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arXiv2026

Inverting Foundation Models of Brain Function with Simulation-Based Inference

Foundation models of brain activity promise a new frontier for in silico neuroscience by emulating neural responses to complex stimuli across tasks and modalities. A natural next step is to ask whether these models can also be used in reverse. Can we recover a stimulus or its properties from synthetic brain activity? We study this question in a proof-of-concept setting using TRIBEv2. We pair the brain emulator with large language models (LLMs) that generate news headlines from linguistic parameters such as valence, arousal, and dominance. We then use simulation-based inference to learn a probabilistic mapping from brain maps to latent stimulus parameters. Our results show that these parameters can be recovered from predicted brain maps, demonstrating that the emulator's synthetic neural encodings preserve information about the controlled stimulus dimensions. They also show that LLMs can serve as controllable stimulus generators for simulated experiments. Together, these findings provide a step toward decoding and inverse design with foundation brain models.

cs.LG↗

arXiv2026

Budget-Constrained Causal Bandits: Bridging Uplift Modeling and Sequential Decision-Making

Treatment allocation under budget constraints is a central challenge in digital advertising. The standard approach trains an offline uplift model on historical data, then solves a constrained optimization to allocate budget. This fails in cold-start settings where little historical data exists. We propose Budget-Constrained Causal Bandits (BCCB), an online framework that learns which users respond to ads while simultaneously spending the budget. BCCB unifies three components: learning individual-level treatment effects, exploring users whose response is uncertain, and pacing the budget over time. We derive the per-arrival decision rule as the KKT condition of a Lagrangian relaxation of the budgeted causal-allocation objective, providing a principled foundation for the algorithm. We evaluate on the Criteo Uplift dataset using 20 random seeds with paired statistical tests. Our central finding is a data-efficiency crossover at n = 7,500 historical observations (paired one-sided t-test, p = 0.043): below this threshold, offline pipelines either fail or produce unreliable allocations, while BCCB operates from the first user. BCCB exhibits 2-4x lower run-to-run variance than offline methods and outperforms all four online baselines (Thompson Sampling, budgeted Thompson Sampling, HTE Greedy, and Uplifting Bandits) at every budget level tested (p < 0.001). These results give practitioners a concrete decision rule for choosing between offline and online paradigms.

cs.LG↗

arXiv2026

Online Learning-to-Defer with Varying Experts

Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. Real-world deployments require handling streaming data, changing expert availability, shifting expert reliability, and feedback observed only for the selected action. We introduce an online multiclass L2D algorithm that combines queried-action bandit feedback with a dynamically varying pool of experts. Let $N=n+n_e$, let $B$ bound the Frobenius norm of the linear score matrix, and let $ρ$ bound the augmented input norm. Assuming linear calibration and zero surrogate minimizability gap for the projected comparator class, our method achieves expected true-deferral regret $O((BN^{3/2}ρ+1)T^{2/3})$, improving to $O(BN^{3/2}ρ\sqrt T+B^2N^3ρ^2)$ under a concentrated-score condition. The analysis combines an online $\mathcal H$-consistency transfer bound with projected online convex optimization. Experiments on synthetic and real-world datasets demonstrate selective routing under varying expert availability and reliability.

stat.ML↗

arXiv2026

More Expressive Feedforward Layers: Part I. Token-Adaptive Mixing of Activations

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

cs.LG↗

arXiv2026

Negligible in Size, Significant in Effect: On Scale Vectors in Large Language Models

Normalization layers in modern large language models (LLMs) consist of a deterministic normalization operation and a learnable scale vector. While the normalization operation has been extensively studied, the scale vector remains poorly understood despite its ubiquitous use. In this work, we present a systematic study of scale vectors in LLMs from the perspectives of expressivity, optimization, and architectural structure. First, we show empirically that although scale vectors constitute only a negligible fraction of model parameters, removing them substantially degrades LLM pre-training. Our theory further shows that, in Pre-Norm architectures, scale vectors do not increase expressivity; instead, they improve optimization through a self-amplifying preconditioning effect on subsequent linear mappings. Second, we investigate the role of weight decay for scale vectors. By distinguishing Input-Norm and Output-Norm layers, we theoretically show that weight decay is beneficial for the former but harmful for the latter, due to their distinct roles in optimization and expressivity. Third, motivated by this understanding, we propose three lightweight and complementary improvements to scale vectors: branch-specific heterogeneity, improved placement around linear mappings, and magnitude-direction reparameterization. Both theory and experiments show that each improvement yields consistent gains. Finally, we combine these improvements into a unified scale-vector strategy and evaluate it through extensive LLM pre-training experiments on dense and mixture-of-experts models ranging from 0.12B to 2B parameters, across multiple optimizers and learning rate schedules, under industrial-scale token budgets. The unified strategy consistently achieves lower terminal loss than well-tuned baselines and exhibits more favorable scaling behavior, while adding negligible parameter and computational overhead.

cs.LG↗

arXiv2026

Generalized Gibbs Ensemble Weighting for Forecast Combination

Forecast combination is a reliable way to improve predictive performance when several forecasting models are available. Simple aggregation rules such as the mean, median, trimmed mean, inverse-loss weighting, and exponential weighting are often strong baselines, but their relative performance can vary across datasets, forecast horizons, deployment settings, and levels of disagreement among base forecasters. We develop Generalized Gibbs Ensemble Weighting (GGEW), a probabilistic framework that treats forecasting models as experts and assigns ensemble weights using a Gibbs-style exponential transformation of normalized predictive loss. The framework extends this basic weighting rule through numerical stabilization, diversity-aware score corrections, and online hyperparameter adaptation. GGEW produces a family of related methods, including Stable Gibbs weighting, Directional Gibbs-NCL, and Symmetric Gibbs-NCL. These variants share one core algorithm and differ only in the score used inside the exponential weighting rule. For sequential deployment, we adopt a UCB-style bandit mechanism, called online Local-UCB, to adapt the learning rate, diversity strength, and Gibbs variant without evaluating the full hyperparameter grid at every prediction step. We evaluate GGEW on official M4 competition forecast submissions and external rolling-origin deployment experiments using Monash Traffic Hourly, Electricity Hourly, and Solar Weekly datasets. Results suggest that Gibbs-style adaptive weighting is a useful and competitive tool across several benchmark settings, although its relative performance varies across datasets, forecast horizons, deployment protocols, and forecast disagreement groups. The contribution is not a universal dominance claim, but a framework and empirical study motivating further investigation of when adaptive Gibbs-style forecast combination is useful.

cs.LG↗

arXiv2026

Conformal Risk-Averse Decision Making with Optimized Certainty Equivalent Risk Control

We study risk-averse decision making, in which an agent selects actions while being uncertain about the true system state. The risk is measured via optimized certainty equivalent (OCE) metrics, which generalize popular criteria such as mean-variance risk and conditional value-at-risk (CVaR). We characterize the optimal policy under known distributions, and show that it reduces to a prediction set-based solution for the CVaR. This provides an operational interpretation of conformal prediction-type prediction sets. For unknown distributions, we develop a data-driven calibration strategy, based on a synthetic model for the likelihood and held-out calibration data, yielding high-probability control of the OCE risk. The approach is evaluated on two wireless beamforming settings.

stat.ML↗

arXiv2026

I-FLOP: Fast Learning of Order and Parents from Interventional Data

We extend the FLOP (fast learning of order and parents) algorithm recently proposed by Wienöbst et al. (2026) from observational to interventional data. In particular, we use the interventional BIC score of Hauser and Bühlmann (2012), adapting it to be used with the iterative Cholesky-based score updates that are partly responsible for FLOP's speed. We show that, in the sample limit, I-FLOP recovers a DAG in the same interventional Markov equivalence class as the data-generating DAG. We compare I-FLOP to existing causal structure learning algorithms on real and simulated interventional data, where it performs favorably in terms of both performance and run time.

stat.ML↗

arXiv2026

Localizing Global Discrepancies: Marginal Contributions and Contextual Anomaly Detection

Global goodness-of-fit and discrepancy statistics can establish that a sample departs from a reference distribution without identifying which observations drive the departure. We develop a framework for this localization problem by assigning to each observation its conditional or marginal contribution across random statistical contexts. This connects resampling diagnostics and data valuation to projection theory and event-level anomaly detection. For symmetric statistics, fixed-size replacement is exactly equivalent to centered conditional localization. For U-statistics, the addition score equals the first Hoeffding/Hájek contribution; for smooth distributional functionals it is related at leading order to the influence function; and for unbiased known-background MMD it reduces exactly to the MMD witness. This viewpoint also yields more efficient estimators. Matched-context subtraction removes fluctuations unrelated to the observation, while for pairwise MMD the event-containing terms give a simple localizer. On the LHC Olympics anomaly-detection benchmark, the pair estimator converges to the direct empirical MMD witness with the predicted 1/(Rm^2) scaling, where m is batch size and R the number of batches. At m=1000 and R=5x106 it reaches correlation 0.9993 with essentially identical AUC. We also ask when context contains information beyond an event's own features. In a shared-latent toy model, the full single-event signal and background distributions are identical by construction, forcing isolated-event AUC=0.5. Discriminating information survives only in cross-event dependence induced by the shared latent parameter; the ensemble recovers this information, whereas an independent-latent control does not. This separates two roles of context: efficient localization of a global discrepancy and genuinely additional class information when the alternative contains shared structure.

stat.ML↗

arXiv2026

Generalized Splines and Gaussian Processes

For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.

math.ST↗

arXiv2026

Learning between the peaks: sharp asymptotics for kernel ridge regression under power-law anisotropy

We study kernel ridge regression under anisotropic Gaussian data, where the input covariance decays as a power law with exponent $α\geq 0$ for polynomial inner-product kernels. We derive asymptotically sharp expressions for the kernel spectrum and the generalization error in the polynomial high-dimensional regime $n=Θ(d^κ)$, revealing how anisotropy reshapes the learning curves. For weak anisotropy ($0<α<1$), the problem remains effectively high-dimensional and retains some features of the isotropic case, while departing from it in others: the variance still peaks at integer sample complexities $κ\in\mathbb{N}$, but these peaks are progressively damped as $α$ grows; meanwhile, for targets strongly aligned with the data's principal directions, the bias drops at fractional sample complexities, decoupling the bias transitions from the interpolation peaks. For strong anisotropy ($α> 1$), the effective dimension of the problem is constant, and the variance stops depending on sample size altogether, plateauing under ridgeless interpolation or vanishing at an explicit rate under fixed ridge penalty. The bias undergoes a sharp transition governed by the target's decay rate: below a threshold, learning is abrupt rather than gradual; above it, the bias decays as a power law that recovers the classical source and capacity rates. We finally specialize these results to single-index targets, showing how the alignment of the index with the data's principal directions determines the effect of anisotropy on learning. Together, our results clarify how the input geometry shapes the kernel features and fundamentally impacts its generalization properties.

stat.ML↗

arXiv2026

Learning a Size-Weight Frontier for Synthetic-Augmented Inference

Synthetic data can improve statistical inference when real data are scarce, but naively treating synthetic samples as real data can introduce bias and lead to unreliable inference. We develop a general framework for synthetic-augmented inference across a population of related tasks. It characterizes synthetic augmentation by the number of synthetic observations and their weight. Central to our framework is a size-weight frontier that specifies, for each weight, the largest synthetic sample size for which all smaller sizes attain the target task-marginal coverage. We estimate this frontier from historical tasks, and establish a finite-sample coverage guarantee simultaneously for all size-weight configurations on or below the estimated frontier. In experiments using large language model responses to augment opinion survey data, our procedure achieves target coverage and substantially narrows confidence intervals.

stat.ME↗

arXiv2026

Quantitative Target Convergence and Uniform-in-Time Propagation of Chaos for Langevin-Regularized SVGD

We establish quantitative convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein variational gradient descent. The Stein interaction need not be small relative to the confining Langevin drift and does not generally yield a contractive particle coupling. At the mean-field level, the Stein and Langevin components dissipate the same relative entropy in the kernel-induced Stein and $2$-Wasserstein geometries, producing the squared kernel Stein discrepancy and relative Fisher information. Under a log-Sobolev inequality for the target, this yields exponential last-iterate convergence. We also derive a finite-particle entropy identity relative to the product target, giving exponential-in-time convergence of the empirical measure up to polynomial sampling errors. For propagation of chaos, we develop two complementary finite-time approaches. A synchronous coupling, combined with exponential moment estimates for the nonlinear mean-field diffusion, yields explicit single-exponential bounds in Wasserstein distance and kernel Stein discrepancy (KSD). Moving-product entropy gives joint-law relative entropy control relative to the evolving mean-field product law and, through entropy superadditivity and concentration, fixed-marginal relative entropy and total variation bounds and empirical KSD estimates. Under an additional $T_2$ inequality for the initial law, it also yields Wasserstein bounds. Combining these finite-time estimates with target convergence at a logarithmic cutoff time gives polynomial uniform-in-time propagation of chaos rates in expectation for empirical KSD and $W_2^2$, and for fixed-marginal total variation and $W_2^2$. All bounds control the last iterate in physical time. We also compare the two finite-time mechanisms and identify regimes in which each gives the sharper polynomial exponent.

stat.ML↗

arXiv2026

Representation Learning with Quantum Signal Processing

Representation learning begins when training changes the features that define similarity between data. A frozen-kernel model only reweights a fixed geometry. We establish quantum signal processing (QSP) as a solvable quantum model of the representation-learning regime. At arbitrary depth, we compute the exact mean and variance of its quantum neural tangent kernel, revealing an input-dependent angular geometry whose diagonal remains non-self-averaging even when the underlying unitary approaches Haar randomness. We also prove a sparse-data guarantee for the full nonlinear gradient flow without freezing or ensemble-averaging the kernel: the realized dynamics converges to an integrable scalar flow with a time-dependent kernel closure and explicit convergence times. A finite-depth speed limit holds for every data set and trajectory. At higher data density, numerical results show coupled evolution beyond both the scalar and frozen-kernel descriptions. These results give a controlled theory of learned quantum data geometry with provable training dynamics beyond the frozen limit.

quant-ph↗

arXiv2026

The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal scheduling

We study a class of product-reference diffusion algorithms for sampling from a discrete distribution. We show that their sampling performance can be characterized using a path-based measure of data geometry that we call the interaction growth complexity (IGC). We show that a bivariate IGC kernel gives an exact representation of both the KL discretization error and a simple one-step upper bound. The simpler univariate IGC density can be used to study the effect of stepsize choices on the iteration complexity required to obtain $ε$-accurate samples in KL divergence. Samplers that traverse the path with equi-spaced steps in log-squared-reliability-odds have performance that depends on the aggregate IGC mass, whereas refined choices of stepsizes have a lower complexity depending on a square-root functional. In the fine-grid limit, both of these characterizations become sharp. We also allow general product reference distributions and show that the reference law can substantially reshape the IGC profile and the resulting sampling complexity; in particular, references far from both the uniform and the data marginals can yield dimension-dependent improvements. Finally, the aggregate IGC mass admits bounds in terms of total correlation and dual total correlation, thereby connecting the pathwise geometry to classical measures of multivariate dependence.

stat.ML↗

arXiv2026

Prequential posteriors

Data assimilation is a fundamental task in updating forecasting models upon observing new data, with applications ranging from weather prediction to online reinforcement learning. Deep generative forecasting models (DGFMs) have shown excellent performance in these areas, but assimilating data into such models is challenging due to their intractable likelihood functions. This limitation restricts the use of standard Bayesian data assimilation methodologies for DGFMs. To overcome this, we introduce prequential posteriors, based upon a predictive-sequential (prequential) loss function; an approach naturally suited for temporally dependent data which is the focus of forecasting tasks. Since the true data-generating process often lies outside the assumed model class, we adopt an alternative notion of consistency and prove that, under mild conditions, both the prequential loss minimizer and the prequential posterior concentrate around parameters with optimal predictive performance. For scalable inference, we employ easily parallelizable wastefree sequential Monte Carlo (SMC) samplers with preconditioned gradient-based kernels, enabling efficient exploration of high-dimensional parameter spaces such as those in DGFMs. We validate our method on both a synthetic multi-dimensional time series and a real-world meteorological dataset; highlighting its practical utility for data assimilation for complex dynamical systems.

stat.ML↗

arXiv2026

Trajectory balance: Improved credit assignment in GFlowNets

Generative flow networks (GFlowNets) are a method for learning a stochastic policy for generating compositional objects, such as graphs or strings, from a given unnormalized density by sequences of actions, where many possible action sequences may lead to the same object. We find previously proposed learning objectives for GFlowNets, flow matching and detailed balance, which are analogous to temporal difference learning, to be prone to inefficient credit propagation across long action sequences. We thus propose a new learning objective for GFlowNets, trajectory balance, as a more efficient alternative to previously used objectives. We prove that any global minimizer of the trajectory balance objective can define a policy that samples exactly from the target distribution. In experiments on four distinct domains, we empirically demonstrate the benefits of the trajectory balance objective for GFlowNet convergence, diversity of generated samples, and robustness to long action sequences and large action spaces.

cs.LG↗

arXiv2026

Let the Flows Tell: Solving Graph Combinatorial Optimization Problems with GFlowNets

Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space. On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates. In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space. Efficient training techniques are also developed to benefit long-range credit assignment. Through extensive experiments on a variety of different CO tasks with synthetic and realistic data, we demonstrate that GFlowNet policies can efficiently find high-quality solutions. Our implementation is open-sourced at https://github.com/zdhNarsil/GFlowNet-CombOpt.

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