We develop a natural Bayesian multiplicity-correcting prior distribution within the probabilistic forward stepwise representation of model space priors for regression problems. The proposed prior, obtained from making an analogy to the Holm procedure, exhibits behavior closely aligned with that of the Matryoshka doll prior. We compare both priors to several other priors, including some recently put forward as objective choices for model space prior probabilities. Our comparisons indicate that adequate multiplicity correction requires a degree of sparsity that many recommended priors do not provide, and we argue that multiplicity correction itself offers a principled and transparent criterion for specifying model space priors in regression.

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.

We test whether the optimal learning-rate schedule depends on bit-width during from-initialisation quantisation-aware training (QAT) for sub-100M decoder language models. A 720-run factorial grid (Phase 2) over bit-width x warmdown fraction x LR magnitude x model size x seed (FP16/INT8/INT6, 15M-100M, 5 seeds) finds the optimal warmdown is 33% at every (bit-width, size) cell. The primary hypothesis -- that INT6 QAT requires a different schedule than higher-precision training -- is falsified at FP16/INT8/INT6. A 625-run follow-up (Phase 5) probes the null along five axes: optimiser (AdamW), schedule shape (cosine), training length (up to 9x more iterations), an extended size sweep (5M-350M), and an INT4 sweep from 3M to 100M. The null is robust under all three setup changes. The INT6 penalty follows a log-linear scaling law whose fit on Phase 2 predicts the five held-out Phase 5 sizes (5M, 8M, 175M, 250M, 350M) within their 95% prediction intervals (5/5). For INT4 the picture is sharper than the higher precisions: at 50M and 100M, wd33 is decisively optimal (paired z ~ 12-15, 10/10 seeds); below 50M, across the six tested sizes from 3M to 30M, no individual size shows a statistically significant schedule preference and the per-size mean penalty oscillates within seed-level noise. The boundary is therefore a transition between a noise-dominated regime below 50M and a decisive wd33 regime at and above 50M, not a clean wd10 region. A weight-to-grid-distance probe falsifies the simplest mechanism for the FP16/INT8/INT6 null result (rapid grid-snapping): pre-warmdown, INT6-QAT weights sit at essentially the same distance from the INT6 grid as FP16 weights (ratio ~ 1.04). Practical recommendation: at sub-100M scale, tune the LR schedule once at FP16 and apply unchanged to INT8/INT6 QAT; for INT4 at 50M+ use wd33; for INT4 below 50M the schedule choice is in the noise.

In this paper, we present a dual representation of the influence functions, whose computational complexity scales with dataset size rather than model size. Both analytically and experimentally, we show that this representation can be an efficient alternative to the original influence functions for estimating changes in parameters, model outputs and loss due to data point removal, when model size is large relative to dataset size, or when evaluating the original influence functions in parameter space is infeasible. The dual representation, however, is limited to linearizable models, which are models whose behavior can be approximated by their linearizations throughout training, and requires materializing a matrix, whose size grows with the product of model output dimension and dataset size.

Tree ensembles such as random forests (RFs) and gradient boosting machines (GBMs) are among the most widely used supervised learners, yet their theoretical properties remain incompletely understood. We adopt a spectral perspective on these algorithms, with two main contributions. First, we derive minimax-optimal convergence for RF regression, showing that, under mild regularity conditions on tree growth, the eigenvalue decay of the induced kernel operator governs the statistical rate. Second, we exploit this spectral viewpoint to develop compression schemes for tree ensembles. For RFs, leading eigenfunctions of the kernel operator capture the dominant predictive directions; for GBMs, leading singular vectors of the smoother matrix play an analogous role. Learning nonlinear maps for these spectral representations yields distilled models that are orders of magnitude smaller than the originals while maintaining competitive predictive performance. Our methods compare favorably to state of the art algorithms for forest pruning and rule extraction, with applications to resource constrained computing.

A.1 Hyperparameters for QLORA We do a hyperparameter search for LoRA over the following variables: LoRA dropout { 0.0, 0.05, 0.1}, LoRA r { 8, 16, 32, 64, 128, 256}, LoRA layers {key+query, all attention layers, all FFN layers, all layers, attention + FFN output layers}. We keep LoRA α fixed and search the learning rate, since LoRA α is always proportional to the learning rate. We find that LoRA dropout 0.05 is useful for small models (7B, 13B), but not for larger models (33B, 65B). Each dot represents a combination of hyperparameters and for each LoRA r we run 3 random seed with each hyperparameter combination. The performance of specific LoRA r values appears to be independent of other hyperparameters.

We propose Conditional Adapter (CODA), a parameter-efficient transfer learning method that also improves inference efficiency. CODA generalizes beyond standard adapter approaches to enable a new way of balancing speed and accuracy using conditional computation. Starting with an existing dense pretrained model, CODA adds sparse activation together with a small number of new parameters and a light-weight training phase. Our experiments demonstrate that the CODA approach provides an unexpectedly efficient way to transfer knowledge. Across a variety of language, vision, and speech tasks, CODA achieves a 2x to 8x inference speed-up compared to the state-of-the-art Adapter approaches with moderate to no accuracy loss and the same parameter efficiency.