We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual-curvature term. In the linear case, where the curvature term vanishes, this recovers the classical effective dimension of the Jacobian kernel covariance, but evaluated at the trained model rather than at initialization as is typical in neural tangent kernel analyses. We further bound this effective dimension via covering complexity of the gradient features, leading to guarantees that depend on learned geometry rather than parameter count. In particular, for manifold-supported data and piecewise Lipschitz Jacobians, the bounds scale with intrinsic dimension, while for one-hidden-layer ReLU networks, the mechanism can be made explicit through counts of activation-stable regions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets illustrate trained-Jacobian compression, the tightness of the residual-curvature linearization, and agreement between the stability bound and observed generalization gaps. A key feature of our bounds is the simplicity of their derivation, which follows from first principles using the Brascamp-Lieb inequality under strongly log-concave noise.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.

Embodied agents equipped with large language models (LLMs) and online constructed navigation maps can perform ObjNav in a zero-shot manner. However, existing agents heavily rely on giant LLMs on the cloud, e.g., GPT-4, while directly switching to small LLMs, e.g., LLaMA3.2-11b,

Recent advances in Scientific Machine Learning have shown that second-order methods can enhance the training of Physics-Informed Neural Networks (PINNs), making them a suitable alternative to traditional numerical methods for Partial Differential Equations (PDEs). However, second-order methods induce large memory requirements, making them scale poorly with the model size. In this paper, we define a local Mixture of Experts (MoE) combining the parameter-efficiency of ensemble models and sparse coding to enable the use of second-order training. Our model -- PINN Balls -- also features a fully learnable domain decomposition structure, achieved through the use of Adversarial Adaptive Sampling (AAS), which adapts the DD to the PDE and its domain. PINN Balls achieves better accuracy than the state-of-the-art in scientific machine learning, while maintaining invaluable scalability properties and drawing from a sound theoretical background.

Large Language Models (LLMs), such as OpenAI's o1 and DeepSeek's R1, excel at advanced reasoning tasks like math and coding via Reinforcement Learning with Verifiable Rewards (RLVR), but still struggle with puzzles solvable by humans without domain knowledge. We introduce ENIGMATA, the first comprehensive suite tailored for improving LLMs with puzzle reasoning skills. It includes 36 tasks across 7 categories, each with: 1) a generator that produces unlimited examples with controllable difficulty, and 2) a rule-based verifier for automatic evaluation. This generator-verifier design supports scalable, multi-task RL training, fine-grained analysis, and seamless RLVR integration. We further propose ENIGMATA-Eval, a rigorous benchmark, and develop optimized multi-task RLVR strategies.

Apple has announced the latest version of macOS. It's all about the reintroduction of Siri, which is now accessible from anywhere on the Mac desktop. The official name of the Mac's operating system is macOS 27 Golden Gate, keeping the California naming scheme around. This year's update is focused on the relaunched Siri (now known as Siri AI), which really strives to transform into a proper AI chatbot along the lines of ChatGPT or Google Gemini--with a unique Apple twist. Is Your Mac Compatible With macOS Golden Gate?

Anthropic Offers Mythos Upgrade for Cyber Partners and a'Safe' Version for the Rest of You Anthropic is releasing Claude Mythos 5 to trusted organizations and Claude Fable 5 to the public, a version it says can't be used for cyberattacks. Anthropic released two new AI models called Claude Fable 5 and Claude Mythos 5 on Tuesday, which the company says have greater capabilities than the Mythos Preview model it released in April to a limited set of tech industry partners. Anthropic has said the initial, limited release stemmed from concerns that the model's capabilities could be exploited by bad actors to develop hacking tools that could catch defenders off guard. Anthropic is currently only releasing Claude Mythos 5 to a limited set of industry partners, many of which received access to Mythos Preview, and the company says it is collaborating with the US government on the rollout. Claude Fable 5, which is being publicly released, uses the same underlying model as Mythos 5, but will have "guardrails" in place at launch, the company said Tuesday, that will block the model from answering many user questions related to cybersecurity, biology, and chemistry.

PCWorld reports that OpenAI launched Lockdown Mode for ChatGPT to combat prompt injection attacks that can hijack AI systems and steal personal information. These attacks have previously compromised AI browsers like Perplexity and controlled smart home devices through Google Gemini by tricking systems with malicious instructions. Lockdown Mode restricts features like live web browsing and Deep Research across all ChatGPT plans, though OpenAI acknowledges risks from uploaded files remain. OpenAI has launched a new security feature in ChatGPT called Lockdown Mode, designed to provide additional protection against so-called "prompt injection attacks." A prompt injection attack is when someone crafts a deceptive prompt in an attempt to trick the LLM into following malicious instructions and/or revealing sensitive information.

Large reasoning models (LRMs), such as OpenAI o1 and DeepSeek-R1, have significantly enhanced their reasoning capabilities by generating longer chains of thought, demonstrating outstanding performance across a variety of tasks. However, this performance gain comes at the cost of a substantial increase in redundant reasoning during the generation process, leading to high computational overhead and exacerbating the issue of overthinking. Although numerous existing approaches aim to address the problem of overthinking, they often rely on external interventions.