Rapidly evolving market conditions call for real-time risk monitoring, but its online estimation remains challenging. In this paper, we study the online estimation of one of the most widely used risk measures, Value at Risk (VaR). Its accurate and reliable estimation is essential for timely risk control and informed decision-making. We propose to use the quantile regression forest in the offline-simulation-online-estimation (OSOA) framework. Specifically, the quantile regression forest is trained offline to learn the relationship between the online VaR and risk factors, and real-time VaR estimates are then produced online by incorporating observed risk factors. To further ensure reliability, we develop a conformalized estimator that calibrates the online VaR estimates. To the best of our knowledge, we are the first to leverage conformal calibration to estimate real-time VaR reliably based on the OSOA formulation. Theoretical analysis establishes the consistency and coverage validity of the proposed estimators. Numerical experiments confirm the proposed method and demonstrate its effectiveness in practice.

Importance weighted variational inference (VI) approximates densities known up to a normalizing constant by optimizing bounds that tighten with the number of Monte Carlo samples $N$. Standard optimization relies on reparameterized gradient estimators, which are well-studied theoretically yet restrict both the choice of the data-generating process and the variational approximation. While REINFORCE gradient estimators do not suffer from such restrictions, they lack rigorous theoretical justification. In this paper, we provide the first comprehensive analysis of REINFORCE gradient estimators in importance weighted VI, leveraging this theoretical foundation to diagnose and resolve fundamental deficiencies in current state-of-the-art estimators. Specifically, we introduce and examine a generalized family of variational inference for Monte Carlo objectives (VIMCO) gradient estimators. We prove that state-of-the-art VIMCO gradient estimators exhibit a vanishing signal-to-noise ratio (SNR) as $N$ increases, which prevents effective optimization. To overcome this issue, we propose the novel VIMCO-$\star$ gradient estimator and show that it averts the SNR collapse of existing VIMCO gradient estimators by achieving a $\sqrt{N}$ SNR scaling instead. We demonstrate its superior empirical performance compared to current VIMCO implementations in challenging settings where reparameterized gradients are typically unavailable.

Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as in $α$-stable Lévy-driven fractional Langevin algorithms. While the target distribution can be estimated from data via score-based or energy-based models, the $α$-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis--Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime and can exhibit substantial finite-time errors and poor empirical control of tail behavior. We introduce the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA), an MH-inspired, fully score-based correction mechanism. MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric $α$-stable noise and learns an acceptance function via Score Balance Matching. We empirically illustrate the strong performance of MAFLA on a series of tasks including combinatorial optimization problems where the method significantly improves finite time sampling accuracy over unadjusted fractional Langevin dynamics.

Time series encountered in practice are rarely stationary. When the data distribution changes, a forecasting model trained on past observations can lose accuracy. We study a small-footprint test-time adaptation (TTA) framework for causal timeseries forecasting and direction classification. The backbone is frozen, and only normalization affine parameters are updated using recent unlabeled windows. For classification we minimize entropy and enforce temporal consistency; for regression we minimize prediction variance across weak time-preserving augmentations and optionally distill from an EMA teacher. A quadratic drift penalty and an uncertainty triggered fallback keep updates stable. We evaluate this framework in two stages: synthetic regime shifts on ETT benchmarks, and daily equity and FX series (SPY, QQQ, EUR/USD) across pandemic, high-inflation, and recovery regimes. On synthetic gradual drift, normalization-based TTA improves forecasting error, while in financial markets a simple batch-normalization statistics update is a robust default and more aggressive norm-only adaptation can even hurt. Our results provide practical guidance for deploying TTA on non-stationary time series.

We establish an optimal sample complexity of $O(ε^{-2})$ for obtaining an $ε$-optimal global policy using a single-timescale actor-critic (AC) algorithm in infinite-horizon discounted Markov decision processes (MDPs) with finite state-action spaces, improving upon the prior state of the art of $O(ε^{-3})$. Our approach applies STORM (STOchastic Recursive Momentum) to reduce variance in the critic updates. However, because samples are drawn from a nonstationary occupancy measure induced by the evolving policy, variance reduction via STORM alone is insufficient. To address this challenge, we maintain a buffer of small fraction of recent samples and uniformly sample from it for each critic update. Importantly, these mechanisms are compatible with existing deep learning architectures and require only minor modifications, without compromising practical applicability.

How can we understand gradient-based training over non-convex landscapes? The edge of stability phenomenon, introduced in Cohen et al. (2021), indicates that the answer is not so simple: namely, gradient descent (GD) with large step sizes often diverges away from the gradient flow. In this regime, the "Central Flow", recently proposed in Cohen et al. (2025), provides an accurate ODE approximation to the GD dynamics over many architectures. In this work, we propose Rod Flow, an alternative ODE approximation, which carries the following advantages: (1) it rests on a principled derivation stemming from a physical picture of GD iterates as an extended one-dimensional object -- a "rod"; (2) it better captures GD dynamics for simple toy examples and matches the accuracy of Central Flow for representative neural network architectures, and (3) is explicit and cheap to compute. Theoretically, we prove that Rod Flow correctly predicts the critical sharpness threshold and explains self-stabilization in quartic potentials. We validate our theory with a range of numerical experiments.

The Shapley value is a ubiquitous framework for attribution in machine learning, encompassing feature importance, data valuation, and causal inference. However, its exact computation is generally intractable, necessitating efficient approximation methods. While the most effective and popular estimators leverage the paired sampling heuristic to reduce estimation error, the theoretical mechanism driving this improvement has remained opaque. In this work, we provide an elegant and fundamental justification for paired sampling: we prove that the Shapley value depends exclusively on the odd component of the set function, and that paired sampling orthogonalizes the regression objective to filter out the irrelevant even component. Leveraging this insight, we propose OddSHAP, a novel consistent estimator that performs polynomial regression solely on the odd subspace. By utilizing the Fourier basis to isolate this subspace and employing a proxy model to identify high-impact interactions, OddSHAP overcomes the combinatorial explosion of higher-order approximations. Through an extensive benchmark evaluation, we find that OddSHAP achieves state-of-the-art estimation accuracy.

Self-attention has greatly contributed to the success of the widely used Transformer architecture by enabling learning from data with long-range dependencies. In an effort to improve performance, a gated attention model that leverages a gating mechanism within the multi-head self-attention has recently been proposed as a promising alternative. Gated attention has been empirically demonstrated to increase the expressiveness of low-rank mapping in standard attention and even to eliminate the attention sink phenomenon. Despite its efficacy, a clear theoretical understanding of gated attention's benefits remains lacking in the literature. To close this gap, we rigorously show that each entry in a gated attention matrix or a multi-head self-attention matrix can be written as a hierarchical mixture of experts. By recasting learning as an expert estimation problem, we demonstrate that gated attention is more sample-efficient than multi-head self-attention. In particular, while the former needs only a polynomial number of data points to estimate an expert, the latter requires exponentially many data points to achieve the same estimation error. Furthermore, our analysis also provides a theoretical justification for why gated attention yields higher performance when a gate is placed at the output of the scaled dot product attention or the value map rather than at other positions in the multi-head self-attention architecture.

Motivated by recent work on benign overfitting in overparameterized machine learning, we study the generalization behavior of functions in Sobolev spaces $W^{k, p}(\mathbb{R}^d)$ that perfectly fit a noisy training data set. Under assumptions of label noise and sufficient regularity in the data distribution, we show that approximately norm-minimizing interpolators, which are canonical solutions selected by smoothness bias, exhibit harmful overfitting: even as the training sample size $n \to \infty$, the generalization error remains bounded below by a positive constant with high probability. Our results hold for arbitrary values of $p \in [1, \infty)$, in contrast to prior results studying the Hilbert space case ($p = 2$) using kernel methods. Our proof uses a geometric argument which identifies harmful neighborhoods of the training data using Sobolev inequalities.

Scientific machine learning (SciML) increasingly requires models that capture multimodal conditional uncertainty arising from ill-posed inverse problems, multistability, and chaotic dynamics. While recent work has favored highly expressive implicit generative models such as diffusion and flow-based methods, these approaches are often data-hungry, computationally costly, and misaligned with the structured solution spaces frequently found in scientific problems. We demonstrate that Mixture Density Networks (MDNs) provide a principled yet largely overlooked alternative for multimodal uncertainty quantification in SciML. As explicit parametric density estimators, MDNs impose an inductive bias tailored to low-dimensional, multimodal physics, enabling direct global allocation of probability mass across distinct solution branches. This structure delivers strong data efficiency, allowing reliable recovery of separated modes in regimes where scientific data is scarce. We formalize these insights through a unified probabilistic framework contrasting explicit and implicit distribution networks, and demonstrate empirically that MDNs achieve superior generalization, interpretability, and sample efficiency across a range of inverse, multistable, and chaotic scientific regression tasks.