Transition-related financial markets are increasingly exposed to abrupt repricing episodes, elevated volatility, and heterogeneous macro-financial shocks. Under such conditions, conventional Gaussian-linear forecasting frameworks may provide an incomplete representation of the dependence structure linking fossil-energy, renewable-energy, technology, and utility-sector assets. This paper investigates whether transition-related financial returns exhibit residual non-linear predictability after controlling for heavy-tailed multivariate linear dynamics. To address this question, we develop a hybrid forecasting framework combining Student-t Vector Autoregressions with nonlinear recurrent residual learning architectures. The empirical analysis considers six major exchange-traded funds representing broad equity markets and key transition-sensitive sectors. The results reveal substantial departures from Gaussian-linear behavior, including excess kurtosis, volatility clustering, and remaining nonlinear dependence after econometric filtering. Out-of-sample forecasting experiments show that the proposed framework consistently improves predictive accuracy relative to conventional VAR models, standalone machine-learning methods, and alternative hybrid specifications. The forecasting gains become more pronounced during periods of macro-financial stress, particularly during the COVID-19 crisis and the Ukraine-related energy shock. Overall, the findings suggest that transition-related financial systems exhibit regime-sensitive and heavy-tailed predictive dynamics that are insufficiently captured by standard Gaussian-linear models alone.

Maintaining predictive accuracy in non-stationary environments requires online model selection to adapt autonomously to unknown distribution shifts. However, existing tuning-free algorithms face a fundamental trade-off between robustness and agility. Specifically, to ensure dynamic regret bounds, they must restrict learning rates to small constants (e.g., $O(1)$). This restriction inevitably causes significant adaptation lag during abrupt changes. To resolve this, we propose a novel optimistic online mirror descent that utilizes safeguarded large learning rates up to $Θ(T)$, where $T$ is the number of rounds. Our key technical contribution is a post-hoc penalty mechanism that dynamically monitors unstable updates and excludes learning rates incurring excessive regret, eliminating the need for restrictive a priori constraints. We show that the cumulative penalty remains $O(\log T)$, allowing our algorithm to match near-optimal worst-case guarantees while achieving superior rates in benign cases. Empirical evaluations on synthetic and eleven diverse real-world datasets demonstrate that our approach reduces the adaptation lag from hundreds of rounds to a few rounds, consistently outperforming tuning-free baselines.

Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise process. We show that the signal-to-noise ratio (SNR) and the training sample size influence the alignment in qualitatively similar ways in networks trained on real-world datasets and in an extremely simple $linear$ network with a single hidden layer, for which the alignment can be estimated analytically. Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error. These findings reveal a non-trivial dependence of alignment on data quality and quantity, decoupled from generalization performance.

Bayesian experimental design (BED) is a principled framework for data-efficient design of sequential experiments. However, existing BED methods are unable to adapt to dynamic constraints inherent in real-world tasks due to budget limitations, varying costs, or physical constraints that restrict how designs evolve over time. In this paper, we introduce a novel approach to BED that enables constrained optimization of experimental designs by combining offline pre-training of an amortized policy and a posterior network with online multi-step lookahead planning using scenario trees. We empirically demonstrate that our method yields substantially more informative design sequences than existing methods across a range of constrained BED tasks, while incurring only a modest additional computational overhead.

Sampling from learned high-dimensional distributions is a foundational computational problem. We introduce U-turn chains: Markov chains obtained by iterating short forward-backward steps of a diffusion model, in which each step proposes a move that remains on the learned data manifold and, paired with a Metropolis-Hastings correction, samples from energy-modified targets. For synthetic languages, we show that minimal U-turn dynamics undergoes an ergodicity-breaking phase transition driven by fragmentation of the data manifold; ergodicity is restored at larger U-turn magnitude. In the non-ergodic regime, low-level features relax faster than high-level ones, an ordering that inverts only at sufficiently large U-turn magnitude. We test these predictions on natural language and natural images. In both modalities, minimal U-turns relax slowly, especially for high-level features approximated by deep representations in CNNs or LLMs. The layer-ordering inversion appears only at large noise when mixing is efficient -- signatures consistent with strongly constrained, weakly mixing local dynamics. We discuss the implications of these results for sampling with diffusion models.

Large language models (LLMs) are prone to hallucinations, i.e., statements unsupported by the input or training data, hindering reliable deployment. In parallel, numerous uncertainty estimation (UE) methods have been proposed to quantify model confidence and are often implicitly treated as proxies for model failure. However, the relationship between uncertainty and hallucinations remains insufficiently characterized. We present a systematic empirical study of the association between uncertainty estimators and hallucinations in LLMs. Rather than assuming this association, we evaluate directly when and to what extent it holds. We consider a diverse set of uncertainty estimators, including information-theoretic, sampling-based, and reflexive estimators, and examine their behavior across hallucination settings. Our experiments cover both intrinsic hallucinations (violations of input faithfulness) and extrinsic hallucinations (unsupported claims relative to training data), using four complementary benchmarks, including RAGTruth and HalluLens. We find that the association is highly variable and often weak, depending on the hallucination type and the LLM under evaluation. These results challenge the use of uncertainty as a direct signal of hallucination and clarify when it provides actionable information.

Gaussian Process (GP) models are widely used as probabilistic models for nonlinear functions because they combine flexible function modelling with uncertainty quantification (Rasmussen and Williams, 2006; Williams, 1998; MacKay, 1992; Neal, 1995). Their predictive performance depends heavily on how kernel hyperparameters are learnt (Sundararajan and Keerthi, 2001). This becomes especially important in higher-dimensional multivariate settings, where many input-specific hyperparameters may be present and where only some inputs may contribute meaningful predictive structure (MacKay, 1992; Neal, 1995; Rasmussen and Williams, 2006; Linkletter et al., 2006; Paananen et al., 2019). In standard Bayesian formulations of GP learning, prior specification is usually imposed directly on kernel hyperparameters such as lengthscales, amplitude parameters, and noise terms (Rasmussen and Williams, 2006; Williams, 1998). This is natural from a modelling point of view, but it does not always give useful control over the covariance structure that those hyperparameters induce over the observed design points (Barnard et al., 2000; Gelman, 2006; Daniels and Kass, 1999; Huang and Wand, 2013). However, it is this induced covariance matrix that directly governs likelihood evaluation, numerical stability, and predictive behaviour (Rasmussen and Williams, 2006; Stein, 1999). 1

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as $m \gtrsim \log(n)$, where $m$ is the network width and $n$ is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared $\ell_2$-norm scaling as $\sqrt{n}$, which is within a constant factor of the minimal $\ell_2$-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.

AsInstrategic classification, aninstitution(e.g., a bank) anticipates adaptation from userswe develop better algorithms under varying assumpwho change their features to increase utilitytions about adaptation (Levanon and Rosenfeld, 2022; in a classification task (e.g., loan repayment). Kleinberg and Raghavan, 2018), there are growing Since a key challenge is the distribution shiftconcerns about negative social impact on the agents who adapt to these systems, whether outcomes areinduced by users, we turn to causal models, which have been shown to bound the worst-static (Milli et al., 2019) or dynamic (G ois et al., case out-of-distribution (OOD) risk, and es-2025). When agents adapt, depending on the untablish several new results that link causal-derlying causal model (Horowitz and Rosenfeld, 2018; ity and strategic classification. First, we Miller et al., 2020), some changes improve agent outcomes while others constitute gaming the classifier,show that causal classification leads to optimal classification error after any sufficientlyworsening classification error. In this paper, we study large adaptation, when the noise is boundedwhether classifiers can maintain accuracy without sacin a certain way. Second, when these as-rificing alignment with predicted agent's goals.

Collaborative analysis of decentralized confidential datasets is important, but direct sharing of original datasets is often restricted by privacy and institutional constraints. Data collaboration (DC) analysis transforms each dataset into privacy-preserving intermediate representations via party-specific obfuscation functions and integrates them into common collaboration representations using an anchor dataset. However, many existing DC analysis methods rely on linear transformations for data obfuscation and integration, which may increase reconstruction risk. Although nonlinear dimensionality reduction can mitigate this risk, conventional linear integration methods cannot accurately align intermediate representations produced by nonlinear transformations. Moreover, existing integration methods mainly minimize discrepancies among parties and do not explicitly incorporate geometric or target-variable information useful for downstream analysis. To overcome these limitations, we first formulate linear kernel integration (LKI) as a linear integration method and then kernelize it to obtain nonlinear kernel integration (NKI). NKI admits a globally optimal solution via kernel ridge regression and an eigenvalue problem. We also introduce graph regularization and a centering constraint so that the target representation can capture geometric and target-variable information useful for downstream analysis. Experiments on image classification tasks demonstrate that NKI improves classification accuracy over existing linear integration methods under nonlinear dimensionality reduction, with further gains from target-variable-aware graph regularization and centering. The results also show that dimensionality reduction choices substantially affect both classification accuracy and reconstruction risk.