I propose a novel framework that integrates stochastic differential equations (SDEs) with deep generative models to improve uncertainty quantification in machine learning applications involving structured and temporal data. This approach, termed Stochastic Latent Differential Inference (SLDI), embeds an Itô SDE in the latent space of a variational autoencoder, allowing for flexible, continuous-time modeling of uncertainty while preserving a principled mathematical foundation. The drift and diffusion terms of the SDE are parameterized by neural networks, enabling data-driven inference and generalizing classical time series models to handle irregular sampling and complex dynamic structure. A central theoretical contribution is the co-parameterization of the adjoint state with a dedicated neural network, forming a coupled forward-backward system that captures not only latent evolution but also gradient dynamics. I introduce a pathwise-regularized adjoint loss and analyze variance-reduced gradient flows through the lens of stochastic calculus, offering new tools for improving training stability in deep latent SDEs. My paper unifies and extends variational inference, continuous-time generative modeling, and control-theoretic optimization, providing a rigorous foundation for future developments in stochastic probabilistic machine learning.

We introduce Guided Harmonic Path-Integral Diffusion (GH-PID), a linearly-solvable framework for guided Stochastic Optimal Transport (SOT) with a hard terminal distribution and soft, application-driven path costs. A low-dimensional guidance protocol shapes the trajectory ensemble while preserving analytic structure: the forward and backward Kolmogorov equations remain linear, the optimal score admits an explicit Green-function ratio, and Gaussian-Mixture Model (GMM) terminal laws yield closed-form expressions. This enables stable sampling and differentiable protocol learning under exact terminal matching. We develop guidance-centric diagnostics -- path cost, centerline adherence, variance flow, and drift effort -- that make GH-PID an interpretable variational ansatz for empirical SOT. Three navigation scenarios illustrated in 2D: (i) Case A: hand-crafted protocols revealing how geometry and stiffness shape lag, curvature effects, and mode evolution; (ii) Case B: single-task protocol learning, where a PWC centerline is optimized to minimize integrated cost; (iii) Case C: multi-expert fusion, in which a commander reconciles competing expert/teacher trajectories and terminal beliefs through an exact product-of-experts law and learns a consensus protocol. Across all settings, GH-PID generates geometry-aware, trust-aware trajectories that satisfy the prescribed terminal distribution while systematically reducing integrated cost.

Diffusion-based samplers -- Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions -- match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule -- selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.

Benchmarks are the primary tool for assessing progress in artificial intelligence (AI), yet current practice evaluates models on isolated test suites and provides little guidance for reasoning about generality or autonomous self-improvement. Here we introduce a geometric framework in which all psychometric batteries for AI agents are treated as points in a structured moduli space, and agent performance is described by capability functionals over this space. First, we define an Autonomous AI (AAI) Scale, a Kardashev-style hierarchy of autonomy grounded in measurable performance on batteries spanning families of tasks (for example reasoning, planning, tool use and long-horizon control). Second, we construct a moduli space of batteries, identifying equivalence classes of benchmarks that are indistinguishable at the level of agent orderings and capability inferences. This geometry yields determinacy results: dense families of batteries suffice to certify performance on entire regions of task space. Third, we introduce a general Generator-Verifier-Updater (GVU) operator that subsumes reinforcement learning, self-play, debate and verifier-based fine-tuning as special cases, and we define a self-improvement coefficient $κ$ as the Lie derivative of a capability functional along the induced flow. A variance inequality on the combined noise of generation and verification provides sufficient conditions for $κ> 0$. Our results suggest that progress toward artificial general intelligence (AGI) is best understood as a flow on moduli of benchmarks, driven by GVU dynamics rather than by scores on individual leaderboards.

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Hebbian and anti-Hebbian plasticity are widely observed in the biological brain, yet their theoretical understanding remains limited. In this work, we find that when a learning method is regularized with L2 weight decay, its learning signal will gradually align with the direction of the Hebbian learning signal as it approaches stationarity. This Hebbian-like behavior is not unique to SGD: almost any learning rule, including random ones, can exhibit the same signature long before learning has ceased. We also provide a theoretical explanation for anti-Hebbian plasticity in regression tasks, demonstrating how it can arise naturally from gradient or input noise, and offering a potential reason for the observed anti-Hebbian effects in the brain. Certainly, our proposed mechanisms do not rule out any conventionally established forms of Hebbian plasticity and could coexist with them extensively in the brain. A key insight for neurophysiology is the need to develop ways to experimentally distinguish these two types of Hebbian observations.

Generative robotic motion planning requires not only the synthesis of smooth and collision-free trajectories but also feasibility across diverse tasks and dynamic constraints. Prior planning methods, both traditional and generative, often struggle to incorporate high-level semantics with low-level constraints, especially the nexus between task configurations and motion controllability. In this work, we present XFlowMP, a task-conditioned generative motion planner that models robot trajectory evolution as entropic flows bridging stochastic noises and expert demonstrations via Schrodinger bridges given the inquiry task configuration. Specifically, our method leverages Schrodinger bridges as a conditional flow matching coupled with a score function to learn motion fields with high-order dynamics while encoding start-goal configurations, enabling the generation of collision-free and dynamically-feasible motions. Through evaluations, XFlowMP achieves up to 53.79% lower maximum mean discrepancy, 36.36% smoother motions, and 39.88% lower energy consumption while comparing to the next-best baseline on the RobotPointMass benchmark, and also reducing short-horizon planning time by 11.72%. On long-horizon motions in the LASA Handwriting dataset, our method maintains the trajectories with 1.26% lower maximum mean discrepancy, 3.96% smoother, and 31.97% lower energy. We further demonstrate the practicality of our method on the Kinova Gen3 manipulator, executing planning motions and confirming its robustness in real-world settings.

Alignment faking is a form of strategic deception in AI in which models selectively comply with training objectives when they infer that they are in training, while preserving different behavior outside training. The phenomenon was first documented for Claude 3 Opus and later examined across additional large language models. In these setups, the word "training" refers to simulated training via prompts without parameter updates, so the observed effects are context conditioned shifts in behavior rather than preference learning. We study the phenomenon using an evaluation framework that compares preference optimization methods (BCO, DPO, KTO, and GRPO) across 15 models from four model families, measured along three axes: safety, harmlessness, and helpfulness. Our goal is to identify what causes alignment faking and when it occurs.

A significant task in statistics and machine learning currently revolves around generating samples from a target distribution that is only accessible via a dataset.

Rapid urbanization in cities like Bangalore has led to severe traffic congestion, making efficient Traffic Signal Control (TSC) essential. Multi-Agent Reinforcement Learning (MARL), often modeling each traffic signal as an independent agent using Q-learning, has emerged as a promising strategy to reduce average commuter delays. While prior work Prashant L A et. al has empirically demonstrated the effectiveness of this approach, a rigorous theoretical analysis of its stability and convergence properties in the context of traffic control has not been explored. This paper bridges that gap by focusing squarely on the theoretical basis of this multi-agent algorithm. We investigate the convergence problem inherent in using independent learners for the cooperative TSC task. Utilizing stochastic approximation methods, we formally analyze the learning dynamics. The primary contribution of this work is the proof that the specific multi-agent reinforcement learning algorithm for traffic control is proven to converge under the given conditions extending it from single agent convergence proofs for asynchronous value iteration.