Directed acyclic graphs (DAGs) constitute a central modeling tool to enable principled reasoning about cause-effect interactions in complex systems. However, since the causal structure underlying a group of variables is often unknown and interventions may be infeasible or ethically challenging to implement, there is a need to address the task of inferring DAGs from observational data. However, most classical structure identification approaches face two key obstacles: the combinatorial challenge of enforcing acyclicity, which severely limits scalability, and identifiability challenges arising from latent confounding or heterogeneous noise. This tutorial offers an overview of recent signal processing and optimization advances that address these issues by recasting DAG structure learning as a continuous, score-based estimation problem over adjacency matrices. We begin with a didactic introduction to structural equation models and the formulation of causal graph recovery, followed by a historical survey of score-based methods ranging from early combinatorial search schemes and greedy heuristics to modern continuous frameworks that leverage smooth characterizations of acyclicity. Building on this foundation, we describe concomitant DAG estimation methods that jointly infer sparse causal structure and exogenous noise levels, improving robustness under heteroscedasticity and distribution shifts by rendering the estimator noise adaptive. All in all, the tutorial introduces readers to challenges and opportunities for signal processing research at the crossroads of causal inference, high-dimensional statistics, and scalable graph learning, while outlining emerging directions including online, nonlinear, and neural causal discovery.

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.

Traditional neural networks provide deterministic predictions without inherent uncertainty estimates. While Bayesian Neural Networks (BNNs) offer a principled approach to uncertainty quantification, their computational complexity limits scalability. Monte Carlo (MC) Dropout, initially introduced as a regularization technique, has been shown to approximate Bayesian inference by enabling probabilistic modeling through multiple stochastic forward passes. In this work, we enhance uncertainty estimation in deep learning by integrating a Dirichlet-based framework within MC Dropout. Specifically, we leverage the formulation proposed by Sensoy et al. (2018), where class probabilities are modeled using a Dirichlet distribution, allowing for a more informative uncertainty representation. The proposed approach maintains the computational efficiency of MC Dropout while improving the quality of uncertainty estimates. We discuss the theoretical foundations of our method and compare it with existing uncertainty quantification techniques. The results highlight the effectiveness of the proposed method in producing well-calibrated uncertainty estimates, offering a practical solution for uncertainty-aware deep learning models.

We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples. In particular, we prove $k^2/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_2/k$ regularization, and $k/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_1/k$ regularization. For $\|\cdot\|_2^2/k$ regularization, the sampling complexity depends mainly on a bounded derivative property: if $|g'(x)|\leq g(x)$, and $g(0)>0$, and $g$ is monotonic or convex, then it admits linear in $k$ sampling complexity; otherwise the general bound is $k^2/\varepsilon^2$. However, if $g(0)=0$, our results indicate that no dimension-free bounds are possible, and even sublinear bounds are ruled out. All upper bounds are complemented by matching lower bounds up to polylogarithmic terms. Moreover, our work relies conceptually and algorithmically on simple uniform or (squared) norm sampling and hereby improves over recent cubic $k^3/\varepsilon^2$ sensitivity sampling bounds of (Alishahi and Phillips, ICML'24). This is achieved by refined arguments involving higher moment bounds and empirical process analyses to avoid overcounting that appears in the de-facto standard VC-dimension and sensitivity framework.

Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.

We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.

We consider the problem of sampling from an unnormalized Boltzmann/ Gibbs density, π(θ) exp V(θ),θ Θ Rd, where the normalization constant is unknown (and/or intractable) and only the potential function V (and typically its derivatives) can be evaluated. This problem arises across various domains in Bayesian inference, statistical physics, and modern machine learning. A common variational perspective on sampling is to characterize the target distribution π as the unique minimizer of a functional (typically a divergence functional) over the space of probability measures. From this viewpoint, sampling can be formulated as evolving an initial distribution ρ0 toward π via the gradient flow of this functional under a suitable geometric structure on the space of probability measures. In this paper, we focus on a gradient flow based sampling methodology built from the spherical Hellinger Kantorovich (SHK), also known as the Wasserstein Fisher Rao (WFR), geometry on the space of probability measures (Kondratyev and Vorotnikov, 2019; Liero et al., 2018; Chizat et al., 2015). When the variational objective is the exclusive KL divergence ρ 7 KL(ρ π), the SHK gradient flow generates a time-indexed family of marginals {ρt}t 0 (initialized at ρ0 P2(Θ)) that evolves according to the continuity reaction equation (4). This evolution is equivalent to the birth-death Langevin dynamics introduced in Lu et al. (2019) .

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