The Numerical Linear Algebra of Large Language Models
15d
- Publicado
- Coletado
Autores
Abdelkader Baggag, Yousef Saad
Categorias
Numerical Linear Algebra (NLA) has consistently played a vital role in advancing science by providing tools to solve fundamental problems encountered in scientific and engineering applications. Over the decades, it has continually evolved to meet the demands driven by successive waves of scientific discovery. For instance, during the 1950s and 1960s, substantial efforts were devoted to developing methods for solving eigenvalue problems that emerged from the rapidly growing field of aerodynamics. This led to the discovery of the LR and QR algorithms. Later the attention turned to the solution of sparse linear systems that were common in applications like computational aerodynamics. Today we are experiencing yet another wave of major scientific advancement and NLA is once more at the heart of its development. This Machine Learning (ML) wave is proving to be utterly disruptive in science and engineering. Many tools in ML particularly Large Language Models (LLMs) are grounded in matrix and tensor methods. As we are approaching Artificial General Intelligence (AGI), it is clear that matrix methods will be called to play an even more significant role. For the numerical linear practitioner the speed of the current change makes it particularly challenging to adapt. This is a survey article that centers on machine learning techniques, with a particular focus on large language models. It has two main objectives. The first is to clarify the core concepts behind Large Language Models in a manner accessible to specialists in numerical methods. The second is to examine the key Numerical Linear Algebra concepts employed by LLM techniques, while also highlighting several significant recent contributions of NLA to the field.
O overfeed.news indexa e aponta. Publicamos um trecho curto — o artigo completo fica em arXiv AI Papers.
Mais de arXiv AI Papers
Entre para seguir esta fonte
LIBERO-PeRM: Benchmarking Personalized Robotic Manipulation
Personalization is essential for general-purpose robots to transition into household environments and achieve true practicality. While general robotic manipulation has been extensively studied, personalized robotic manipulation (PeRM) lacks a controlled benchmark for separating task completion from preference satisfaction. We introduce LIBERO-PeRM, a benchmark for preference learning in embodied agents. LIBERO-PeRM studies five questions: 1) how policies follow preferences in zero-shot settings; 2) how they learn single preferences from demonstrations; 3) how efficiently they acquire preferences as demonstrations increase; 4) how learned preferences generalize across layouts and tasks; and 5) how policies compose preferences and behave under conflicts. We propose a controlled formulation and taxonomy of observable preferences in simulation, covering motion, skill, and user-profile choices. This formulation is restricted to language-expressible preferences that can be instantiated in LIBERO without changing the task goal, evaluated automatically, and reused across tasks. Building on LIBERO, our procedural pipeline generates paired demonstrations with and without preferences. We create five task suites (312 tasks) with high-quality demonstrations, rich metadata, and preference-satisfaction metrics. Experiments show that current VLA policies struggle with zero-shot personalization, while SFT makes preferences learnable. Mixed training achieves better no-preference retention than sequential adaptation and improves performance with contextual prompts compared with preference-only training, but can weaken motion-preference learning; preference generalization, decomposition, and conflict handling remain difficult. The code, datasets, and demos can be found on the project website: https://libero-perm.github.io/.
Discovering state-of-the-art quantum circuits with large language models
Significant efforts are currently being directed toward the implementation of large-scale quantum information processors, but realizing useful quantum advantage also requires the development of efficient quantum algorithms. While for many practical problems the dominant resource costs are associated with a small set of computational primitives, optimizing them in the quantum setting presents unique challenges, including a reversibility constraint absent from ordinary classical computation. Here we introduce an evolutionary framework in which large language models (LLMs) propose circuit designs and an independent evaluator verifies and scores them. We apply this framework to tasks spanning numerical computation, chemistry, optimization, quantum simulation, and number theory. In seven of the eight tasks, the discovered circuits significantly reduce both Toffoli and qubit counts relative to the state of the art. In the eighth, the search discovers a new reversible architecture for prime-field modular inversion. By handling reversibility differently, the architecture makes divsteps -- a classical algorithm previously impractical for reversible integer arithmetic -- viable and substantially improves the Toffoli-qubit trade-off for modular inversion. These results open the door for evaluator-guided LLM co-design of new algorithmic structures for quantum computation.
T2-z: Test-Time Optimization of Environment Latents for Robotic World Models
Predicting how the world responds to an action requires more than what can be seen: robots must account for environmental factors that are neither directly observable nor explicitly specified, from physical properties such as friction and mass to human-defined environmental rules. We introduce T2-Z for test-time optimization of environment latents, which augments a robotic world model with a learnable environment latent to condition world predictions. This compact latent captures underlying environmental variation without supervision of what environmental factors exist or what values they take. During training, T2-Z alternates between two stages: a world-model optimization stage that fixes the latents and learns general action-conditioned dynamics across environments; and an environment-latent optimization stage that freezes the world model and optimizes the conditioning latents, allowing latent structure to emerge purely through world prediction. At test time, the world model remains frozen and a newly initialized environment latent is optimized from one or more test-time interactions, adapting the model to previously unseen environmental configurations. Across 7 simulated settings and 4 real-world robotic tasks, T2-Z reduces object prediction error metric from 15.20 to 7.00 and raises action success from 33.73% to 72.96%. Its emergent latents capture structured representations of individual and compounded factors across diverse visual scenes. Ablations further show that T2-Z can efficiently infer environment latents from as little as a single test-time interaction trajectory.
CoDA: Co-Evolution of Data Engines and Action Policies
Generative simulation can automate robot environment construction, but physically valid environments are not necessarily useful for training. The challenge is to identify environments that address a policy's failures and supervision that helps it overcome them. We introduce CoDA, a framework in which a simulation data engine and an action policy co-evolve through a shared heterogeneous memory of environment construction, policy failures, and learning progress. The memory compiles validated diagnoses from a vision-language critic into generation programs that recreate difficulties and training programs that help the policy overcome them. We further propose Text2potential to convert these diagnoses into staged potentials for action-chunk value learning, providing dense supervision for policy updates while preserving the task's optimal policy under standard shaping assumptions. Measured learning progress guides further generation and program refinement. Across four manipulation task families, CoDA improves out-of-distribution success from 5.3% after supervised fine-tuning to 27.6%, exceeding the strongest closed-loop baseline by 9.5 percentage points with the same generation budget. On LIBERO-Pro, 15 rounds of generation and training raise success from 25% to 84% under position perturbations and from 1% to 76% under instruction perturbations.