arXiv subjects

Minghui Song

Publications and source records attributed to Minghui Song.

At least 19 recordsLinked to original sources

arXiv2026

Nonparametric Schr\"odinger Bridge Time Series Generator: Algorithm, Convergence Analysis and Applications

We conduct a convergence analysis for the Schr\"odinger Bridge Time Series (SBTS) data generator. Starting from a regularized formulation in which the data ensemble is mixed with a standard multivariate Gaussian distribution with a prescribed probability, we prove that the Euler-Maruyama discretization converges to the mixed target distribution with half-order convergence rate, provided that the ensemble size and kernel bandwidth are chosen appropriately. We further show that the regularized distribution converges to the original target distribution as the mixing probability tends to zero. The analysis simultaneously accounts for the ensemble approximation error, kernel approximation error, and time-discretization error, and therefore provides a full distributional convergence result for the SBTS generator. Empirically, we further examine the flexibility of the method by replacing the Wiener reference measure with the path measure induced by a more general SDE. The numerical experiments show that the schemes based on both the original Wiener reference measure and the SDE-induced reference measure achieve comparable performance, demonstrating the robustness and stability of the SBTS framework.

math.NA↗

arXiv2026

Group-Graph Policy Optimization for Long-Horizon Agentic Reinforcement Learning

Group-based Reinforcement Learning (RL) has significantly enhanced Large Language Models (LLMs) in agentic scenarios. To achieve finer-grained policy updates, recent agentic RL frameworks have shifted from trajectory-level to step-level training. However, long-horizon agentic RL suffers from severe reward sparsity and delay, as feedback is often deferred for dozens of interaction steps. While existing step-level frameworks refine training granularity, their credit assignment remains coarse-grained and still treats agent exploration as isolated, linear trajectories. This oversimplified perspective ignores the inherent graph structure of state transitions, leading to high-variance state-value estimation and myopic, localized credit assignment. To overcome these critical bottlenecks, we propose Group-Graph Policy Optimization (G2PO), a novel group-based RL algorithm tailored for multi-turn agentic tasks. G2PO explicitly transforms linear interaction trajectories into a global state-transition graph. By aggregating identical observations across different trajectories, we introduce group-aggregation state-value estimation that reduces sampling variance and trajectory-dependent bias. Furthermore, we redefine agent actions as transitions between state nodes and propose an edge-centric advantage estimation strategy. By globally standardizing Temporal Difference (TD) errors across the entire graph, G2PO explicitly identifies and prioritizes critical transitions that drive absolute task progress. Extensive experiments on representative long-horizon benchmarks-WebShop, ALFWorld, and AppWorld-demonstrate that G2PO substantially outperforms state-of-the-art prompt-based and RL baselines, achieving remarkable success rate improvements of up to 22.2% over GRPO.

cs.LG↗

arXiv2026

Exponential ergodicity of exact and numerical solutions for McKean-Vlasov SDEs driven by L\'evy noise

This paper investigates the exponential ergodicity of the exact solution and the tamed Euler solution for McKean-Vlasov stochastic differential equations driven by L\'evy noise. First, we establish exponential ergodicity for both the original equation and the tamed Euler method. Then we prove the convergence of the numerical invariant measure to the exact invariant measure, which is obtained by combining the propagation of chaos (PoC) result with the strong convergence of the tamed Euler scheme. Furthermore, we derive a convergence rate for the numerical invariant measure by establishing uniform-in-time PoC and uniform-in-time convergence of the tamed Euler method. Finally, numerical experiments are presented to illustrate the theoretical results.

math.NA↗

arXiv2026

Uniform-in-time error estimates for McKean-Vlasov SDEs with common noise and stochastic algorithms

In this work, by construct an asymptotic coupling by reflection, we first explore the uniform-in-time estimate on probability distance for two measure-valued processes induced by a McKean-Vlasov SDE with common noise and an interacting particle system, where the drift terms are dissipative merely in the long distance. As direct applications of this estimate, we establish the uniform-in-time error estimates for the numerical solutions derived via backward/tamed/adaptive Euler-Maruyama methods. Moreover, as another direct application, the uniform-in-time conditional propagation of chaos is quantified.

math.PR↗

arXiv2025

A Computationally Efficient Finite Element Method for Shape Reconstruction of Inverse Conductivity Problems

The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such approaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. {\it A prior} error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

math.NA↗

arXiv2025

A High-order Backpropagation Algorithm for Neural Stochastic Differential Equation Model

Neural stochastic differential equation model with a Brownian motion term can capture epistemic uncertainty of deep neural network from the perspective of a dynamical system. The goal of this paper is to improve the convergence rate of the sample-wise backpropagation algorithm in neural stochastic differential equation model which has been proposed in [Archibald et al., SIAM Journal on Numerical Analysis, 62 (2024), pp. 593-621]. It is necessary to emphasize that, improving the convergence order of the algorithm consisting of forward backward stochastic differential equations remains challenging, due to the loss of information of Z term in backward equations under sample-wise approximation and the limitations of the forward network form. In this paper, we develop a high-order backpropagation algorithm to improve the training accuracy. Under the convexity assumption, the result indicates that the first-order convergence is achieved when the number of training steps is proportional to the cubic number of layers. Finally, numerical examples illustrate our theoretical results.

math.NA↗

arXiv2025

Dimension-independent convergence rate of propagation of chaos and numerical analysis for McKean-Vlasov stochastic differential equations with coefficients nonlinearly dependent on measure

In contrast to ordinary stochastic differential equations (SDEs), the numerical simulation of McKean-Vlasov stochastic differential equations (MV-SDEs) requires approximating the distribution law first. Based on the theory of propagation of chaos, particle approximation method is widely used. Then, a natural question is to investigate the convergence rate of the method (also referred to as the convergence rate of PoC). In fact, the PoC convergence rate is well understood for MV-SDEs with coefficients linearly dependent on the measure, but the rate deteriorates with dimension $d$ under the $L^p$-Wasserstein metric for nonlinear measure-dependent coefficients, even when Lipschitz continuity with respect to the measure is assumed. The main objective of this paper is to establish a dimension-independent convergence result of PoC for MV-SDEs whose coefficients are nonlinear with respect to the measure component but Lipschitz continuous. As a complement we further give the time discretization of the equations and thus verify the convergence rate of PoC using numerical experiments.

math.NA↗

arXiv2025

Strong convergence of the adaptive Milstein method for nonlinear stochastic differential equations with piecewise continuous arguments

In this work, an adaptive time-stepping Milstein method is constructed for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift is one-sided Lipschitz continuous and the diffusion does not impose the commutativity condition. It is widely recognized that explicit Euler or Milstein methods may blow up when the system exhibits superlinear growth, and modifications are needed. Hence we propose an adaptive variant to deal with the case of superlinear growth drift coefficient. To the best of our knowledge, this is the first work to develop a numerical method with variable step sizes for nonlinear SDEPCAs. It is proven that the adaptive Milstein method is strongly convergent in the sense of $L_p, p\ge 2$, and the convergence rate is optimal, which is consistent with the order of the explicit Milstein scheme with globally Lipschitz coefficients. Finally, several numerical experiments are presented to support the theoretical analysis.

math.NA↗

arXiv2024

Bootstrap Model Averaging

Model averaging has gained significant attention in recent years due to its ability of fusing information from different models. The critical challenge in frequentist model averaging is the choice of weight vector. The bootstrap method, known for its favorable properties, presents a new solution. In this paper, we propose a bootstrap model averaging approach that selects the weights by minimizing a bootstrap criterion. Our weight selection criterion can also be interpreted as a bootstrap aggregating. We demonstrate that the resultant estimator is asymptotically optimal in the sense that it achieves the lowest possible squared error loss. Furthermore, we establish the convergence rate of bootstrap weights tending to the theoretically optimal weights. Additionally, we derive the limiting distribution for our proposed model averaging estimator. Through simulation studies and empirical applications, we show that our proposed method often has better performance than other commonly used model selection and model averaging methods, and bootstrap variants.

stat.ME↗

arXiv2024

Time-domain direct sampling method for inverse electromagnetic scattering with a single incident source

In this paper, we consider an inverse electromagnetic medium scattering problem of reconstructing unknown objects from time-dependent boundary measurements. A novel time-domain direct sampling method is developed for determining the locations of unknown scatterers by using only a single incident source. Notably, our method imposes no restrictions on the the waveform of the incident wave. Based on the Fourier-Laplace transform, we first establish the connection between the frequency-domain and the time-domain direct sampling method. Furthermore, we elucidate the mathematical mechanism of the imaging functional through the properties of modified Bessel functions. Theoretical justifications and stability analyses are provided to demonstrate the effectiveness of the proposed method. Finally, several numerical experiments are presented to illustrate the feasibility of our approach.

math.NA↗

arXiv2024

E5-V: Universal Embeddings with Multimodal Large Language Models

Multimodal large language models (MLLMs) have shown promising advancements in general visual and language understanding. However, the representation of multimodal information using MLLMs remains largely unexplored. In this work, we introduce a new framework, E5-V, designed to adapt MLLMs for achieving universal multimodal embeddings. Our findings highlight the significant potential of MLLMs in representing multimodal inputs compared to previous approaches. By leveraging MLLMs with prompts, E5-V effectively bridges the modality gap between different types of inputs, demonstrating strong performance in multimodal embeddings even without fine-tuning. We propose a single modality training approach for E5-V, where the model is trained exclusively on text pairs. This method demonstrates significant improvements over traditional multimodal training on image-text pairs, while reducing training costs by approximately 95%. Additionally, this approach eliminates the need for costly multimodal training data collection. Extensive experiments across four types of tasks demonstrate the effectiveness of E5-V. As a universal multimodal model, E5-V not only achieves but often surpasses state-of-the-art performance in each task, despite being trained on a single modality.

cs.CL↗

arXiv2024

ASI++: Towards Distributionally Balanced End-to-End Generative Retrieval

Generative retrieval, a promising new paradigm in information retrieval, employs a seq2seq model to encode document features into parameters and decode relevant document identifiers (IDs) based on search queries. Existing generative retrieval solutions typically rely on a preprocessing stage to pre-define document IDs, which can suffer from a semantic gap between these IDs and the retrieval task. However, end-to-end training for both ID assignments and retrieval tasks is challenging due to the long-tailed distribution characteristics of real-world data, resulting in inefficient and unbalanced ID space utilization. To address these issues, we propose ASI++, a novel fully end-to-end generative retrieval method that aims to simultaneously learn balanced ID assignments and improve retrieval performance. ASI++ builds on the fully end-to-end training framework of vanilla ASI and introduces several key innovations. First, a distributionally balanced criterion addresses the imbalance in ID assignments, promoting more efficient utilization of the ID space. Next, a representation bottleneck criterion enhances dense representations to alleviate bottlenecks in learning ID assignments. Finally, an information consistency criterion integrates these processes into a joint optimization framework grounded in information theory. We further explore various module structures for learning ID assignments, including neural quantization, differentiable product quantization, and residual quantization. Extensive experiments on both public and industrial datasets demonstrate the effectiveness of ASI++ in improving retrieval performance and achieving balanced ID assignments.

cs.IR↗

arXiv2024

An adaptive Euler-Maruyama scheme for SDDEs

This paper proposes an adaptive numerical method for stochastic delay differential equations (SDDEs) with a non-global Lipschitz drift term and a non-constant delay, building upon the work of Wei Fang and others. The method adapts the step size based on the growth of the drift term. Differing slightly from the conventional Euler-Maruyama format, this paper addresses the estimation of the delay term by substituting it with the numerically obtained solution closest to the left endpoint.This approach overcomes the challenge of numerical nodes not falling within the nodes after subtracting the delay. The paper proves the convergence of the numerical method for a class of non-global Lipschitz continuous SDDEs under the assumption that the step size function satisfies certain conditions.

math.NA↗

arXiv2024

ResLoRA: Identity Residual Mapping in Low-Rank Adaption

As one of the most popular parameter-efficient fine-tuning (PEFT) methods, low-rank adaptation (LoRA) is commonly applied to fine-tune large language models (LLMs). However, updating the weights of LoRA blocks effectively and expeditiously is challenging due to the long calculation path in the original model. To address this, we propose ResLoRA, an improved framework of LoRA. By adding residual paths during training and using merging approaches to eliminate these extra paths during inference, our method can achieve better results in fewer training steps without any extra trainable parameters or inference cost compared to LoRA. The experiments on NLG, NLU, and text-to-image tasks demonstrate the effectiveness of our method. To the best of our knowledge, ResLoRA is the first work that combines the residual path with LoRA. The code of our method is available at https://github.com/microsoft/LMOps/tree/main/reslora .

cs.CL↗

arXiv2023

Democratizing Reasoning Ability: Tailored Learning from Large Language Model

Large language models (LLMs) exhibit impressive emergent abilities in natural language processing, but their democratization is hindered due to huge computation requirements and closed-source nature. Recent research on advancing open-source smaller LMs by distilling knowledge from black-box LLMs has obtained promising results in the instruction-following ability. However, the reasoning ability which is more challenging to foster, is relatively rarely explored. In this paper, we propose a tailored learning approach to distill such reasoning ability to smaller LMs to facilitate the democratization of the exclusive reasoning ability. In contrast to merely employing LLM as a data annotator, we exploit the potential of LLM as a reasoning teacher by building an interactive multi-round learning paradigm. This paradigm enables the student to expose its deficiencies to the black-box teacher who then can provide customized training data in return. Further, to exploit the reasoning potential of the smaller LM, we propose self-reflection learning to motivate the student to learn from self-made mistakes. The learning from self-reflection and LLM are all tailored to the student's learning status, thanks to the seamless integration with the multi-round learning paradigm. Comprehensive experiments and analysis on mathematical and commonsense reasoning tasks demonstrate the effectiveness of our method. The code will be available at https://github.com/Raibows/Learn-to-Reason.

cs.CL↗

arXiv2023

Auto Search Indexer for End-to-End Document Retrieval

Generative retrieval, which is a new advanced paradigm for document retrieval, has recently attracted research interests, since it encodes all documents into the model and directly generates the retrieved documents. However, its power is still underutilized since it heavily relies on the "preprocessed" document identifiers (docids), thus limiting its retrieval performance and ability to retrieve new documents. In this paper, we propose a novel fully end-to-end retrieval paradigm. It can not only end-to-end learn the best docids for existing and new documents automatically via a semantic indexing module, but also perform end-to-end document retrieval via an encoder-decoder-based generative model, namely Auto Search Indexer (ASI). Besides, we design a reparameterization mechanism to combine the above two modules into a joint optimization framework. Extensive experimental results demonstrate the superiority of our model over advanced baselines on both public and industrial datasets and also verify the ability to deal with new documents.

cs.IR↗

arXiv2022

The rate of Lp-convergence for the Euler-Maruyama method of the stochastic differential equations with Markovian switching

This work deals with the Euler-Maruyama (EM) scheme for stochastic differential equations with Markovian switching (SDEwMSs). We focus on the Lp-convergence rate (p is greater than or equal to 2) of the EM method given in this paper. As far as we know, the skeleton process of the Markov chain is used in the continuous numerical methods in most papers. By contrast, the continuous EM method in this paper is to use the Markov chain directly. To the best of our knowledge, there are only two papers that consider the rate of Lp-convergence, which is no more than 1/p (p is greater than or equal to 2) in these papers. The contribution of this paper is that the rate of Lp-convergence of the EM method can reach 1/2. We believe that the technique used in this paper to construct the EM method can also be used to construct other methods for SDEwMSs.

math.NA↗

arXiv2020

Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods

In this paper, we consider the equivalence of the $p$th moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is $p$th moment exponentially stable, then any of them is $p$th moment exponentially stable for a sufficiently small step size $h$ and $τ$ under the global Lipschitz assumption on the drift and diffusion coefficients

math.NA↗