We quantify the efficiency of temporal difference (TD) learning over the direct, or Monte Carlo (MC), estimator for policy evaluation in reinforcement learning, with an emphasis on estimation of quantities related to rare events. Policy evaluation is complicated in the rare event setting by the long timescale of the event and by the need for \emph{relative accuracy} in estimates of very small values. Specifically, we focus on least-squares TD (LSTD) prediction for finite state Markov chains, and show that LSTD can achieve relative accuracy far more efficiently than MC. We prove a central limit theorem for the LSTD estimator and upper bound the \emph{relative asymptotic variance} by simple quantities characterizing the connectivity of states relative to the transition probabilities between them. Using this bound, we show that, even when both the timescale of the rare event and the relative accuracy of the MC estimator are exponentially large in the number of states, LSTD maintains a fixed level of relative accuracy with a total number of observed transitions of the Markov chain that is only \emph{polynomially} large in the number of states.

Rare events such as financial crashes, climate extremes, and biological anomalies are notoriously difficult to model due to their scarcity and heavy-tailed distributions. Classical deep generative models often struggle to capture these rare occurrences, either collapsing low-probability modes or producing poorly calibrated uncertainty estimates. In this work, we propose the Quantum-Enhanced Generative Model (QEGM), a hybrid classical-quantum framework that integrates deep latent-variable models with variational quantum circuits. The framework introduces two key innovations: (1) a hybrid loss function that jointly optimizes reconstruction fidelity and tail-aware likelihood, and (2) quantum randomness-driven noise injection to enhance sample diversity and mitigate mode collapse. Training proceeds via a hybrid loop where classical parameters are updated through backpropagation while quantum parameters are optimized using parameter-shift gradients. We evaluate QEGM on synthetic Gaussian mixtures and real-world datasets spanning finance, climate, and protein structure. Results demonstrate that QEGM reduces tail KL divergence by up to 50 percent compared to state-of-the-art baselines (GAN, VAE, Diffusion), while improving rare-event recall and coverage calibration. These findings highlight the potential of QEGM as a principled approach for rare-event prediction, offering robustness beyond what is achievable with purely classical methods.

The objective of this work is to develop a method for detecting rare gamma quanta against the background of charged particles in the fluxes from sources in the Universe with the help of the deep learning and normalizing flows based method designed for anomaly detection. It is shown that the suggested method has a potential for the gamma detection. The method was tested on model data from the TAIGA-IACT experiment. The obtained quantitative performance indicators are still inferior to other approaches, and therefore possible ways to improve the implementation of the method are proposed.

Abstract--Extreme events frequently occur in real-world time series and often carry significant practical implications. In domains such as climate and healthcare, these events, such as floods, heatwaves, or acute medical episodes, can lead to serious consequences. Accurate forecasting of such events is therefore of substantial importance. Most existing time series forecasting models are optimized for overall performance within the prediction window, but often struggle to accurately predict extreme events, such as high temperatures or heart rate spikes. The main challenges are data imbalance and the neglect of valuable information contained in intermediate events that precede extreme events. In this paper, we propose xTime, a novel framework for extreme event forecasting in time series. In addition, we introduce a mixture of experts (MoE) mechanism that dynamically selects and fuses outputs from expert models across different rarity levels, which further improves the forecasting performance for extreme events. Experiments on multiple datasets show that xTime achieves consistent improvements, with forecasting accuracy on extreme events improving from 3% to 78%. Time series forecasting plays a fundamental role across a broad spectrum of critical applications, such as stock market analysis, weather and climate modeling, and electricity demand prediction.

Scarcity of labeled data, especially for rare events, hinders training effective machine learning models. This paper proposes SYNAPSE-G (Synthetic Augmentation for Positive Sampling via Expansion on Graphs), a novel pipeline leveraging Large Language Models (LLMs) to generate synthetic training data for rare event classification, addressing the cold-start problem. This synthetic data serve as seeds for semi-supervised label propagation on a similarity graph constructed between the seeds and a large unlabeled dataset. This identifies candidate positive examples, subsequently labeled by an oracle (human or LLM). The expanded dataset then trains/fine-tunes a classifier. We theoretically analyze how the quality (validity and diversity) of the synthetic data impacts the precision and recall of our method. Experiments on the imbalanced SST2 and MHS datasets demonstrate SYNAPSE-G's effectiveness in finding positive labels, outperforming baselines including nearest neighbor search.

Causal effect estimation seeks to determine the impact of an intervention from observational data. However, the existing causal inference literature primarily addresses treatment effects on frequently occurring events. But what if we are interested in estimating the effects of a policy intervention whose benefits, while potentially important, can only be observed and measured in rare yet impactful events, such as extreme climate events? The standard causal inference methodology is not designed for this type of inference since the events of interest may be scarce in the observed data and some degree of extrapolation is necessary. Extreme Value Theory (EVT) provides methodologies for analyzing statistical phenomena in such extreme regimes. We introduce a novel framework for assessing treatment effects in extreme data to capture the causal effect at the occurrence of rare events of interest. In particular, we employ the theory of multivariate regular variation to model extremities. We develop a consistent estimator for extreme treatment effects and present a rigorous non-asymptotic analysis of its performance. We illustrate the performance of our estimator using both synthetic and semi-synthetic data.

Machine learning has emerged as an alternative approach for solving partial differential equations, reproducing trajectories of dynamical systems, emulating statistical properties of chaotic systems, etc. Neural networks and deep learning play a particularly important role in developing new techniques for understanding and solving various dynamical systems. Reservoir computing [15, 30] is a particular class of machine learning models; it utilizes a large recurrent network (reservoir), and only a linear output layer is trained to match the trajectory. Echo-State Networks refer to reservoirs that have the Echo-State Property (see e.g.

We investigate the choice patterns of Large Language Models (LLMs) in the context of Decisions from Experience tasks that involve repeated choice and learning from feedback, and compare their behavior to human participants. We find that on the aggregate, LLMs appear to display behavioral biases similar to humans: both exhibit underweighting rare events and correlation effects. However, more nuanced analyses of the choice patterns reveal that this happens for very different reasons. LLMs exhibit strong recency biases, unlike humans, who appear to respond in more sophisticated ways. While these different processes may lead to similar behavior on average, choice patterns contingent on recent events differ vastly between the two groups. Specifically, phenomena such as ``surprise triggers change" and the ``wavy recency effect of rare events" are robustly observed in humans, but entirely absent in LLMs. Our findings provide insights into the limitations of using LLMs to simulate and predict humans in learning environments and highlight the need for refined analyses of their behavior when investigating whether they replicate human decision making tendencies.

Estimating treatment effects in time series data presents a significant challenge, especially when the control group is always unobservable. For example, in analyzing the effects of Christmas on retail sales, we lack direct observation of what would have occurred in late December without the Christmas's impact. To address this, we try to recover the control group in the event period while accounting for confounders and temporal dependencies. Experimental results on the M5 Walmart retail sales data demonstrate robust estimation of the potential outcome of the control group as well as accurate predicted holiday effect. Furthermore, we provided theoretical guarantees for the estimated treatment effect, proving its consistency and asymptotic normality. The proposed methodology is applicable not only to this always-missing control scenario but also in other conventional time series causal inference settings.