In this section, we discuss the hyperbolic operations used in HNN formulations and set up the meta-learning problem. This particular setup is also known as the N-ways K-shot learning problem. This section provides the theoretical proofs of the theorems presented in our main paper. Note that points in the local tangent space follow Euclidean algebra. The columns present the number of tasks in each batch (# Tasks), HNN update learning rate (), meta update learning rate (), and size of hidden dimensions (d).

We introduce SOCK, a benchmark command line interface (CLI) that measures large language models' (LLMs) ability to self-replicate without human intervention. In this benchmark, self-replication is defined not only as an LLM's ability to create a functioning and running copy of itself, but also the ability for that self-replication to persist and occur across different computational contexts. Accordingly, we've developed a system to categorize LLMs based on broad self-replication capabilities in two general classes, Replication-Capability Levels (RCL) and Persistence-Capability Levels (PCL). Using a five-task suite based on practically manipulable modern CLI utilities and computer processes, experiments are orchestrated in a controlled environment with an LLM acting agentically. The performance of the LLM on agent tasks is then computed to produce an R-score (a quantitative evaluation of overall self-replication ability) and data used to categorize LLMs into specific RCL-PCL matrices. SOCK offers two primary contributions: (1) Provides the first formalized definitions and benchmark suite for evaluating LLM self-replication, with the goal of establishing a standard for future research, to our knowledge; (2) Allows the industry to track the effectiveness of future multi-agent systems and mitigate potential self-replication threat vectors within them. The results compiled from evaluating a variety of open-weight and proprietary frontier models reveal significant obstacles to persistent self-replication and multi-agent systems, including context retention and multi-agent decision-making. We propose future research directions to safely reduce the severity of these obstacles, potentially lowering future risk of more functional multi-agent systems.

Neural scaling laws--power-law relationships between generalization errors and characteristics of deep learning models--are vital tools for developing reliable models while managing limited resources. Although the success of large language models highlights the importance of these laws, their application to deep regression models remains largely unexplored. Here, we empirically investigate neural scaling laws in deep regression using a parameter estimation model for twisted van der Waals magnets. We observe power-law relationships between the loss and both training dataset size and model capacity across a wide range of values, employing various architectures--including fully connected networks, residual networks, and vision transformers. Furthermore, the scaling exponents governing these relationships range from 1 to 2, with specific values depending on the regressed parameters and model details. The consistent scaling behaviors and their large scaling exponents suggest that the performance of deep regression models can improve substantially with increasing data size.

Neural Information Processing Systems http://nips.cc/

PSD factorizations are fundamentally linked to understanding the expressiveness of semidefinite programs as well as the power and limitations of quantum resources in information theory.

This proof draws closely to the proof of the temporal policy gradient theorem. We shall now prove that, under Assumption 4.2, the B&B process can be formulated as a Second, Lemma A.1, together with Assumption 4.2, ensures the existence of (deterministic) distributions This concludes the proof.Proposition 4.4. In Depth-First-Search B&B (DFS B&B), that is, when nodes are processed depth-first and left-first by the algorithm, Assumption 4.2 holds. Solid lines show the moving average. The results are averaged over the solving runs that finished successfully for all methods.

To simulate the ground truth, physics-based equations derived using Lagrangian mechanics is employed. The equations for n -pendulum and spring systems are given in detail below.

We introduce GraphNet, a dataset of 2.7K real-world deep learning computational graphs with rich metadata, spanning six major task categories across multiple deep learning frameworks. To evaluate tensor compiler performance on these samples, we propose the benchmark metric Speedup Score S(t), which jointly considers runtime speedup and execution correctness under tunable tolerance levels, offering a reliable measure of general optimization capability. Furthermore, we extend S(t) to the Error-aware Speedup Score ES(t), which incorporates error information and helps compiler developers identify key performance bottlenecks. In this report, we benchmark the default tensor compilers, CINN for PaddlePaddle and TorchInductor for PyTorch, on computer vision (CV) and natural language processing (NLP) samples to demonstrate the practicality of GraphNet. The full construction pipeline with graph extraction and compiler evaluation tools is available at https://github.com/PaddlePaddle/GraphNet .

Group Relative Policy Optimization (GRPO) has significantly enhanced the reasoning capability of large language models by optimizing the arithmetic mean of token-level rewards. Unfortunately, GRPO is observed to suffer from unstable policy updates when facing tokens with outlier importance-weighted rewards, which manifest as extreme importance sampling ratios during training. In this study, we propose Geometric-Mean Policy Optimization (GMPO), with the aim to improve the stability of GRPO through suppressing token reward outliers. Instead of optimizing the arithmetic mean, GMPO maximizes the geometric mean of token-level rewards, which is inherently less sensitive to outliers and maintains a more stable range of importance sampling ratio. GMPO is plug-and-play-simply replacing GRPO's arithmetic mean with the geometric mean of token-level rewards, as the latter is inherently less sensitive to outliers. GMPO is theoretically plausible-analysis reveals that both GMPO and GRPO are weighted forms of the policy gradient while the former enjoys more stable weights, which consequently benefits policy optimization and performance. Experiments on multiple mathematical reasoning benchmarks show that GMPO-7B improves the average Pass@1 of GRPO by up to 4.1%, outperforming many state-of-the-art approaches. Code is available at https://github.com/callsys/GMPO.