Lack of samples per minute or per hour (metrics 1 and 2) are helping to understand 4G networking issues.

How much does a trained RL policy actually use its past observations? We propose \emph{Temporal Range}, a model-agnostic metric that treats first-order sensitivities of multiple vector outputs across a temporal window to the input sequence as a temporal influence profile and summarizes it by the magnitude-weighted average lag. Temporal Range is computed via reverse-mode automatic differentiation from the Jacobian blocks $\partial y_s/\partial x_t\in\mathbb{R}^{c\times d}$ averaged over final timesteps $s\in\{t+1,\dots,T\}$ and is well-characterized in the linear setting by a small set of natural axioms. Across diagnostic and control tasks (POPGym; flicker/occlusion; Copy-$k$) and architectures (MLPs, RNNs, SSMs), Temporal Range (i) remains small in fully observed control, (ii) scales with the task's ground-truth lag in Copy-$k$, and (iii) aligns with the minimum history window required for near-optimal return as confirmed by window ablations. We also report Temporal Range for a compact Long Expressive Memory (LEM) policy trained on the task, using it as a proxy readout of task-level memory. Our axiomatic treatment draws on recent work on range measures, specialized here to temporal lag and extended to vector-valued outputs in the RL setting. Temporal Range thus offers a practical per-sequence readout of memory dependence for comparing agents and environments and for selecting the shortest sufficient context.

This paper presents a new class of gradient methods for distributed machine learning that adaptively skip the gradient calculations to learn with reduced communication and computation. Simple rules are designed to detect slowly-varying gradients and, therefore, trigger the reuse of outdated gradients. The resultant gradient-based algorithms are termed Lazily A ggregated G radient -- justifying our acronym LAG used henceforth. Theoretically, the merits of this contribution are: i) the convergence rate is the same as batch gradient descent in strongly-convex, convex, and nonconvex cases; and, ii) if the distributed datasets are heterogeneous (quantified by certain measurable constants), the communication rounds needed to achieve a targeted accuracy are reduced thanks to the adaptive reuse of lagged gradients. Numerical experiments on both synthetic and real data corroborate a significant communication reduction compared to alternatives.

This paper presents a new class of gradient methods for distributed machine learning that adaptively skip the gradient calculations to learn with reduced communication and computation. Simple rules are designed to detect slowly-varying gradients and, therefore, trigger the reuse of outdated gradients. The resultant gradient-based algorithms are termed Lazily A ggregated G radient -- justifying our acronym LAG used henceforth. Theoretically, the merits of this contribution are: i) the convergence rate is the same as batch gradient descent in strongly-convex, convex, and nonconvex cases; and, ii) if the distributed datasets are heterogeneous (quantified by certain measurable constants), the communication rounds needed to achieve a targeted accuracy are reduced thanks to the adaptive reuse of lagged gradients. Numerical experiments on both synthetic and real data corroborate a significant communication reduction compared to alternatives.

Third, we provide a general "reduction"

As discussed in the main text, there could be many ways to incorporate anatomical priors into our formulation. Here, we demonstrate one example--assuming that brain regions are sparsely connected, and therefore many blocks of the dynamics matrices will be zero. This can be implemented using a block-wise spike-and-slab prior on the dynamics matrices. A.1 Formulation We want a block-sparse prior for dynamics matrices. We break down this matrix into a B B set of blocks, where B is the number of neural populations.

The ionosphere critically influences Global Navigation Satellite Systems (GNSS), satellite communications, and Low Earth Orbit (LEO) operations, yet accurate prediction of its variability remains challenging due to nonlinear couplings between solar, geomagnetic, and thermospheric drivers. Total Electron Content (TEC), a key ionospheric parameter, is derived from GNSS observations, but its reliable forecasting is limited by the sparse nature of global measurements and the limited accuracy of empirical models, especially during strong space weather conditions. In this work, we present a machine learning framework for ionospheric TEC forecasting that leverages Temporal Fusion Transformers (TFT) to predict sparse ionosphere data. Our approach accommodates heterogeneous input sources, including solar irradiance, geomagnetic indices, and GNSS-derived vertical TEC, and applies preprocessing and temporal alignment strategies. Experiments spanning 2010-2025 demonstrate that the model achieves robust predictions up to 24 hours ahead, with root mean square errors as low as 3.33 TECU. Results highlight that solar EUV irradiance provides the strongest predictive signals. Beyond forecasting accuracy, the framework offers interpretability through attention-based analysis, supporting both operational applications and scientific discovery. To encourage reproducibility and community-driven development, we release the full implementation as the open-source toolkit \texttt{ionopy}.

Transformers have exhibited exceptional capabilities in sequence modeling tasks, leveraging self-attention and in-context learning. Critical to this success are induction heads, attention circuits that enable copying tokens based on their previous occurrences. In this work, we introduce a novel framework that showcases transformers' ability to dynamically handle causal structures. Existing works rely on Markov Chains to study the formation of induction heads, revealing how transformers capture causal dependencies and learn transition probabilities in-context. However, they rely on a fixed causal structure that fails to capture the complexity of natural languages, where the relationship between tokens dynamically changes with context. To this end, our framework varies the causal structure through interleaved Markov chains with different lags while keeping the transition probabilities fixed. This setting unveils the formation of Selective Induction Heads, a new circuit that endows transformers with the ability to select the correct causal structure in-context. We empirically demonstrate that transformers learn this mechanism to predict the next token by identifying the correct lag and copying the corresponding token from the past. We provide a detailed construction of a 3-layer transformer to implement the selective induction head, and a theoretical analysis proving that this mechanism asymptotically converges to the maximum likelihood solution. Our findings advance the understanding of how transformers select causal structures, providing new insights into their functioning and interpretability.

The proliferation of fine-tuned language model experts for specific tasks and domains signals the need for efficient selection and combination methods. We propose LoRA-Augmented Generation (LAG) for leveraging large libraries of knowledge and task-specific LoRA adapters. LAG requires no additional training or access to data, and efficiently filters, retrieves, and applies experts on a per-token and layer basis. We evaluate LAG on various knowledge-intensive tasks, achieving superior performance over existing data-free methods. We explore scenarios where additional data is available, demonstrating LAG's compatibility with alternative solutions such as retrieval-augmented generation (RAG).