From Spectra to Joint Schedules in LLM Pre-training: 3+3(+2) Scaling-Law Regimes

Power-law learning curves are often treated as fixed properties of a model and its data, although learning-rate and batch-size schedules can change the observed loss. We study this dependence in noisy online SGD with linear random features. Conditional on the representation, an exact Volterra equation separates two response components: a forcing term that propagates unresolved target error and a memory kernel that propagates stochastic-error injections. We prove that either component follows a power law if and only if its cumulative weighted spectral mass has the corresponding low-spectrum scaling; individual eigenvalues and target coefficients need not obey coordinatewise power laws. Under a joint schedule, intrinsic time $T_t=_{s

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arXiv AI Papers·2 days ago☆

Scalable and Adaptive Distribution Regression via Random Features↗

Distribution regression seeks to learn mappings from probability distributions to response variables and has emerged as a unifying framework for kernel methods, multi-instance learning, and set-valued learning. Despite its success, kernel-based distribution regression remains computationally expensive and sensitive to kernel choice, motivating scalable and adaptive representations. Building on random feature approximations, we develop a finite-sample analysis of Random Feature Distribution Regression (RaDR) under two-stage sampling. Our perturbation analysis separates the effects of distribution-level sampling, finite within-bag sampling, and random feature approximation. We establish sufficient feature-budget conditions under which RaDR preserves the order of our excess-risk upper bound for exact two-stage kernel distribution regression while maintaining computational efficiency. We further propose Adaptive Random Feature Distribution Regression (A-RaDR), a unified score-guided framework that adaptively allocates random features according to downstream learning objectives. Experiments on synthetic benchmarks and 41 real-world datasets demonstrate that RaDR delivers substantial computational savings while maintaining competitive predictive performance, whereas A-RaDR further improves learning accuracy in distribution regression while retaining the computational speed up. Together, these results establish a practical and theoretically grounded framework for adaptive, scalable distribution learning

arXiv AI Papers·2 days ago☆

Query-Dependent Reading for Quantized Inner-Product Estimation: A Minimax Separation↗

We study quantization for inner-product estimation when storage and reading have separate budgets. In the static bit-probe model [Yao, 1981], an encoder stores w bits about x before seeing a query y, and a reader probes at most $r

arXiv AI Papers·2 days ago☆

Measuring Data Diversity with Embeddings: A Taxonomy and Benchmark for Diversity Measures↗

Diversity is a critical property of text corpora, with direct implications for training data curation, test set design, and model generalization. However, its measurement remains fragmented and poorly understood. Existing embedding-based diversity measures vary widely in how they are computed and can yield substantially different diversity scores on the same data, yet no systematic comparison of their empirical behaviour exists. We make two contributions toward addressing this gap. First, we present a taxonomy of 21 embedding-based diversity measures, organizing them into four families based on their underlying computations. Second, we introduce EmbDivBench, a benchmark that evaluates these measures along three conceptually distinct components (variety, balance and disparity) on both simulated and natural text data. Although Mean Pairwise Distance is the most commonly used measure, our experiments show that it is outperformed by several alternatives on the three diversity components, with Graph Entropy being the best choice overall. Lastly, we distil these findings into practical recommendations for diversity measure selection.

arXiv AI Papers·2 days ago☆

Sample-Efficient Reinforcement Learning with General Utilities via Neural Dual Critics↗

Reinforcement learning with general utilities optimizes nonlinear functionals of policy-induced occupancy measures. We develop a natural actor-critic method for concave RLGU objectives admitting an exact variational dual representation, with explicit constructions for a broad class of separable occupancy utilities. Rather than estimating the occupancy measure and then forming the policy-gradient pseudo-reward, our method learns this pseudo-reward directly as a function of the state-action pair using a neural critic. The critic is trained by projected stochastic updates, while multilevel Monte Carlo combines outputs across training horizons to control its bias at logarithmic expected cost. Our analysis separates conditional actor bias from estimator second moments and accounts explicitly for dual optimization, critic approximation, and neural linearization errors. Under policy-transfer, dual-stability, and local critic-regularity conditions, the method achieves {O}(T^{-1/2}) global convergence above policy, critic-class, and finite-width approximation errors using {O}(T) expected environment transitions and reference samples. For a fixed critic representation, this yields {O}(ε^{-2}) sampling complexity above the approximation floor, covering infinite state-action spaces and occupancy utilities beyond fixed finite vectors of returns.