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.

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.

L3GS: Layered 3D Gaussian Splats for Efficient 3D Scene Delivery

Traditional 3D content representations include dense point clouds that consume large amounts of data and hence network bandwidth, while newer representations such as neural radiance fields suffer from poor frame rates due to their non-standard volumetric rendering pipeline. 3D Gaussian splats (3DGS) can be seen as a generalization of point clouds that meet the best of both worlds, with high visual quality and efficient rendering for real-time frame rates. However, delivering 3DGS scenes from a hosting server to client devices is still challenging due to high network data consumption (e.g., 1.5 GB for a single scene). The goal of this work is to create an efficient 3D content delivery framework that allows users to view high quality 3D scenes with 3DGS as the underlying data representation. The main contributions of the paper are: (1) Creating new layered 3DGS scenes for efficient delivery, (2) Scheduling algorithms to choose what splats to download at what time, and (3) Trace-driven experiments from users wearing virtual reality headsets to evaluate the visual quality and latency. Our system for Layered 3D Gaussian Splats delivery L3GS demonstrates high visual quality, achieving 16.9% higher average SSIM compared to baselines, and also works with other compressed 3DGS representations.

Test-Time Training Provably Improves Transformers as In-context Learners

Test-time training (TTT) methods explicitly update the weights of a model to adapt to the specific test instance, and they have found success in a variety of settings, including most recently language modeling and reasoning. To demystify this success, we investigate a gradient-based TTT algorithm for in-context learning, where we train a transformer model on the in-context demonstrations provided in the test prompt. Specifically, we provide a comprehensive theoretical characterization of linear transformers when the update rule is a single gradient step. Our theory (i) delineates the role of alignment between pretraining distribution and target task, (ii) demystifies how TTT can alleviate distribution shift, and (iii) quantifies the sample complexity of TTT including how it can significantly reduce the eventual sample size required for in-context learning. As our empirical contribution, we study the benefits of TTT for TabPFN, a tabular foundation model. In line with our theory, we demonstrate that TTT significantly reduces the required sample size for tabular classification (3 to 5 times fewer) unlocking substantial inference efficiency with a negligible training cost.

Selective Attention: Enhancing Transformer through Principled Context Control

The attention mechanism within the transformer architecture enables the model to weigh and combine tokens based on their relevance to the query. While self-attention has enjoyed major success, it notably treats all queries $q$ in the same way by applying the mapping $V^\top\text{softmax}(Kq)$, where $V,K$ are the value and key embeddings respectively. In this work, we argue that this uniform treatment hinders the ability to control contextual sparsity and relevance. As a solution, we introduce the $\textit{Selective Self-Attention}$ (SSA) layer that augments the softmax nonlinearity with a principled temperature scaling strategy. By controlling temperature, SSA adapts the contextual sparsity of the attention map to the query embedding and its position in the context window. Through theory and experiments, we demonstrate that this alleviates attention dilution, aids the optimization process, and enhances the model's ability to control softmax spikiness of individual queries. We also incorporate temperature scaling for value embeddings and show that it boosts the model's ability to suppress irrelevant/noisy tokens. Notably, SSA is a lightweight method which introduces less than 0.5% new parameters through a weight-sharing strategy and can be fine-tuned on existing LLMs. Extensive empirical evaluations demonstrate that SSA-equipped models achieve a noticeable and consistent accuracy improvement on language modeling benchmarks.

On the Power of Convolution Augmented Transformer

The transformer architecture has catalyzed revolutionary advances in language modeling. However, recent architectural recipes, such as state-space models, have bridged the performance gap. Motivated by this, we examine the benefits of Convolution-Augmented Transformer (CAT) for recall, copying, and length generalization tasks. CAT incorporates convolutional filters in the K/Q/V embeddings of an attention layer. Through CAT, we show that the locality of the convolution synergizes with the global view of the attention. Unlike comparable architectures, such as Mamba or transformer, CAT can provably solve the associative recall (AR) and copying tasks using a single layer while also enjoying guaranteed length generalization. We also establish computational tradeoffs between convolution and attention by characterizing how convolution can mitigate the need for full attention by summarizing the context window and creating salient summary tokens to attend. Evaluations on real datasets corroborate our findings and demonstrate that CAT and its variations indeed enhance the language modeling performance.

TREACLE: Thrifty Reasoning via Context-Aware LLM and Prompt Selection

Recent successes in natural language processing have led to the proliferation of large language models (LLMs) by multiple providers. Each LLM offering has different inference accuracy, monetary cost, and latency, and their accuracy further depends on the exact wording of the question (i.e., the specific prompt). At the same time, users often have a limit on monetary budget and latency to answer all their questions, and they do not know which LLMs to choose for each question to meet their accuracy and long-term budget requirements. To navigate this rich design space, we propose TREACLE (Thrifty Reasoning via Context-Aware LLM and Prompt Selection), a reinforcement learning policy that jointly selects the model and prompting scheme while respecting the user's monetary cost and latency constraints. TREACLE uses the problem context, including question text embeddings (reflecting the type or difficulty of a query) and the response history (reflecting the consistency of previous responses) to make smart decisions. Our evaluations on standard reasoning datasets (GSM8K, CSQA, and LLC ) with various LLMs and prompts show that TREACLE enables cost savings of up to 85% compared to baselines while maintaining high accuracy. Importantly, it provides the user with the ability to gracefully trade off accuracy for cost.

Class-attribute Priors: Adapting Optimization to Heterogeneity and Fairness Objective

Modern classification problems exhibit heterogeneities across individual classes: Each class may have unique attributes, such as sample size, label quality, or predictability (easy vs difficult), and variable importance at test-time. Without care, these heterogeneities impede the learning process, most notably, when optimizing fairness objectives. Confirming this, under a gaussian mixture setting, we show that the optimal SVM classifier for balanced accuracy needs to be adaptive to the class attributes. This motivates us to propose CAP: An effective and general method that generates a class-specific learning strategy (e.g. hyperparameter) based on the attributes of that class. This way, optimization process better adapts to heterogeneities. CAP leads to substantial improvements over the naive approach of assigning separate hyperparameters to each class. We instantiate CAP for loss function design and post-hoc logit adjustment, with emphasis on label-imbalanced problems. We show that CAP is competitive with prior art and its flexibility unlocks clear benefits for fairness objectives beyond balanced accuracy. Finally, we evaluate CAP on problems with label noise as well as weighted test objectives to showcase how CAP can jointly adapt to different heterogeneities.

FedYolo: Augmenting Federated Learning with Pretrained Transformers

The growth and diversity of machine learning applications motivate a rethinking of learning with mobile and edge devices. How can we address diverse client goals and learn with scarce heterogeneous data? While federated learning aims to address these issues, it has challenges hindering a unified solution. Large transformer models have been shown to work across a variety of tasks achieving remarkable few-shot adaptation. This raises the question: Can clients use a single general-purpose model, rather than custom models for each task, while obeying device and network constraints? In this work, we investigate pretrained transformers (PTF) to achieve these on-device learning goals and thoroughly explore the roles of model size and modularity, where the latter refers to adaptation through modules such as prompts or adapters. Focusing on federated learning, we demonstrate that: (1) Larger scale shrinks the accuracy gaps between alternative approaches and improves heterogeneity robustness. Scale allows clients to run more local SGD epochs which can significantly reduce the number of communication rounds. At the extreme, clients can achieve respectable accuracy locally highlighting the potential of fully-local learning. (2) Modularity, by design, enables $>$100$\times$ less communication in bits. Surprisingly, it also boosts the generalization capability of local adaptation methods and the robustness of smaller PTFs. Finally, it enables clients to solve multiple unrelated tasks simultaneously using a single PTF, whereas full updates are prone to catastrophic forgetting. These insights on scale and modularity motivate a new federated learning approach we call "You Only Load Once" (FedYolo): The clients load a full PTF model once and all future updates are accomplished through communication-efficient modules with limited catastrophic-forgetting, where each task is assigned to its own module.

Max-Margin Token Selection in Attention Mechanism

Attention mechanism is a central component of the transformer architecture which led to the phenomenal success of large language models. However, the theoretical principles underlying the attention mechanism are poorly understood, especially its nonconvex optimization dynamics. In this work, we explore the seminal softmax-attention model $f(\boldsymbol{X})=\langle \boldsymbol{Xv}, \texttt{softmax}(\boldsymbol{XWp})\rangle$, where $\boldsymbol{X}$ is the token sequence and $(\boldsymbol{v},\boldsymbol{W},\boldsymbol{p})$ are trainable parameters. We prove that running gradient descent on $\boldsymbol{p}$, or equivalently $\boldsymbol{W}$, converges in direction to a max-margin solution that separates $\textit{locally-optimal}$ tokens from non-optimal ones. This clearly formalizes attention as an optimal token selection mechanism. Remarkably, our results are applicable to general data and precisely characterize $\textit{optimality}$ of tokens in terms of the value embeddings $\boldsymbol{Xv}$ and problem geometry. We also provide a broader regularization path analysis that establishes the margin maximizing nature of attention even for nonlinear prediction heads. When optimizing $\boldsymbol{v}$ and $\boldsymbol{p}$ simultaneously with logistic loss, we identify conditions under which the regularization paths directionally converge to their respective hard-margin SVM solutions where $\boldsymbol{v}$ separates the input features based on their labels. Interestingly, the SVM formulation of $\boldsymbol{p}$ is influenced by the support vector geometry of $\boldsymbol{v}$. Finally, we verify our theoretical findings via numerical experiments and provide insights.