Tootfinder

A Universal Banach--Bregman Framework for Stochastic Iterations: Unifying Stochastic Mirror Descent, Learning and LLM Training
Johnny R. Zhang (Independent Researcher), Xiaomei Mi (University of Manchester), Gaoyuan Du (Amazon), Qianyi Sun (Microsoft), Shiqi Wang (Meta), Jiaxuan Li (Amazon), Wenhua Zhou (Independent Researcher)
https://arx…

A Universal Banach--Bregman Framework for Stochastic Iterations: Unifying Stochastic Mirror Descent, Learning and LLM Training
Stochastic optimization powers the scalability of modern artificial intelligence, spanning machine learning, deep learning, reinforcement learning, and large language model training. Yet, existing theory remains largely confined to Hilbert spaces, relying on inner-product frameworks and orthogonality. This paradigm fails to capture non-Euclidean settings, such as mirror descent on simplices, Bregman proximal methods for sparse learning, natural gradient descent in information geometry, or Kullb…

On Tackling High-Dimensional Nonconvex Stochastic Optimization via Stochastic First-Order Methods with Non-smooth Proximal Terms and Variance Reduction
Yue Xie, Jiawen Bi, Hongcheng Liu
https://arxiv.org/abs/2509.13992

On Tackling High-Dimensional Nonconvex Stochastic Optimization via Stochastic First-Order Methods with Non-smooth Proximal Terms and Variance Reduction
When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. To alleviate this linear dependence, we adopt non-Euclidean settings and propose two choices of nonsmooth proximal terms when taking the stochastic gradient steps. This approach leads to stronger convergence metric, incremental computational overhead, and potentially dimension-insensitive …

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Random attractor and SRB measure for stochastic Hopf bifurcation under discretization
Chuchu Chen, Jialin Hong, Yibo Wang
https://arxiv.org/abs/2509.12865 https://

Random attractor and SRB measure for stochastic Hopf bifurcation under discretization
Chaotic phases in stochastic differential equations are characterized by two essential long-time dynamical features: a random attractor capturing asymptotic geometry and a Sinai-Ruelle-Bowen (SRB) measure describing statistical information. This paper investigates whether the stochastic Hopf bifurcation under discretization could inherit both features. We establish that the stochastic Hopf bifurcation under discretization induces a discrete random dynamical system. Further, we prove that this d…

Nonparametric learning of stochastic differential equations from sparse and noisy data
Arnab Ganguly, Riten Mitra, Jinpu Zhou
https://arxiv.org/abs/2508.11597 https://

Nonparametric learning of stochastic differential equations from sparse and noisy data
The paper proposes a systematic framework for building data-driven stochastic differential equation (SDE) models from sparse, noisy observations. Unlike traditional parametric approaches, which assume a known functional form for the drift, our goal here is to learn the entire drift function directly from data without strong structural assumptions, making it especially relevant in scientific disciplines where system dynamics are partially understood or highly complex. We cast the estimation prob…

A cohomological smoothness conjecture for moduli of mixed characteristic local shtukas with one leg
Sean Howe
https://arxiv.org/abs/2508.11595 https://arxi…

A cohomological smoothness conjecture for moduli of mixed characteristic local shtukas with one leg
We give a simple geometric characterization of the locus where the inscribed Banach--Colmez Tangent Spaces of moduli of mixed characteristic local shtukas with one leg and fixed determinant are connected. We conjecture that the structure morphism for the underlying diamond is cohomologically smooth over this locus and, applying the Fargues--Scholze Jacobian criterion, we prove this conjecture in the case of EL infinite level Rapoport--Zink spaces, generalizing a result of Ivanov--Weinstein in t…

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Trust Region Algorithm for Stochastic Minimax Problems with Decision-Dependent Distributions
Yan Gao, Yongchao Liu, Zili Luo
https://arxiv.org/abs/2509.12868 https://

Trust Region Algorithm for Stochastic Minimax Problems with Decision-Dependent Distributions
Stochastic minimax optimization has drawn much attention over the past decade due to its broad applications in machine learning, signal processing and game theory. In some applications, the probability distribution of uncertainty depends on decision variables, as the environment may respond to decisions. In this paper, we propose a trust region algorithm for finding the stationary points of stochastic minimax problems with decision-dependent distributions. At each iteration, the algorithm local…

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BSDE Approach for $\alpha$-Potential Stochastic Differential Games
Xin Guo, Xun Li, Liangquan Zhang
https://arxiv.org/abs/2507.13256 https://

BSDE Approach for $α$-Potential Stochastic Differential Games
In this paper, we examine a class of $α$-potential stochastic differential games with random coefficients via the backward stochastic differential equations (BSDEs) approach. Specifically, we show that the first and second order linear derivatives of the objective function for each player can be expressed through the corresponding first and second-order adjoint equations, which leads to rigorous estimates for $α$. We illustrate the dependence of $α$ on game characteristics through detailed a…