature querying, comparison, and semantic description. Experimental results show that DFR allows LLMs to decode latent spatial patterns and perform accurate zero-shot reasoning across tasks, while significantly improving efficiency compared to text-based baselines. Our results demonstrate that treating embeddings as primary data inputs, provides a more direct, efficient, and scalable approach to multimodal geospatial intelligence.

Learning to Bet for Horizon-Aware Anytime-Valid Testing

We develop horizon-aware anytime-valid tests and confidence sequences for bounded means under a strict deadline $N$. Using the betting/e-process framework, we cast horizon-aware betting as a finite-horizon optimal control problem with state space $(t, \log W_t)$, where $t$ is the time and $W_t$ is the test martingale value. We first show that in certain interior regions of the state space, policies that deviate significantly from Kelly betting are provably suboptimal, while Kelly betting reaches the threshold with high probability. We then identify sufficient conditions showing that outside this region, more aggressive betting than Kelly can be better if the bettor is behind schedule, and less aggressive can be better if the bettor is ahead. Taken together these results suggest a simple phase diagram in the $(t, \log W_t)$ plane, delineating regions where Kelly, fractional Kelly, and aggressive betting may be preferable. Guided by this phase diagram, we introduce a Deep Reinforcement Learning approach based on a universal Deep Q-Network (DQN) agent that learns a single policy from synthetic experience and maps simple statistics of past observations to bets across horizons and null values. In limited-horizon experiments, the learned DQN policy yields state-of-the-art results.

Covariance-Aware Transformers for Quadratic Programming and Decision Making

We explore the use of transformers for solving quadratic programs and how this capability benefits decision-making problems that involve covariance matrices. We first show that the linear attention mechanism can provably solve unconstrained QPs by tokenizing the matrix variables (e.g.~$A$ of the objective $\frac{1}{2}x^\top Ax+b^\top x$) row-by-row and emulating gradient descent iterations. Furthermore, by incorporating MLPs, a transformer block can solve (i) $\ell_1$-penalized QPs by emulating iterative soft-thresholding and (ii) $\ell_1$-constrained QPs when equipped with an additional feedback loop. Our theory motivates us to introduce Time2Decide: a generic method that enhances a time series foundation model (TSFM) by explicitly feeding the covariance matrix between the variates. We empirically find that Time2Decide uniformly outperforms the base TSFM model for the classical portfolio optimization problem that admits an $\ell_1$-constrained QP formulation. Remarkably, Time2Decide also outperforms the classical "Predict-then-Optimize (PtO)" procedure, where we first forecast the returns and then explicitly solve a constrained QP, in suitable settings. Our results demonstrate that transformers benefit from explicit use of second-order statistics, and this can enable them to effectively solve complex decision-making problems, like portfolio construction, in one forward pass.

When and How Unlabeled Data Provably Improve In-Context Learning

Recent research shows that in-context learning (ICL) can be effective even when demonstrations have missing or incorrect labels. To shed light on this capability, we examine a canonical setting where the demonstrations are drawn according to a binary Gaussian mixture model (GMM) and a certain fraction of the demonstrations have missing labels. We provide a comprehensive theoretical study to show that: (1) The loss landscape of one-layer linear attention models recover the optimal fully-supervised estimator but completely fail to exploit unlabeled data; (2) In contrast, multilayer or looped transformers can effectively leverage unlabeled data by implicitly constructing estimators of the form $\sum_{i\ge 0} a_i (X^\top X)^iX^\top y$ with $X$ and $y$ denoting features and partially-observed labels (with missing entries set to zero). We characterize the class of polynomials that can be expressed as a function of depth and draw connections to Expectation Maximization, an iterative pseudo-labeling algorithm commonly used in semi-supervised learning. Importantly, the leading polynomial power is exponential in depth, so mild amount of depth/looping suffices. As an application of theory, we propose looping off-the-shelf tabular foundation models to enhance their semi-supervision capabilities. Extensive evaluations on real-world datasets show that our method significantly improves the semisupervised tabular learning performance over the standard single pass inference.

Extrapolation by Association: Length Generalization Transfer in Transformers

Transformer language models have demonstrated impressive generalization capabilities in natural language domains, yet we lack a fine-grained understanding of how such generalization arises. In this paper, we investigate length generalization--the ability to extrapolate from shorter to longer inputs--through the lens of \textit{task association}. We find that length generalization can be \textit{transferred} across related tasks. That is, training a model with a longer and related auxiliary task can lead it to generalize to unseen and longer inputs from some other target task. We demonstrate this length generalization transfer across diverse algorithmic tasks, including arithmetic operations, string transformations, and maze navigation. Our results show that transformer models can inherit generalization capabilities from similar tasks when trained jointly. Moreover, we observe similar transfer effects in pretrained language models, suggesting that pretraining equips models with reusable computational scaffolding that facilitates extrapolation in downstream settings. Finally, we provide initial mechanistic evidence that length generalization transfer correlates with the re-use of the same attention heads between the tasks. Together, our findings deepen our understanding of how transformers generalize to out-of-distribution inputs and highlight the compositional reuse of inductive structure across tasks.

Continuous Chain of Thought Enables Parallel Exploration and Reasoning

Current language models generate chain-of-thought traces by autoregressively sampling tokens from a finite vocabulary. While this discrete sampling has achieved remarkable success, conducting chain-of-thought with continuously-valued tokens (CoT2) offers a richer and more expressive alternative. Our work examines the benefits of CoT2 through logical reasoning tasks that inherently require search capabilities and provide optimization and exploration methods for CoT2. Theoretically, we show that CoT2 allows the model to track multiple traces in parallel and quantify its benefits for inference efficiency. Notably, one layer transformer equipped with CoT2 can provably solve the combinatorial "subset sum problem" given sufficient embedding dimension. These insights lead to a novel and effective supervision strategy where we match the softmax outputs to the empirical token distributions of a set of target traces. Complementing this, we introduce sampling strategies that unlock policy optimization and self-improvement for CoT2. Our first strategy samples and composes $K$ discrete tokens at each decoding step to control the level of parallelism, and reduces to standard CoT when $K=1$. Our second strategy relies on continuous exploration over the probability simplex. Experiments confirm that policy optimization with CoT2 indeed improves the performance of the model beyond its initial discrete or continuous supervision.

Attention with Trained Embeddings Provably Selects Important Tokens

Token embeddings play a crucial role in language modeling but, despite this practical relevance, their theoretical understanding remains limited. Our paper addresses the gap by characterizing the structure of embeddings obtained via gradient descent. Specifically, we consider a one-layer softmax attention model with a linear head for binary classification, i.e., $\texttt{Softmax}( p^\top E_X^\top ) E_X v = \frac{ \sum_{i=1}^T \exp(p^\top E_{x_i}) E_{x_i}^\top v}{\sum_{j=1}^T \exp(p^\top E_{x_{j}}) }$, where $E_X = [ E_{x_1} , \dots, E_{x_T} ]^\top$ contains the embeddings of the input sequence, $p$ is the embedding of the $\mathrm{\langle cls \rangle}$ token and $v$ the output vector. First, we show that, already after a single step of gradient training with the logistic loss, the embeddings $E_X$ capture the importance of tokens in the dataset by aligning with the output vector $v$ proportionally to the frequency with which the corresponding tokens appear in the dataset. Then, after training $p$ via gradient flow until convergence, the softmax selects the important tokens in the sentence (i.e., those that are predictive of the label), and the resulting $\mathrm{\langle cls \rangle}$ embedding maximizes the margin for such a selection. Experiments on real-world datasets (IMDB, Yelp) exhibit a phenomenology close to that unveiled by our theory.

Making Small Language Models Efficient Reasoners: Intervention, Supervision, Reinforcement

Recent research enhances language model reasoning by scaling test-time compute via longer chain-of-thought traces. This often improves accuracy but also introduces redundancy and high computational cost, especially for small language models distilled with supervised fine-tuning (SFT). In this work, we propose new algorithms to improve token-efficient reasoning with small-scale models by effectively trading off accuracy and computation. We first show that the post-SFT model fails to determine the optimal stopping point of the reasoning process, resulting in verbose and repetitive outputs. Verbosity also significantly varies across wrong vs correct responses. To address these issues, we propose two solutions: (1) Temperature scaling (TS) to control the stopping point for the thinking phase and thereby trace length, and (2) TLDR: a length-regularized reinforcement learning method based on GRPO that facilitates multi-level trace length control (e.g. short, medium, long reasoning). Experiments on four reasoning benchmarks, MATH500, AMC, AIME24 and OlympiadBench, demonstrate that TS is highly effective compared to s1's budget forcing approach and TLDR significantly improves token efficiency by about 50% with minimal to no accuracy loss over the SFT baseline. Moreover, TLDR also facilitates flexible control over the response length, offering a practical and effective solution for token-efficient reasoning in small models. Ultimately, our work reveals the importance of stopping time control, highlights shortcomings of pure SFT, and provides effective algorithmic recipes.