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Zebang Shen

Publications and source records attributed to Zebang Shen.

At least 19 recordsLinked to original sources
arXiv2026

Support Before Frequency in Discrete Diffusion

Discrete diffusion models are increasingly competitive for language modeling, yet it remains unclear how their denoising objectives organize learning. Although these objectives target the full data distribution, we show that the exact reverse process induces a hierarchy between coarse support information and finer frequency information. For uniform and absorbing (a.k.a. masking) diffusion, we prove that, in the small-noise regime of the final denoising steps, each single-token reverse edit decomposes into a leading scale, determined by whether it moves toward the data support (e.g., grammatically valid sentences), and a finer coefficient, determining relative probabilities within the same scale. Thus, recovering validity structure only requires learning the correct order of magnitude of reverse probabilities, whereas recovering data frequencies requires coefficient-level estimation. The separation is mechanism-dependent: uniform diffusion exhibits a trichotomy into validity-improving, validity-preserving, and validity-worsening edits, while absorbing diffusion places its leading-order mass on validity-improving moves. Experiments on a masked language diffusion model and synthetic regular-language tasks support these predictions: support-localization emerges earlier than within-support frequency ranking, and the contrast between uniform and absorbing diffusion matches the predicted rate separation. Together, our results suggest that discrete diffusion models learn data support before data frequencies.

cs.LG↗
arXiv2026

Select-then-differentiate: Solving Bilevel Optimization with Manifold Lower-level Solution Sets

We study optimistic bilevel optimization when the lower-level problem has a non-isolated manifold of minimizers. In this setting, the hyper-objective may be non-differentiable because the upper-level criterion must choose among multiple lower-level solutions. Under a local Polyak--{\L}ojasiewicz (P{\L}) condition, we show that differentiability does not require the lower-level solution set to be a singleton: uniqueness of the optimistic selection is sufficient. This yields an explicit pseudoinverse-based hyper-gradient formula extending the classical singleton-minimizer result. We further characterize the regularity of the hyper-objective: non-degeneracy of the selected minimizer along the solution manifold yields local smoothness, while failure of uniqueness can create many non-differentiable points and failure of non-degeneracy can destroy all positive H\"older regularity of the hyper-gradient. Motivated by this theory, we propose HG-MS, a select-then-differentiate method combining explicit optimistic selection with efficient pseudoinverse-based hyper-gradient computation. Despite the nonconvex nature of optimistic selection over the lower-level solution manifold, we show that HG-MS converges to a stationary point of the optimistic objective with complexity governed by the intrinsic dimension of the solution manifold rather than its ambient dimension. Empirically, we test a practical variant of HG-MS for matched-budget LLM source reweighting. This variant preserves the select-then-differentiate principle and obtains the best GSM8K/MATH scores across the tested backbones, along with competitive or best MT-Bench instruction-following results.

math.OC↗
arXiv2026

On the Connectedness of Sublevel Sets in Invex Optimization

Understanding the topology of sublevel sets yields crucial insights into the optimization landscape of non-convex functions. If sublevel sets are connected, local search algorithms are less likely to be trapped in isolated valleys, facilitating convergence to global minimizers. However, few results exist to establish connectedness in the nonconvex setting. In this work, we present a mathematical toolkit based on the topological mountain pass theorem and use it to study invex functions, a class of functions that includes those satisfying the Polyak-{\L}ojasiewicz inequality and generalizations thereof. We show that their sublevel sets are connected under mild assumptions. We further leverage our result to establish the connectedness of different solution sets for invex-incave minimax problems and incave games.

math.OC↗
arXiv2026

A Schr\"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon $T$, with performance often deteriorating exponentially. We overcome these limitations for a subclass of linearly-solvable SOC problems-those whose uncontrolled drift is the gradient of a potential. In this setting, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator $\mathcal{L}$. We prove that, under the gradient drift assumption, $\mathcal{L}$ is unitarily equivalent to a Schr\"odinger operator $\mathcal{S} = -\Delta + \mathcal{V}$ with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of $\mathcal{L}$. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), $\mathcal{S}$ matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with \emph{arbitrary} terminal cost. Second, in a more general setting, we learn the eigensystem of $\mathcal{L}$ using neural networks. We identify implicit reweighting issues with existing eigenfunction learning losses that degrade performance in control tasks, and propose a novel loss function to mitigate this. We evaluate our method on several long-horizon benchmarks, achieving an order-of-magnitude improvement in control accuracy compared to state-of-the-art methods, while reducing memory usage and runtime complexity from $\mathcal{O}(Td)$ to $\mathcal{O}(d)$.

cs.LG↗
arXiv2026

Manifold Generalization Provably Proceeds Memorization in Diffusion Models

Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the \emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the \emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$\mu_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $\mu_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the \emph{regularity of the manifold support} and attain a near-parametric rate toward a \emph{different} target distribution. This target distribution has density uniformly comparable to that of~$\mu_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-\beta/(4k)}\bigr)$-neighborhood of the manifold, where $\beta$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that \emph{generalization} -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$\mu_{\scriptscriptstyle\mathrm{data}}$.

cs.LG↗
arXiv2025

Flow Density Control: Generative Optimization Beyond Entropy-Regularized Fine-Tuning

Adapting large-scale foundation flow and diffusion generative models to optimize task-specific objectives while preserving prior information is crucial for real-world applications such as molecular design, protein docking, and creative image generation. Existing principled fine-tuning methods aim to maximize the expected reward of generated samples, while retaining knowledge from the pre-trained model via KL-divergence regularization. In this work, we tackle the significantly more general problem of optimizing general utilities beyond average rewards, including risk-averse and novelty-seeking reward maximization, diversity measures for exploration, and experiment design objectives among others. Likewise, we consider more general ways to preserve prior information beyond KL-divergence, such as optimal transport distances and Renyi divergences. To this end, we introduce Flow Density Control (FDC), a simple algorithm that reduces this complex problem to a specific sequence of simpler fine-tuning tasks, each solvable via scalable established methods. We derive convergence guarantees for the proposed scheme under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we validate our method on illustrative settings, text-to-image, and molecular design tasks, showing that it can steer pre-trained generative models to optimize objectives and solve practically relevant tasks beyond the reach of current fine-tuning schemes.

cs.LG↗
arXiv2025

Scalable Neural Incentive Design with Parameterized Mean-Field Approximation

Designing incentives for a multi-agent system to induce a desirable Nash equilibrium is both a crucial and challenging problem appearing in many decision-making domains, especially for a large number of agents $N$. Under the exchangeability assumption, we formalize this incentive design (ID) problem as a parameterized mean-field game (PMFG), aiming to reduce complexity via an infinite-population limit. We first show that when dynamics and rewards are Lipschitz, the finite-$N$ ID objective is approximated by the PMFG at rate $\mathscr{O}(\frac{1}{\sqrt{N}})$. Moreover, beyond the Lipschitz-continuous setting, we prove the same $\mathscr{O}(\frac{1}{\sqrt{N}})$ decay for the important special case of sequential auctions, despite discontinuities in dynamics, through a tailored auction-specific analysis. Built on our novel approximation results, we further introduce our Adjoint Mean-Field Incentive Design (AMID) algorithm, which uses explicit differentiation of iterated equilibrium operators to compute gradients efficiently. By uniting approximation bounds with optimization guarantees, AMID delivers a powerful, scalable algorithmic tool for many-agent (large $N$) ID. Across diverse auction settings, the proposed AMID method substantially increases revenue over first-price formats and outperforms existing benchmark methods.

cs.GT↗
arXiv2025

When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis

Score-based methods, such as diffusion models and Bayesian inverse problems, are often interpreted as learning the data distribution in the low-noise limit ($\sigma \to 0$). In this work, we propose an alternative perspective: their success arises from implicitly learning the data manifold rather than the full distribution. Our claim is based on a novel analysis of scores in the small-$\sigma$ regime that reveals a sharp separation of scales: information about the data manifold is $\Theta(\sigma^{-2})$ stronger than information about the distribution. We argue that this insight suggests a paradigm shift from the less practical goal of distributional learning to the more attainable task of geometric learning, which provably tolerates $O(\sigma^{-2})$ larger errors in score approximation. We illustrate this perspective through three consequences: i) in diffusion models, concentration on data support can be achieved with a score error of $o(\sigma^{-2})$, whereas recovering the specific data distribution requires a much stricter $o(1)$ error; ii) more surprisingly, learning the uniform distribution on the manifold-an especially structured and useful object-is also $O(\sigma^{-2})$ easier; and iii) in Bayesian inverse problems, the maximum entropy prior is $O(\sigma^{-2})$ more robust to score errors than generic priors. Finally, we validate our theoretical findings with preliminary experiments on large-scale models, including Stable Diffusion.

stat.ML↗
arXiv2025

Landing with the Score: Riemannian Optimization through Denoising

Under the data manifold hypothesis, high-dimensional data are concentrated near a low-dimensional manifold. We study the problem of Riemannian optimization over such manifolds when they are given only implicitly through the data distribution, and the standard manifold operations required by classical algorithms are unavailable. This formulation captures a broad class of data-driven design problems that are central to modern generative AI. Our key idea is to introduce a link function that connects the data distribution to the geometric operations needed for optimization. We show that this function enables the recovery of essential manifold operations, such as retraction and Riemannian gradient computation. Moreover, we establish a direct connection between our construction and the score function in diffusion models of the data distribution. This connection allows us to leverage well-studied parameterizations, efficient training procedures, and even pretrained score networks from the diffusion model literature to perform optimization. Building on this foundation, we propose two efficient inference-time algorithms -- Denoising Landing Flow (DLF) and Denoising Riemannian Gradient Descent (DRGD) -- and provide theoretical guarantees for both feasibility (approximate manifold adherence) and optimality (small Riemannian gradient norm). Finally, we demonstrate the effectiveness of our approach on finite-horizon reference tracking tasks in data-driven control, highlighting its potential for practical generative and design applications.

cs.LG↗
arXiv2025

Adjoint-Based Aerodynamic Shape Optimization with a Manifold Constraint Learned by Diffusion Models

We introduce an adjoint-based aerodynamic shape optimization framework that integrates a diffusion model trained on existing designs to learn a smooth manifold of aerodynamically viable shapes. This manifold is enforced as an equality constraint to the shape optimization problem. Central to our method is the computation of adjoint gradients of the design objectives (e.g., drag and lift) with respect to the manifold space. These gradients are derived by first computing shape derivatives with respect to conventional shape design parameters (e.g., Hicks-Henne parameters) and then backpropagating them through the diffusion model to its latent space via automatic differentiation. Our framework preserves mathematical rigor and can be integrated into existing adjoint-based design workflows with minimal modification. Demonstrated on extensive transonic RANS airfoil design cases using off-the-shelf and general-purpose nonlinear optimizers, our approach eliminates ad hoc parameter tuning and variable scaling, maintains robustness across initialization and optimizer choices, and achieves superior aerodynamic performance compared to conventional approaches. This work establishes how AI generated priors integrates effectively with adjoint methods to enable robust, high-fidelity aerodynamic shape optimization through automatic differentiation.

cs.CE↗
arXiv2025

GS-Occ3D: Scaling Vision-only Occupancy Reconstruction with Gaussian Splatting

Occupancy is crucial for autonomous driving, providing essential geometric priors for perception and planning. However, existing methods predominantly rely on LiDAR-based occupancy annotations, which limits scalability and prevents leveraging vast amounts of potential crowdsourced data for auto-labeling. To address this, we propose GS-Occ3D, a scalable vision-only framework that directly reconstructs occupancy. Vision-only occupancy reconstruction poses significant challenges due to sparse viewpoints, dynamic scene elements, severe occlusions, and long-horizon motion. Existing vision-based methods primarily rely on mesh representation, which suffer from incomplete geometry and additional post-processing, limiting scalability. To overcome these issues, GS-Occ3D optimizes an explicit occupancy representation using an Octree-based Gaussian Surfel formulation, ensuring efficiency and scalability. Additionally, we decompose scenes into static background, ground, and dynamic objects, enabling tailored modeling strategies: (1) Ground is explicitly reconstructed as a dominant structural element, significantly improving large-area consistency; (2) Dynamic vehicles are separately modeled to better capture motion-related occupancy patterns. Extensive experiments on the Waymo dataset demonstrate that GS-Occ3D achieves state-of-the-art geometry reconstruction results. By curating vision-only binary occupancy labels from diverse urban scenes, we show their effectiveness for downstream occupancy models on Occ3D-Waymo and superior zero-shot generalization on Occ3D-nuScenes. It highlights the potential of large-scale vision-based occupancy reconstruction as a new paradigm for scalable auto-labeling. Project Page: https://gs-occ3d.github.io/

cs.CV↗
arXiv2025

Provable Maximum Entropy Manifold Exploration via Diffusion Models

Exploration is critical for solving real-world decision-making problems such as scientific discovery, where the objective is to generate truly novel designs rather than mimic existing data distributions. In this work, we address the challenge of leveraging the representational power of generative models for exploration without relying on explicit uncertainty quantification. We introduce a novel framework that casts exploration as entropy maximization over the approximate data manifold implicitly defined by a pre-trained diffusion model. Then, we present a novel principle for exploration based on density estimation, a problem well-known to be challenging in practice. To overcome this issue and render this method truly scalable, we leverage a fundamental connection between the entropy of the density induced by a diffusion model and its score function. Building on this, we develop an algorithm based on mirror descent that solves the exploration problem as sequential fine-tuning of a pre-trained diffusion model. We prove its convergence to the optimal exploratory diffusion model under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we empirically evaluate our approach on both synthetic and high-dimensional text-to-image diffusion, demonstrating promising results.

cs.LG↗
arXiv2025

Superquantile-Gibbs Relaxation for Minima-selection in Bilevel Optimization

Bilevel optimization (BLO) becomes more challenging when the lower-level objective admits multiple minimizers. Compared with the commonly studied unique-minimizer setting, this introduces two difficulties: (1) evaluating the hyper-objective $F_{\max}$ requires minima selection over the lower-level solution set; and (2) $F_{\max}$ may be discontinuous without additional structure. We address both issues under a parameter-uniform local Polyak-Lojasiewicz (PL) condition on the lower-level objective, denoted by $\mathrm{PL}^{\circ}$. Unlike the global PL condition often assumed in BLO, the PL inequality in $\mathrm{PL}^{\circ}$ is imposed only near the local minima. This formulation accommodates bounded, non-singleton minimizer sets and is motivated by hyperparameter tuning in over-parameterized learning. We show that $F_{\max}$ is Lipschitz continuous and that the lower-level minimizer sets are connected, compact, embedded submanifolds with a common intrinsic dimension $k$. This intrinsic dimension determines the explicit accuracy exponents in our complexity bound. We propose a randomized method whose output satisfies an $(\mathcal O(\epsilon),\rho)$-Goldstein stationarity condition for $F_{\max}$ in expectation. In the small-accuracy regime, suppressing logarithmic and fixed-problem factors, it uses at most $\widetilde{\mathcal O}(m^{15+12k}\epsilon^{-2}(\epsilon\rho)^{-12k-14})$ queries to the gradient oracle of the lower-level objective, where $m$ is the upper-level dimension. Its key ingredient is a Superquantile-Gibbs relaxation that provides an approximate function-evaluation oracle for $F_{\max}$ and turns minima selection into a sampling problem addressed by standard Langevin dynamics. To our knowledge, this is the first rigorous treatment of minima selection in BLO that explicitly relates overall complexity to the lower-level intrinsic dimension.

math.OC↗
arXiv2025

Poincar\'e Inequality for Local Log-Polyak-Lojasiewicz Measures : Non-asymptotic Analysis in Low-temperature Regime

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such landscapes, we propose to study the class of \logPLmeasure\ measures $\mu_\epsilon \propto \exp(-V/\epsilon)$, where the potential $V$ satisfies a local Polyak-{\L}ojasiewicz (P\L) inequality, and its set of local minima is provably \emph{connected}. Notably, potentials in this class can exhibit local maxima and we characterize its optimal set S to be a compact $\mathcal{C}^2$ \emph{embedding submanifold} of $\mathbb{R}^d$ without boundary. The \emph{non-contractibility} of S distinguishes our function class from the classical convex setting topologically. Moreover, the embedding structure induces a naturally defined Laplacian-Beltrami operator on S, and we show that its first non-trivial eigenvalue provides an \emph{$\epsilon$-independent} lower bound for the \Poincare\ constant in the \Poincare\ inequality of $\mu_\epsilon$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $\epsilon$ converges to its equilibrium $\mu_\epsilon$ at a rate of $\tilde{\mathcal{O}}(1/\epsilon)$, provided $\epsilon$ is sufficiently small. Here $\tilde{\mathcal{O}}$ hides logarithmic terms.

cs.LG↗
arXiv2024

Poincare Inequality for Local Log-Polyak-\L ojasiewicz Measures: Non-asymptotic Analysis in Low-temperature Regime

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such landscapes, we propose to study the class of log-P{\L}$^\circ$ measures $\mu_\epsilon \propto \exp(-V/\epsilon)$, where the potential $V$ satisfies a local Polyak-{\L}ojasiewicz (P{\L}) inequality, and its set of local minima is provably connected. Notably, potentials in this class can exhibit local maxima and we characterize its optimal set $S$ to be a compact ${C}^2$ embedding submanifold of ${R}^d$ without boundary. The non-contractibility of $S$ distinguishes our function class from the classical convex setting topologically. Moreover, the embedding structure induces a naturally defined Laplacian-Beltrami operator on $S$, and we show that its first non-trivial eigenvalue provides an $\epsilon$-independent lower bound for the Poincar\'e constant in the Poincar\'e inequality of $\mu_\epsilon$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $\epsilon$ converges to its equilibrium $\mu_\epsilon$ at a rate of $\tilde{O}(1/\epsilon)$, provided $\epsilon$ is sufficiently small. Here $\tilde{O}$ hides logarithmic terms.

math.PR↗
arXiv2024

Learning to Steer Markovian Agents under Model Uncertainty

Designing incentives for an adapting population is a ubiquitous problem in a wide array of economic applications and beyond. In this work, we study how to design additional rewards to steer multi-agent systems towards desired policies \emph{without} prior knowledge of the agents' underlying learning dynamics. Motivated by the limitation of existing works, we consider a new and general category of learning dynamics called \emph{Markovian agents}. We introduce a model-based non-episodic Reinforcement Learning (RL) formulation for our steering problem. Importantly, we focus on learning a \emph{history-dependent} steering strategy to handle the inherent model uncertainty about the agents' learning dynamics. We introduce a novel objective function to encode the desiderata of achieving a good steering outcome with reasonable cost. Theoretically, we identify conditions for the existence of steering strategies to guide agents to the desired policies. Complementing our theoretical contributions, we provide empirical algorithms to approximately solve our objective, which effectively tackles the challenge in learning history-dependent strategies. We demonstrate the efficacy of our algorithms through empirical evaluations.

cs.LG↗
arXiv2024

A Hessian-Aware Stochastic Differential Equation for Modelling SGD

Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME achieves the order-best approximation error guarantee among existing SDE models in the literature, while significantly reducing the dependence on the smoothness parameter of the objective. Empirical experiments on neural network-based loss functions further validate this improvement. Further, for quadratic objectives, under mild conditions, HA-SME is proved to be the first SDE model that recovers exactly the SGD dynamics in the distributional sense. Consequently, when the local landscape near a stationary point can be approximated by quadratics, HA-SME provides a more precise characterization of the local escaping behaviors of SGD. With the enhanced approximation guarantee, we further conduct an escape time analysis using HA-SME, showcasing how it can be employed to analytically study the escaping behavior of SGD for general function classes.

stat.ML↗
arXiv2023

Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs

We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.

math.NA↗