Local prediction-error-based curiosity rewards focus on the current transition without considering the world model's cumulative prediction error across all visited transitions. We introduce Curiosity-Critic, which grounds its intrinsic reward in the improvement of this cumulative objective, and show that it reduces to a tractable per-step form: the difference between the current prediction error and the asymptotic error baseline of the current state transition. We estimate this baseline online with a learned critic co-trained alongside the world model; regressing a single scalar, the critic converges well before the world model saturates, redirecting exploration toward learnable transitions without oracle knowledge of the noise floor. The reward is higher for learnable transitions and collapses toward the baseline for stochastic ones, effectively separating epistemic (reducible) from aleatoric (irreducible) prediction error online. Prior prediction-error curiosity formulations, from Schmidhuber (1991) to learned-feature-space variants, emerge as special cases corresponding to specific approximations of this baseline. Experiments on a stochastic grid world show that Curiosity-Critic outperforms prediction-error and visitation-count baselines in convergence speed and final world model accuracy.

The low-degree polynomial framework has emerged as a powerful tool for providing evidence of statistical-computational gaps in high-dimensional inference. For detection problems, the standard approach bounds the low-degree advantage through an explicit orthonormal basis. However, this method does not extend naturally to estimation tasks, and thus fails to capture the \emph{detection-recovery gap phenomenon} that arises in many high-dimensional problems. Although several important advances have been made to overcome this limitation \cite{SW22, SW25, CGGV25+}, the existing approaches often rely on delicate, model-specific combinatorial arguments. In this work, we develop a general approach for obtaining \emph{conditional computational lower bounds} for recovery problems from mild bounds on low-degree testing advantage. Our method combines the notion of algorithmic contiguity in \cite{Li25} with a cross-validation reduction in \cite{DHSS25} that converts successful recovery into a hypothesis test with lopsided success probabilities. In contrast to prior unconditional lower bounds, our argument is conceptually simple, flexible, and largely model-independent. We apply this framework to several canonical inference problems, including planted submatrix, planted dense subgraph, stochastic block model, multi-frequency angular synchronization, orthogonal group synchronization, and multi-layer stochastic block model. In the first three settings, our method recovers existing low-degree lower bounds for recovery in \cite{SW22, SW25} via a substantially simpler argument. In the latter three, it gives new evidence for conjectured computational thresholds including the persistence of detection-recovery gaps. Together, these results suggest that mild control of low-degree advantage is often sufficient to explain computational barriers for recovery in high-dimensional statistical models.

Empirical studies of trained models often report a transient regime in which signal is detectable in a finite gradient descent time window before overfitting dominates. We provide an analytically tractable random-matrix model that reproduces this phenomenon for gradient flow in a linear teacher--student setting. In this framework, learning occurs when an isolated eigenvalue separates from a noisy bulk, before eventually disappearing in the overfitting regime. The key ingredient is anisotropy in the input covariance, which induces fast and slow directions in the learning dynamics. In a two-block covariance model, we derive the full time-dependent bulk spectrum of the symmetrized weight matrix through a $2\times 2$ Dyson equation, and we obtain an explicit outlier condition for a rank-one teacher via a rank-two determinant formula. This yields a transient Baik-Ben Arous-Péché (BBP) transition: depending on signal strength and covariance anisotropy, the teacher spike may never emerge, emerge and persist, or emerge only during an intermediate time interval before being reabsorbed into the bulk. We map the corresponding phase diagrams and validate the theory against finite-size simulations. Our results provide a minimal solvable mechanism for early stopping as a transient spectral effect driven by anisotropy and noise.

As deep neural networks are deployed in safety-critical domains such as autonomous driving and medical diagnosis, stakeholders need explanations that are interpretable but also trustworthy with formal guarantees. Existing XAI methods fall short: heuristic attribution techniques (e.g., LIME, Integrated Gradients) highlight influential features but offer no mathematical guarantees about decision boundaries, while formal methods verify robustness yet remain untargeted, analyzing the nearest boundary regardless of whether it represents a critical risk. In safety-critical systems, not all misclassifications carry equal consequences; confusing a "Stop" sign for a "60 kph" sign is far more dangerous than confusing it with a "No Passing" sign. We introduce ViTaX (Verified and Targeted Explanations), a formal XAI framework that generates targeted semifactual explanations with mathematical guarantees. For a given input (class y) and a user-specified critical alternative (class t), ViTaX: (1) identifies the minimal feature subset most sensitive to the y->t transition, and (2) applies formal reachability analysis to guarantee that perturbing these features by epsilon cannot flip the classification to t. We formalize this through Targeted epsilon-Robustness, certifying whether a feature subset remains robust under perturbation toward a specific target class. ViTaX is the first method to provide formally guaranteed explanations of a model's resilience against user-identified alternatives. Evaluations on MNIST, GTSRB, EMNIST, and TaxiNet demonstrate over 30% fidelity improvement with minimal explanation cardinality.

In-context learning enables transformers to adapt to new tasks from a few examples at inference time, while grokking highlights that this generalization can emerge abruptly only after prolonged training. We study task generalization and grokking in in-context learning using a Bayesian perspective, asking what enables the delayed transition from memorization to generalization. Concretely, we consider modular arithmetic tasks in which a transformer must infer a latent linear function solely from in-context examples and analyze how predictive uncertainty evolves during training. We combine approximate Bayesian techniques to estimate the posterior distribution and we study how uncertainty behaves across training and under changes in task diversity, context length, and context noise. We find that epistemic uncertainty collapses sharply when the model groks, making uncertainty a practical label-free diagnostic of generalization in transformers. Additionally, we provide theoretical support with a simplified Bayesian linear model, showing that asymptotically both delayed generalization and uncertainty peaks arise from the same underlying spectral mechanism, which links grokking time to uncertainty dynamics.

High-dimensional and complex discrete distributions often exhibit multimodal behavior due to inherent discontinuities, posing significant challenges for sampling. Gradient-based discrete samplers, while effective, frequently become trapped in local modes when confronted with rugged or disconnected energy landscapes. This limits their ability to achieve adequate mixing and convergence in high-dimensional multimodal discrete spaces. To address these challenges, we propose \emph{Hyperbolic Secant-squared Gibbs-Sampling (HiSS)}, a novel family of sampling algorithms that integrates a \emph{Metropolis-within-Gibbs} framework to enhance mixing efficiency. HiSS leverages a logistic convolution kernel to couple the discrete sampling variable with the continuous auxiliary variable in a joint distribution. This design allows the auxiliary variable to encapsulate the true target distribution while facilitating easy transitions between distant and disconnected modes. We provide theoretical guarantees of convergence and demonstrate empirically that HiSS outperforms many popular alternatives on a wide variety of tasks, including Ising models, binary neural networks, and combinatorial optimization.

We analyze the score field of a diffusion generative model through a Burgers-type evolution law. For VE diffusion, the heat-evolved data density implies that the score obeys viscous Burgers in one dimension and the corresponding irrotational vector Burgers system in $\R^d$, giving a PDE view of \emph{speciation transitions} as the sharpening of inter-mode interfaces. For any binary decomposition of the noised density into two positive heat solutions, the score separates into a smooth background and a universal $\tanh$ interfacial term determined by the component log-ratio; near a regular binary mode boundary this yields a normal criterion for speciation. In symmetric binary Gaussian mixtures, the criterion recovers the critical diffusion time detected by the midpoint derivative of the score and agrees with the spectral criterion of Biroli, Bonnaire, de~Bortoli, and Mézard (2024). After subtracting the background drift, the inter-mode layer has a local Burgers $\tanh$ profile, which becomes global in the symmetric Gaussian case with width $σ_τ^2/a$. We also quantify exponential amplification of score errors across this layer, show that Burgers dynamics preserves irrotationality, and use a change of variables to reduce the VP-SDE to the VE case, yielding a closed-form VP speciation time. Gaussian-mixture formulas are verified to machine precision, and the local theorem is checked numerically on a quartic double-well.

Since the random language model was proposed by E. DeGiuli [Phys. Rev. Lett. 122, 128301], language models have been investigated intensively from the viewpoint of statistical mechanics. Recently, the existence of a Berezinskii--Kosterlitz--Thouless transition was numerically demonstrated in models with long-range interactions between symbols. In statistical mechanics, it has long been known that long-range interactions can induce phase transitions. Therefore, it has remained unclear whether phase transitions observed in language models originate from genuinely linguistic properties that are absent in conventional spin models. In this study, we construct a random language model with short-range interactions and numerically investigate its statistical properties. Our model belongs to the class of context-sensitive grammars in the Chomsky hierarchy and allows explicit reference to contexts. We find that a phase transition occurs even when the model refers only to contexts whose length remains constant with respect to the sentence length. This result indicates that finite-temperature phase transitions in language models are genuinely induced by the intrinsic nature of language, rather than by long-range interactions.

We propose a topological framework for the detection of Hopf bifurcations directly from time series, based on persistent homology applied to phase space reconstructions via Takens embedding within the framework of Topological Data Analysis. The central idea is that changes in the dynamical regime are reflected in the emergence or disappearance of a dominant one-dimensional homological features in the reconstructed attractor. To quantify this behavior, we introduce a simple and interpretable scalar topological functional defined as the maximum persistence of homology classes in dimension one. This functional is used to construct a computable criterion for identifying critical parameters in families of dynamical systems without requiring knowledge of the underlying equations. The proposed approach is validated on representative systems of increasing complexity, showing consistent detection of the bifurcation point. The results support the interpretation of dynamical transitions as topological phase transitions and demonstrate the potential of topological data analysis as a model-free tool for the quantitative analysis of nonlinear time series.

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $γ$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $γ$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.