We study the problem of entropy calibration, which asks whether a language model's entropy over generations matches its log loss on human text. Past work found that models are miscalibrated, with entropy per step increasing as generations grow longer, due to error accumulation. To calibrate the model and improve text quality, it has become standard practice to truncate the distribution, but this approach reduces output diversity, which we would like to avoid. Therefore, in this paper, we ask: does miscalibration improve automatically with scale, and if not, is it theoretically possible to calibrate without tradeoffs? To build intuition, we first study a simplified theoretical setting to characterize the scaling behavior of miscalibration with respect to dataset size. We find that the rate of scaling depends on the power law exponent of the data distribution -- in particular, for a power law exponent close to 1, the scaling exponent is close to 0, meaning that miscalibration improves very slowly with scale.

Large-scale AI models, such as Large Language Models (LLMs) and Diffusion Models (DMs), have grown rapidly in size, creating significant challenges for efficient deployment on resource-constrained hardware. In this paper, we introduce Dynamic-Length Float (DFloat11), a lossless compression framework that reduces LLM and DM size by 30% while preserving outputs that are bit-for-bit identical to the original model. DFloat11 is motivated by the low entropy in the BFloat16 weight representation of LLMs, which reveals significant inefficiency in the existing storage format. By applying entropy coding, DFloat11 assigns dynamic-length encodings to weights based on frequency, achieving near information-optimal compression without any loss of precision. To facilitate efficient inference with dynamic-length encodings, we develop a custom GPU kernel for fast online decompression. Our design incorporates the following: (i) compact, hierarchical lookup tables (LUTs) that fit within GPUSRAM for efficient decoding, (ii) a two-phase GPU kernel for coordinating thread read/write positions using lightweight auxiliary variables, and (iii) transformer-block-level decompression to minimize latency. Experiments on Llama 3.3, Qwen 3, Mistral 3, FLUX.1, and others validate our hypothesis that DFloat11 achieves around 30% model size reduction while preserving bit-for-bit identical outputs. Compared to a potential alternative of offloading parts of an uncompressed model to the CPU to meet memory constraints, DFloat11 achieves 2.3-46.2

Neural scaling laws are observed in a range of domains, to date with no universal understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language. One theory suggests that language scaling laws emerge when Zipf-distributed task quanta are learned in descending order of frequency. In this paper we examine power-law scaling in AlphaZero, a reinforcement learning algorithm, using a model of language-model scaling. We find that game states in training and inference data scale with Zipf's law, which is known to arise from the tree structure of the environment, and examine the correlation between scaling-law and Zipf'slaw exponents. In agreement with the quanta scaling model, we find that agents optimize state loss in descending order of frequency, even though this order scales inversely with modelling complexity. We also find that inverse scaling, the failure of models to improve with size, is correlated with unusual Zipf curves where end-game states are among the most frequent states. We show evidence that larger models shift their focus to these less-important states, sacrificing their understanding of important early-game states.

We investigate scaling laws for stochastic momentum algorithms on the power law random features model, parameterized by data complexity, target complexity, and model size. When trained with a stochastic momentum algorithm, our analysis reveals four distinct loss curve shapes determined by varying data-target complexities. While traditional stochastic gradient descent with momentum (SGD-M) yields identical scaling law exponents to SGD, dimension-adapted Nesterov acceleration (DANA) improves these exponents by scaling momentum hyperparameters based on model size and data complexity. This outscaling phenomenon, which also improves compute-optimal scaling behavior, is achieved by DANA across a broad range of data and target complexities, while traditional methods fall short. Extensive experiments on high-dimensional synthetic quadratics validate our theoretical predictions and large-scale text experiments with LSTMs show DANA's improved loss exponents over SGD hold in a practical setting.

We study the problem of entropy calibration, which asks whether a language model's entropy over generations matches its log loss on human text. Past work found that models are miscalibrated, with entropy per step increasing as generations grow longer, due to error accumulation. To calibrate the model and improve text quality, it has become standard practice to truncate the distribution, but this approach reduces output diversity, which we would like to avoid. Therefore, in this paper, we ask: does miscalibration improve automatically with scale, and if not, is it theoretically possible to calibrate without tradeoffs? To build intuition, we first study a simplified theoretical setting to characterize the scaling behavior of miscalibration with respect to dataset size. We find that the rate of scaling depends on the power law exponent of the data distribution --- in particular, for a power law exponent close to 1, the scaling exponent is close to 0, meaning that miscalibration improves very slowly with scale.

We focus on the challenging extensive-width regime $P\gg 1$ and permit diverging condition number in the second-layer, covering as a special case the power-law scaling $a_p\asymp p^{-\beta}$ where $\beta\in\mathbb{R}_{\ge 0}$. We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective, and explicitly identify sharp transition times to recover each signal direction. In the power-law setting, we characterize scaling law exponents for the MSE loss with respect to the number of training samples and SGD steps, as well as the number of parameters in the student neural network. Our analysis entails that while the learning of individual teacher neurons exhibits abrupt transitions, the juxtaposition of $P\gg 1$ emergent learning curves at different timescales leads to a smooth scaling law in the cumulative objective.

The scaling exponent $α$ in neural scaling laws $L(N) \propto N^{-α}$ is commonly treated as a fixed constant set by architecture and data. We present evidence that $α$ depends systematically on the optimizer. In controlled random-feature regression experiments -- the canonical theoretical framework for neural scaling -- we measure $α$ across five optimizer variants and six spectral conditions. Preconditioned optimizers consistently yield steeper scaling (larger $α$), with the $α$-shift increasing across most of the tested spectral range, peaking near $s = 1.5$, and remaining large at $s = 2.0$. At $s \approx 1.0$ (characteristic of natural language), the full natural gradient achieves $α\approx 0.31$ versus $α\approx 0.12$ for gradient descent -- a $2.6\times$ larger fitted exponent that, within the random-feature model, compounds with each model-size doubling. Whether and how this exponent shift transfers to large-scale LLM training -- where recent evidence suggests the advantage may attenuate with scale -- remains an important open question. Our results imply that scaling-law forecasts should account for optimizer choice, and we provide a spectral diagnostic predicting when advanced optimizers will pay off.

Reward modeling is not only a prediction problem: in KL-regularized policy optimization, the learned reward is exponentiated to define the deployed policy, so downstream value depends on errors in reward-tilted regions. We study this feedback in a Gaussian single-index model with $r^*(x) = σ^*(\langle θ^*, x\rangle)$ and $x \sim N(0, I_d)$. We analyze a two-stage neural reward model that first learns the hidden direction $θ^*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression. Exponential reward weighting changes the Hermite signal available to the first layer; for any feature-learning temperature $β_1$ above a dimension-free $O(1)$ threshold, a constant fraction of neurons recover the hidden direction, with weak-recovery complexity governed by the generative exponent. After feature recovery, we derive tilted-policy value-gap bounds for an idealized label-weighted fit with weights $e^{y/β_2}$ and a more practical surrogate-weighted fit with weights $e^{r_{a_0}(x)/β_2}$. Keeping the $β_2$-dependence explicit yields an admissible set of deployment temperatures, balancing the gain from lowering $β_2$ against the learning cost amplified by exponential weighting; in the surrogate-weighted case, proxy-dependent factors shrink this admissible set.

We introduce a model for neural scaling laws under sparse activations. In the model, test loss is often dominated by rare coordinates that are never observed in the training input. This mechanism induces a novel bottleneck absent from dense models. We derive the asymptotic population loss in both the underparameterized and overparameterized regimes, and show that the loss exhibits a double-descent peak near the interpolation threshold -- where the number of parameters is just sufficient to fit the training data -- resulting in a loss curve governed by two distinct scaling exponents -- one for the overparameterized regime and one for the underparameterized regime -- with a gap determined by the degree of sparsity. Additionally, we derive a compute-optimal frontier that favors increasing dataset size over model capacity under fixed compute budgets. We also analyze gradient-descent dynamics and identify a scaling law for the probability that fixed-step gradient descent becomes unstable. We further show that the sparsity-induced effect persists under nonlinear activations.

We develop a mean-field theory of dropout as a perturbation of critical signal propagation at the edge of chaos. Dropout shifts the perfect-alignment fixed point, making the depth scale for information propagation finite even at critical initialization. We derive critical and crossover scaling laws for correlation decay and establish that smooth activations and kinked, ReLU-like activations constitute distinct universality classes, with different critical exponents and a universal two-parameter scaling collapse in detuning and dropout strength. The distinction traces to the analytic structure of the correlation map: smooth activations admit a Taylor expansion near perfect alignment, while kinked activations develop a branch point with universal non-analyticity. As a corollary, the framework yields saturated dropout profiles under fixed budget; a rank-flow tie-breaker then selects front-loaded schedules, substantially reducing held-out test loss at no extra computational cost, with accuracy gains as a consistent secondary effect. We test the predictions in MLPs and Vision Transformers and discuss CNN/ResNet extensions.