This paper investigates the structural dynamics of stock market volatility through the Financial Chaos Index, a tensor- and eigenvalue-based measure designed to capture realized volatility via mutual fluctuations among asset prices. Motivated by empirical evidence of regime-dependent volatility behavior and perceptual time dilation during financial crises, we develop a regime-switching framework based on the Modified Lognormal Power-Law distribution. Analysis of the FCIX from January 1990 to December 2023 identifies three distinct market regimes, low-chaos, intermediate-chaos, and high-chaos, each characterized by differing levels of systemic stress, statistical dispersion and persistence characteristics. Building upon the segmented regime structure, we further examine the informational forces that shape forward-looking market expectations. Using sentiment-based predictors derived from the Equity Market Volatility tracker, we employ an elastic net regression model to forecast implied volatility, as proxied by the VIX index. Our findings indicate that shifts in macroeconomic, financial, policy, and geopolitical uncertainty exhibit strong predictive power for volatility dynamics across regimes. Together, these results offer a unified empirical perspective on how systemic uncertainty governs both the realized evolution of financial markets and the anticipatory behavior embedded in implied volatility measures.

Parametric Markov chains (pMC) are used to model probabilistic systems with unknown or partially known probabilities. Although (universal) pMC verification for reachability properties is known to be coETR-complete, there have been efforts to approach it using potentially easier-to-check properties such as asking whether the pMC is monotonic in certain parameters. In this paper, we first reduce monotonicity to asking whether the reachability probability from a given state is never less than that of another given state. Recent results for the latter property imply an efficient algorithm to collapse same-value equivalence classes, which in turn preserves verification results and monotonicity. We implement our algorithm to collapse "trivial" equivalence classes in the pMC and show empirical evidence for the following: First, the collapse gives reductions in size for some existing benchmarks and significant reductions on some custom benchmarks; Second, the collapse speeds up existing algorithms to check monotonicity and parameter lifting, and hence can be used as a fast pre-processing step in practice.

--This work presents a novel method for leveraging intercepted Radio Frequency (RF) signals to inform a constrained Reinforcement Learning (RL) policy for robust control of a satellite operating in contested environments. Uncooperative satellite engagements with nation-state actors prompts the need for enhanced maneuverability and agility on-orbit. However, robust, autonomous and rapid adversary avoidance capabilities for the space environment is seldom studied. Further, the capability constrained nature of many space vehicles does not afford robust space situational awareness capabilities that can be used for well informed maneuvering. We present a "Cat & Mouse" system for training optimal adversary avoidance algorithms using RL. We propose the novel approach of utilizing intercepted radio frequency communication and dynamic spacecraft state as multi-modal input that could inform paths for a mouse to outmaneuver the cat satellite. Given the current ubiquitous use of RF communications, our proposed system can be applicable to a diverse array of satellites. In addition to providing a comprehensive framework for training and implementing a constrained RL policy capable of providing control for robust adversary avoidance, we also explore several optimization based methods for adversarial avoidance. These methods were then tested on real-world data obtained from the Space Surveillance Network (SSN) to analyze the benefits and limitations of different avoidance methods. In March of 2025, Chinese satellites exhibited dog-fighting capabilities [1], following years of both Russian [2] and Chinese [3] satellites approaching dangerously close to US satellites in geosynchronous orbit. Such uncooperative activity prompts the need for satellite agility and maneuverability which can be facilitated through edge-based autonomy. To achieve this, appropriate sensing would be required to properly characterize the contested environment. Not all satellites have precise space domain awareness (SDA) sensing suites onboard, despite having powerful buses and flight controllers that can facilitate autonomous operations. We propose leveraging an uncooperative space vehicle's communication systems as a means to evaluate safe flight control policies to carefully navigate contested domains in situations where support from the ground is not feasible.

This paper addresses the challenge of navigation in large, visually complex environments with sparse rewards. We propose a method that uses object-oriented macro actions grounded in a topological map, allowing a simple Deep Q-Network (DQN) to learn effective navigation policies. The agent builds a map by detecting objects from RGBD input and selecting discrete macro actions that correspond to navigating to these objects. This abstraction drastically reduces the complexity of the underlying reinforcement learning problem and enables generalization to unseen environments. We evaluate our approach in a photorealistic 3D simulation and show that it significantly outperforms a random baseline under both immediate and terminal reward conditions. Our results demonstrate that topological structure and macro-level abstraction can enable sample-efficient learning even from pixel data.

We develop a geometric account of sequence modelling that links patterns in the data to measurable properties of the loss landscape in transformer networks. First, we cast conditional sequence distributions into a Hilbert-space framework and apply tensor decompositions to identify their principal modes. Truncating the small-amplitude modes yields an effective data distribution that preserves dominant structure while discarding statistical detail. Second, we show theoretically that Local Learning Coefficient (LLC) estimates are insensitive to modes below a data-dependent threshold. Consequently, the LLC calculated in practice characterises the geometry of the effective rather than the true distribution. This insight clarifies why reliable LLC estimates can be obtained even when a network parameter is not a strict minimiser of the population loss, and it highlights how the inverse temperature in SGLD acts as a resolution dial on the landscape structure.

Do We Need Transformers to Play FPS Video Games? Abstract --In this paper, we explore the Transformer based architectures for reinforcement learning in both online and offline settings within the Doom game environment. Our investigation focuses on two primary approaches: Deep Transformer Q-learning Networks (DTQN) for online learning [1] and Decision Transformers (DT) for offline reinforcement learning [2]. DTQN leverages the sequential modelling capabilities of Transformers to enhance Q-learning in partially observable environments,while Decision Transformers repurpose sequence modelling techniques to enable offline agents to learn from past trajectories without direct interaction with the environment. We conclude that while Transformers might have performed well in Atari games, more traditional methods perform better than Transformer based method in both the settings in the VizDoom environment [3] Q-networks traditionally have been used in various Atari game environments [4] and in partially observable environments like in the Doom 1993 game [5].

We study regret minimization under privacy constraints in episodic inhomogeneous linear Markov Decision Processes (MDPs), motivated by the growing use of reinforcement learning (RL) in personalized decision-making systems that rely on sensitive user data. In this setting, both transition probabilities and reward functions are assumed to be linear in a feature mapping $\phi(s, a)$, and we aim to ensure privacy through joint differential privacy (JDP), a relaxation of differential privacy suited to online learning. Prior work has established suboptimal regret bounds by privatizing the LSVI-UCB algorithm, which achieves $\widetilde{O}(\sqrt{d^3 H^4 K})$ regret in the non-private setting. Building on recent advances that improve this to near minimax optimal regret $\widetilde{O}(d\sqrt{H^{3}K})$ via LSVI-UCB++ with Bernstein-style bonuses, we design a new differentially private algorithm by privatizing LSVI-UCB++ and adapting techniques for variance-aware analysis from offline RL. Our algorithm achieves a regret bound of $\widetilde{O}(d \sqrt{H^3 K} + H^{15/4} d^{7/6} K^{1/2} / \epsilon)$, improving over previous private methods. Empirical results show that our algorithm retains near-optimal utility compared to non-private baselines, indicating that privacy can be achieved with minimal performance degradation in this setting.

Traditional gradient-based sampling methods, like standard Hamiltonian Monte Carlo, require that the desired target distribution is continuous and differentiable. This limits the types of models one can define, although the presented models capture the reality in the observations better. In this project, Generalized Randomized Hamiltonian Monte Carlo (GRHMC) processes for sampling continuous densities with discontinuous gradient and piecewise smooth targets are proposed. The methods combine the advantages of Hamiltonian Monte Carlo methods with the nature of continuous time processes in the form of piecewise deterministic Markov processes to sample from such distributions. It is argued that the techniques lead to GRHMC processes that admit the desired target distribution as the invariant distribution in both scenarios. Simulation experiments verifying this fact and several relevant real-life models are presented, including a new parameterization of the spike and slab prior for regularized linear regression that returns sparse coefficient estimates and a regime switching volatility model.

Stochastic Optimal Control (SOC) problems arise in systems influenced by uncertainty, such as autonomous robots or financial models. Traditional methods like dynamic programming are often intractable for high-dimensional, nonlinear systems due to the curse of dimensionality. This dissertation explores the path integral control framework as a scalable, sampling-based alternative. By reformulating SOC problems as expectations over stochastic trajectories, it enables efficient policy synthesis via Monte Carlo sampling and supports real-time implementation through GPU parallelization. We apply this framework to six classes of SOC problems: Chance-Constrained SOC, Stochastic Differential Games, Deceptive Control, Task Hierarchical Control, Risk Mitigation of Stealthy Attacks, and Discrete-Time LQR. A sample complexity analysis for the discrete-time case is also provided. These contributions establish a foundation for simulator-driven autonomy in complex, uncertain environments.

Reasoning is an essential skill to enable Large Language Models (LLMs) to interact with the world. As tasks become more complex, they demand increasingly sophisticated and diverse reasoning capabilities for sequential decision-making, requiring structured reasoning over the context history to determine the next best action. We introduce TALES, a diverse collection of synthetic and human-written text-adventure games designed to challenge and evaluate diverse reasoning capabilities. We present results over a range of LLMs, open- and closed-weights, performing a qualitative analysis on the top performing models. Despite an impressive showing on synthetic games, even the top LLM-driven agents fail to achieve 15% on games designed for human enjoyment. Code and visualization of the experiments can be found at https://microsoft.github.io/tale-suite.