Graphical models have been widely used as parsimonious encoders of assumptions of the underlying causal system and provide a basis for causal inferences. Models encoding stronger constraints tend to require higher expressive power, which are also harder, and sometimes impossible to empirically falsify. In this paper, we introduce two new collections of distributions that include counterfactual quantities which are experimentally accessible under counterfactual randomizations. Correspondingly, we define two new classes of graphical models for encoding empirically testable constraints in these distributions. We further present a sound and complete calculus, based on counterfactual calculus, which licenses inferences in these two new models with rules that are within the empirically falsifiable boundary. Finally, we formulate a hierarchy over several graphical models based on the constraints they encode and study the fundamental trade-off between the expressive power and empirical falsifiability of different models across the hierarchy.

In explainable AI, DNN gradients are used to interpret the prediction; in safetycritical control systems, gradients could encode safety constraints; in scientificcomputing applications, gradients could encode physical invariants. While recent work on provable editing of DNNs has focused on input-output constraints, the problem of enforcing hard constraints on DNN gradients remains unaddressed. We present ProGrad, the first efficient approach for editing the parameters of a DNN to provably enforce hard constraints on the DNN gradients.

Transformers have emerged as a powerful neural network architecture capable of tackling a wide range of learning tasks. In this work, we provide a theoretical analysis of their ability to automatically extract structure from data in an unsupervised setting. In particular, we demonstrate their suitability for clustering when the input data is generated from a Gaussian mixture model. To this end, we study a simplified two-head attention layer and define a population risk whose minimization with unlabeled data drives the head parameters to align with the true mixture centroids. This phenomenon highlights the ability of attention-based layers to capture underlying distributional structure. We further examine an attention layer with key, query, and value matrices fixed to the identity, and show that, even without any trainable parameters, it can perform in-context quantization, revealing the surprising capacity of transformer-based methods to adapt dynamically to input-specific distributions.

Concavity and its refinements underpin tractability in multiplayer games, where players independently choose actions to maximize their own payoffs which depend on other players' actions. In concave games, where players' strategy sets are compact and convex, and their payoffs are concave in their own actions, strong guarantees follow: Nash equilibria always exist and decentralized algorithms converge to equilibria. If the game is furthermore monotone, an even stronger guarantee holds: Nash equilibria are unique under strictness assumptions. Unfortunately, we show that certifying concavity or monotonicity is NP-hard, already for games where utilities are multivariate polynomials and compact, convex basic semialgebraic strategy sets--an expressive class that captures extensive-form games with imperfect recall. On the positive side, we develop two hierarchies of sum-of-squares programs that certify concavity and monotonicity of a given game, and each level of the hierarchies can be solved in polynomial time. We show that almost all concave/monotone games are certified at some finite level of the hierarchies. Subsequently, we introduce the classes of SOS-concave/monotone games, which globally approximate concave/monotone games, and show that for any given game we can compute the closest SOS-concave/monotone game in polynomial time. Finally, we apply our techniques to canonical examples of extensiveform games with imperfect recall.

Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of $O((n+n_e)T^{2/3})$ in general and $O((n+n_e)\sqrt{T})$ under a low-noise condition, where $T$ is the time horizon, $n$ is the number of labels, and $n_e$ is the number of distinct experts observed across rounds. The analysis builds on novel $\mathcal{H}$-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.

Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a single trajectory of the slow component, while the fast dynamics remain unobserved, making statistical learning challenging. Approaches based on partial differential equations (PDE), such as Fokker-Planck formulations, aim to characterize the evolution of probability densities, typically requiring dense space-time data or grid-based solvers. In contrast, we adopt a trajectory-based perspective and develop a data-driven framework for learning effective stochastic dynamics from a single observed path. We model the dynamics by coupled multiscale stochastic differential equations (SDEs) and first obtain a principled model reduction through stochastic averaging. Unlike generic model reduction techniques such as PCA, this respects the dynamical structure of the original system and explicitly incorporates the interaction between slow and fast scales. A central challenge, however, is that the reduced model depends on the invariant distribution of the fast process, which is a solution to an intractable and often unknown PDE. We introduce a novel learning framework that parameterizes the invariant distribution using normalizing flows, enabling expressive density modeling in the latent fast-variable space. The flow is trained end-to-end by optimizing a penalized likelihood objective induced by the reduced stochastic dynamics. Furthermore, we develop a Bayesian variational inference procedure for uncertainty quantification, employing a second normalizing flow to approximate the posterior distribution over model parameters. This yields a scalable approach to capturing epistemic uncertainty in multiscale systems.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

We compare in-context learning with fixed queries and agentic learning with adaptive queries for uniform approximation of task families. We consider two settings: an unrestricted regime, where querying and approximation are arbitrary functions, and a realizable regime, where we require these operations to be implemented by ReLU neural networks. In both settings, adaptivity never hinders approximation performance. However, this advantage can change when one passes from the unrestricted regime to the realizable regime. We identify four distinct approximation scenarios, each witnessed by an explicit task family: (a) no advantage of adaptivity; (b) an advantage in the unrestricted regime that persists under ReLU realizability; (c) an advantage that arises only under realizability; and (d) an advantage that disappears under realizability. This demonstrates that representational constraints interact profoundly with the effect of adaptivity.

Robust uncertainty quantification is increasingly important in modern data analysis and is often formalized under Huber's model, which allows an $\varepsilon$-fraction of arbitrary corruptions. In many experimental sciences, however, the measurement protocol is well controlled, and contamination is more plausibly introduced upstream. Motivated by this noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $θ$ in Efron's Gaussian two-groups model, where an unknown fraction $\varepsilon$ of observations have arbitrarily shifted means, but all samples share the same law of additive Gaussian measurement noise with variance $σ^2$. We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown contamination proportion and all noise-oblivious adversaries. Although prior work has shown that the minimax point estimation rate of theta does not deteriorate when $\varepsilon$ becomes unknown, our results reveal that, with a given $σ^2$, the minimax-optimal length of confidence intervals that are adaptive to unknown $\varepsilon$ is of order $σ(n^{-1/4}+\varepsilon^{1/2}/\max\{1, \log(en \varepsilon^2)\}^{1/2})$, which is polynomially worse than the optimal length when $\varepsilon$ is known. When the variance $σ^2$ is also unknown, we show a further degradation: no adaptive confidence interval can be shorter than $Ω(σn^{-1/8})$. Algorithmically, we introduce a Fourier-based certification procedure built on Carathéodory's positive-semidefiniteness constraints. By scanning candidate points and accepting those whose residual characteristic function is certifiably consistent with a Gaussian location mixture, our algorithm attains the minimax lower bound in the known-variance setting and is computable in polynomial time.

We present a PAC-Bayes-style generalization bound which enables the replacement of the KL-divergence with a variety of Integral Probability Metrics (IPM). We provide instances of this bound with the IPM being the total variation metric and the Wasserstein distance. A notable feature of the obtained bounds is that they naturally interpolate between classical uniform convergence bounds in the worst case (when the prior and posterior are far away from each other), and improved bounds in favorable cases (when the posterior and prior are close). This illustrates the possibility of reinforcing classical generalization bounds with algorithm-and data-dependent components, thus making them more suitable to analyze algorithms that use a large hypothesis space.