If had debuted this year, William Shakespeare might have been accused of writing it with AI. A certain suspicious rhetorical device appears again and again in the play. It's in Act I, Scene ii: "The fault, dear Brutus, is not in our stars, but in ourselves." In Act III, Scene ii: "Not that I loved Caesar less, but that I loved Rome more." And later in that same scene: "I come to bury Caesar, not to praise him."

Richard Sutton's "Bitter Lesson" is usually read as a warning against building too much human knowledge into AI systems. Over the long run, the methods that win are not the ones that encode our clever intuition most directly, but the ones that scale: search, learning, and other general methods that can absorb more compute and data. We take a general architecture, expose it to massive data, and train it with a simple self-supervised objective. Language models predict the next token. Vision models reconstruct masked patches, align views, or match teacher representations.

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

Polynomial Stein discrepancies (PSD) provide a scalable alternative to kernel Stein methods for measuring sample quality and goodness-of-fit testing, but their statistical properties remain poorly understood. We show that increasing polynomial degree primarily amplifies signal without adequately controlling variance, rather than directly optimising the signal-to-noise ratio (SNR). Under suitable assumptions, this might lead to a failure mode in which the $\text{SNR}^2$ can provably decay exponentially with polynomial degree. Motivated by this observation, we reformulate Stein discrepancy construction as an explicit $\text{SNR}^2$ maximisation problem, yielding a Rayleigh quotient over Stein features. This perspective motivates $λ$-PSD, an approximate scalable covariance-aware reweighting scheme defined in a low-dimensional subspace. Under Gaussian settings, we show that $λ$-PSD avoids the exponential $\text{SNR}^2$ collapse and achieves a stable $\text{SNR}^2$. Empirically, $λ$-PSD substantially improves test power while retaining linear-time complexity in the number of samples, highlighting the importance of SNR-aware design for scalable Stein discrepancies.

Classification and Regression Trees (CART) constitute one of the most influential paradigms in statistical learning. Although a variety of impurity measures have been proposed for different statistical models, these criteria are typically introduced on a case-by-case basis and analyzed separately. In this paper, we study CART through the lens of Bregman divergences. This perspective places the classical least-squares criterion, Poisson deviance, Kullback-Leibler-type losses, and other impurity measures associated with exponential-family models within a common framework. As a result, key ingredients of the CART methodology -- including node representatives, impurity measures, and split selection rules -- can be expressed and analyzed through general properties of convex functions rather than through separate model-specific constructions. Beyond the algorithmic formulation, we investigate theoretical properties of Bregman-based CART procedures. In particular, we analyze how geometric properties of the generating convex function influence impurity reductions and stability of recursive partitions. We also establish consistency results within the proposed framework, providing a unified theoretical treatment for a broad family of CART type procedures. Our results provide a geometric interpretation of impurity-based tree construction and show that many classical CART impurity criteria admit a common interpretation within a Bregman framework.

We study the problem of estimating the arrival times of vertices in a uniform random recursive tree from its unlabeled structure. We adopt a pointwise perspective and analyze the distribution of the relative estimation error, and derive tail bounds that are uniform in both the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability of underestimating the arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation tail decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results reveal a tradeoff between upper- and lower-tail performance in arrival-time estimation that is invisible to the previously studied risk functional. Nevertheless, the refined centrality measure attains the optimal order of the risk for all its parameter values.

Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., α-helices, β-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation.