Probabilistic Smoothing with Ratio-Monotone Transforms for Global Optimization
This paper proposes a generalized probabilistic smoothing framework for global optimization that replaces Gaussian kernels with flexible symmetric unimodal kernels combined with monotonic ratio-based transformations. The authors prove that the smoothed objective preserves the global maximizer and that stationary points concentrate near the true optimum under large amplification, without requiring a decreasing smoothing schedule. Explicit complexity bounds for stochastic gradient ascent are derived, and a leave-one-out baseline is shown to provably reduce variance. Experiments on high-dimensional benchmarks and black-box adversarial attacks demonstrate improved robustness over existing methods.
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Hamiltonian Probability Gradient Flow Analysis of the Muon Optimizer
This paper develops a rigorous theoretical framework for the Muon optimizer by interpreting its regularized orthogonalization map as the gradient of a Fenchel-dual smoothing of the nuclear norm, identifying Muon updates as mirror/prox steps with momentum as dual coordinates. The authors lift this structure to probability measures over matrix-valued parameters, deriving a mean-field phase-space equation that constitutes a damped Hamiltonian probability dynamics with monotonically decreasing Hamiltonian energy. Exponential convergence rates are established under gradient-dominance and curvature assumptions, and propagation-of-chaos guarantees are provided for the interacting particle system. The framework extends to transformer mixture-of-experts architectures via blockwise Muon probability flows.
Optimal Mixture Transport (OMT): Biconvex Formulation for Scalable, Stable Optimal Transport
This paper introduces Optimal Mixture Transport (OMT), a framework that reformulates optimal transport between probability distributions as a strictly biconvex optimization problem with a provably unique global minimizer. By operating at the level of mixture components (modeled as exponential-family distributions) rather than individual samples, OMT decouples computational complexity from sample size. The authors provide theoretical stability guarantees showing bounded perturbations yield bounded changes in transport plans, and validate the approach on image data and large-scale single-cell RNA sequencing datasets.
Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization
This paper addresses miscalibration in Gaussian process predictive distributions used for Bayesian optimization, focusing specifically on the lower tail relevant to minimization objectives. The authors introduce a framework for 'goal-oriented' spatial calibration below a threshold t, defining occurrence calibration and thresholded μ-calibration on sublevel sets. They propose tcGP, a post-hoc calibration method, and prove the resulting EI-based optimizer remains dense in the design space. Experiments on standard benchmarks show tcGP improves both lower-tail calibration and overall BO performance compared to standard and globally calibrated GP models.
DRPO: Smooth divergence regularization replaces hard masking in LLM RL training
A new arXiv preprint proposes Divergence Regularized Policy Optimization (DRPO), a method that replaces the hard trust-region mask used in DPPO with a smooth advantage-weighted quadratic regularizer on policy shift. The approach addresses a known weakness in PPO and GRPO where importance ratios poorly proxy distributional shift in long-tailed vocabularies, and in DPPO where gradient signals are discarded rather than corrected at trust-region boundaries. Experiments across model scales, architectures, and precision settings show improved stability and efficiency in LLM RL post-training.
Preference-Shaped Expected Hypervolume and R2 Improvement: Exact Computation and Monotonicity
This paper analyzes preference-shaped expected improvement criteria for Bayesian multiobjective optimization, focusing on hypervolume (EHVI) and R2 indicator families. The authors establish which preference transformations preserve exact computation, Pareto compatibility, and monotonicity, and which alter the underlying geometry. A key result is that exact integral R2 improvement is not generally an objective-space weighted hypervolume but is exactly a scalarization-space volume (Tchebycheff shadow measure), enabling new finite-sum and quadrature algorithms for ER2I. The work also provides an achievement-space Gaussian surrogate formulation reducing ER2I to an integral of scalar Gaussian expected improvements.
Global Convergence Theory for Wasserstein Policy Gradient in Entropy-Regularized RL
This paper establishes the first global convergence theory for Wasserstein Policy Gradient (WPG), a continuous-control RL optimization method that uses optimal-transport geometry over action distributions. The authors show that the Bellman recursion structure of entropy-regularized RL induces a Polyak–Łojasiewicz (PL) geometry that substitutes for classical convexity, enabling global convergence analysis. Key technical contributions include a statewise KL representation of the soft Bellman residual, a Bellman resolvent identity linking value improvement to relative Fisher information, and a uniform log-Sobolev inequality for the evolving Gibbs policy family. The result yields geometric contraction up to discretization bias, providing theoretical grounding for WPG in continuous-action settings.
Kolmogorov Regression lifts diffusion policies to Cameron-Martin space for robust long-horizon control
Researchers introduce a backward Kolmogorov equation framework that reformulates diffusion policy training as a deterministic boundary-value PDE problem in Cameron-Martin space, replacing stochastic score matching. The approach uses a precision-weighted Cameron-Martin loss and a Kolmogorov residual as an inference-time failure detector, yielding convergence guarantees tied to kernel effective rank rather than action dimension. Validation on the PushT manipulation benchmark shows 17% improvement in episode reward and 67.6% reduction in inter-step drift; a 6-station manufacturing scheduling task shows 28.4% lower RMSE than LSTM baselines and 96% reduction in deadlock events via Hamilton-Jacobi reachability certification.
The Matching Principle: A Geometric Theory Unifying Robustness, Domain Adaptation, and Alignment via Nuisance Covariance
This paper proposes the 'matching principle': a unified geometric framework arguing that robustness methods (CORAL, IRM, adversarial training, augmentation, metric learning, Jacobian penalties, alignment constraints) are all estimators of the same object—the covariance of label-preserving deployment nuisance—and that regularizing the encoder Jacobian along this covariance's range is the core statistical problem. The authors prove closed-form optimality results in a linear-Gaussian model, introduce the Trajectory Deviation Index (TDI) as a label-free embedding sensitivity probe, and validate predictions across 13 pre-registered experimental blocks including Qwen2.5-7B. At 7B scale, matched style-PMH improves selective honesty while standard DPO degrades Style TDI, connecting the theory to alignment safety.