arXiv subjects

Niao He

Publications and source records attributed to Niao He.

At least 55 records · Page 3Linked to original sources

arXiv2025

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↗

arXiv2025

Can RLHF be More Efficient with Imperfect Reward Models? A Policy Coverage Perspective

Sample efficiency is critical for online Reinforcement Learning from Human Feedback (RLHF). While existing works investigate sample-efficient online exploration strategies, the potential of utilizing misspecified yet relevant reward models to accelerate learning remains underexplored. This paper studies how to transfer knowledge from those imperfect reward models in online RLHF. We start by identifying a novel property due to KL-regularization in the RLHF objective: \emph{a policy's coverability of the optimal policy is captured by its sub-optimality}. Building on this insight, we propose novel transfer learning principles and a theoretical algorithm -- \emph{\textbf{T}ransfer \textbf{P}olicy \textbf{O}ptimization (\textbf{TPO})} -- with provable benefits compared to standard online learning. Empirically, inspired by our theoretical findings, we develop a win-rate-based transfer policy selection strategy with improved computational efficiency. Moreover, our empirical transfer learning technique is modular and can be integrated with various policy optimization methods, such as DPO, IPO and XPO, to further enhance their performance. We validate the effectiveness of our method through experiments on summarization tasks.

cs.LG↗

arXiv2025

Steering No-Regret Agents in MFGs under Model Uncertainty

Incentive design is a popular framework for guiding agents' learning dynamics towards desired outcomes by providing additional payments beyond intrinsic rewards. However, most existing works focus on a finite, small set of agents or assume complete knowledge of the game, limiting their applicability to real-world scenarios involving large populations and model uncertainty. To address this gap, we study the design of steering rewards in Mean-Field Games (MFGs) with density-independent transitions, where both the transition dynamics and intrinsic reward functions are unknown. This setting presents non-trivial challenges, as the mediator must incentivize the agents to explore for its model learning under uncertainty, while simultaneously steer them to converge to desired behaviors without incurring excessive incentive payments. Assuming agents exhibit no(-adaptive) regret behaviors, we contribute novel optimistic exploration algorithms. Theoretically, we establish sub-linear regret guarantees for the cumulative gaps between the agents' behaviors and the desired ones. In terms of the steering cost, we demonstrate that our total incentive payments incur only sub-linear excess, competing with a baseline steering strategy that stabilizes the target policy as an equilibrium. Our work presents an effective framework for steering agents behaviors in large-population systems under uncertainty.

cs.LG↗

arXiv2025

Primal Methods for Variational Inequality Problems with Functional Constraints

Variational inequality problems are recognized for their broad applications across various fields including machine learning and operations research. First-order methods have emerged as the standard approach for solving these problems due to their simplicity and scalability. However, they typically rely on projection or linear minimization oracles to navigate the feasible set, which becomes computationally expensive in practical scenarios featuring multiple functional constraints. Existing efforts to tackle such functional constrained variational inequality problems have centered on primal-dual algorithms grounded in the Lagrangian function. These algorithms along with their theoretical analysis often require the existence and prior knowledge of the optimal Lagrange multipliers. In this work, we propose a simple primal method, termed Constrained Gradient Method (CGM), for addressing functional constrained variational inequality problems, without requiring any information on the optimal Lagrange multipliers. We establish a non-asymptotic convergence analysis of the algorithm for Minty variational inequality problems with monotone operators under smooth constraints. Remarkably, our algorithms match the complexity of projection-based methods in terms of operator queries for both monotone and strongly monotone settings, while using significantly cheaper oracles based on quadratic programming. Furthermore, we provide several numerical examples to evaluate the efficacy of our algorithms.

math.OC↗

arXiv2025

Automated Design of Affine Maximizer Mechanisms in Dynamic Settings

Dynamic mechanism design is a challenging extension to ordinary mechanism design in which the mechanism designer must make a sequence of decisions over time in the face of possibly untruthful reports of participating agents. Optimizing dynamic mechanisms for welfare is relatively well understood. However, there has been less work on optimizing for other goals (e.g. revenue), and without restrictive assumptions on valuations, it is remarkably challenging to characterize good mechanisms. Instead, we turn to automated mechanism design to find mechanisms with good performance in specific problem instances. In fact, the situation is similar even in static mechanism design. However, in the static case, optimization/machine learning-based automated mechanism design techniques have been successful in finding high-revenue mechanisms in cases beyond the reach of analytical results. We extend the class of affine maximizer mechanisms to MDPs where agents may untruthfully report their rewards. This extension results in a challenging bilevel optimization problem in which the upper problem involves choosing optimal mechanism parameters, and the lower problem involves solving the resulting MDP. Our approach can find truthful dynamic mechanisms that achieve strong performance on goals other than welfare, and can be applied to essentially any problem setting-without restrictions on valuations-for which RL can learn optimal policies.

cs.GT↗

arXiv2025

Poincare Inequality for Local Log-Polyak-Łojasiewicz 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-PŁ$^\circ$ measures $μ_ε\propto \exp(-V/ε)$, where the potential $V$ satisfies a local Polyak-Łojasiewicz (PŁ) 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 $ε$-independent lower bound for the Poincaré constant in the Poincaré inequality of $μ_ε$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $ε$ converges to its equilibrium $μ_ε$ at a rate of $\tilde{O}(1/ε)$, provided $ε$ is sufficiently small. Here $\tilde{O}$ hides logarithmic terms.

math.PR↗

arXiv2025

Poincaré 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 $μ_ε\propto \exp(-V/ε)$, where the potential $V$ satisfies a local Polyak-Łojasiewicz (PŁ) 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{$ε$-independent} lower bound for the \Poincare\ constant in the \Poincare\ inequality of $μ_ε$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $ε$ converges to its equilibrium $μ_ε$ at a rate of $\tilde{\mathcal{O}}(1/ε)$, provided $ε$ is sufficiently small. Here $\tilde{\mathcal{O}}$ hides logarithmic terms.

cs.LG↗

arXiv2025

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↗

arXiv2025

A Variational Inequality Approach to Independent Learning in Static Mean-Field Games

Competitive games involving thousands or even millions of players are prevalent in real-world contexts, such as transportation, communications, and computer networks. However, learning in these large-scale multi-agent environments presents a grand challenge, often referred to as the "curse of many agents". In this paper, we formalize and analyze the Static Mean-Field Game (SMFG) under both full and bandit feedback, offering a generic framework for modeling large population interactions while enabling independent learning. We first establish close connections between SMFG and variational inequality (VI), showing that SMFG can be framed as a VI problem in the infinite agent limit. Building on the VI perspective, we propose independent learning and exploration algorithms that efficiently converge to approximate Nash equilibria, when dealing with a finite number of agents. Theoretically, we provide explicit finite sample complexity guarantees for independent learning across various feedback models in repeated play scenarios, assuming (strongly-)monotone payoffs. Numerically, we validate our results through both simulations and real-world applications in city traffic and network access management.

math.OC↗

arXiv2024

On the Crucial Role of Initialization for Matrix Factorization

This work revisits the classical low-rank matrix factorization problem and unveils the critical role of initialization in shaping convergence rates for such nonconvex and nonsmooth optimization. We introduce Nystrom initialization, which significantly improves the global convergence of Scaled Gradient Descent (ScaledGD) in both symmetric and asymmetric matrix factorization tasks. Specifically, we prove that ScaledGD with Nystrom initialization achieves quadratic convergence in cases where only linear rates were previously known. Furthermore, we extend this initialization to low-rank adapters (LoRA) commonly used for finetuning foundation models. Our approach, NoRA, i.e., LoRA with Nystrom initialization, demonstrates superior performance across various downstream tasks and model scales, from 1B to 7B parameters, in large language and diffusion models.

cs.LG↗

arXiv2024

Stochastic Optimization under Hidden Convexity

In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from optimal control, revenue and inventory management, to convex reinforcement learning all admit such a hidden convex structure. Unfortunately, in the majority of applications considered, the map $c(\cdot)$ is unavailable or implicit; therefore, directly solving the convex reformulation is not possible. On the other hand, the stochastic gradients with respect to the original variable are often easy to obtain. Motivated by these observations, we examine the basic projected stochastic (sub-) gradient methods for solving such problems under hidden convexity. We provide the first sample complexity guarantees for global convergence in smooth and non-smooth settings. Additionally, in the smooth setting, we improve our results to the last iterate convergence in terms of function value gap using the momentum variant of projected stochastic gradient descent.

math.OC↗

arXiv2024

Implicit Regularization of Sharpness-Aware Minimization for Scale-Invariant Problems

Sharpness-aware minimization (SAM) improves generalization of various deep learning tasks. Motivated by popular architectures such as LoRA, we explore the implicit regularization of SAM for scale-invariant problems involving two groups of variables. Instead of focusing on commonly used sharpness, this work introduces a concept termed balancedness, defined as the difference between the squared norm of two variables. This allows us to depict richer global behaviors of SAM. In particular, our theoretical and empirical findings reveal that i) SAM promotes balancedness; and ii) the regularization on balancedness is data-responsive -- outliers have stronger impact. The latter coincides with empirical observations that SAM outperforms SGD in the presence of outliers. Leveraging the implicit regularization, we develop a resource-efficient SAM variant, balancedness-aware regularization (BAR), tailored for scale-invariant problems such as finetuning language models with LoRA. BAR saves 95% computational overhead of SAM, with enhanced test performance across various tasks on RoBERTa, GPT2, and OPT-1.3B.

cs.LG↗

arXiv2024

On the Statistical Efficiency of Mean-Field Reinforcement Learning with General Function Approximation

In this paper, we study the fundamental statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general model-based function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MF-MBED), which characterizes the inherent complexity of mean-field model classes. We show that a rich family of Mean-Field RL problems exhibits low MF-MBED. Additionally, we propose algorithms based on maximal likelihood estimation, which can return an $ε$-optimal policy for MFC or an $ε$-Nash Equilibrium policy for MFG. The overall sample complexity depends only polynomially on MF-MBED, which is potentially much lower than the size of state-action space. Compared with previous works, our results only require the minimal assumptions including realizability and Lipschitz continuity.

cs.LG↗

arXiv2024

Exploiting Approximate Symmetry for Efficient Multi-Agent Reinforcement Learning

Mean-field games (MFG) have become significant tools for solving large-scale multi-agent reinforcement learning problems under symmetry. However, the assumption of exact symmetry limits the applicability of MFGs, as real-world scenarios often feature inherent heterogeneity. Furthermore, most works on MFG assume access to a known MFG model, which might not be readily available for real-world finite-agent games. In this work, we broaden the applicability of MFGs by providing a methodology to extend any finite-player, possibly asymmetric, game to an "induced MFG". First, we prove that $N$-player dynamic games can be symmetrized and smoothly extended to the infinite-player continuum via explicit Kirszbraun extensions. Next, we propose the notion of $α,β$-symmetric games, a new class of dynamic population games that incorporate approximate permutation invariance. For $α,β$-symmetric games, we establish explicit approximation bounds, demonstrating that a Nash policy of the induced MFG is an approximate Nash of the $N$-player dynamic game. We show that TD learning converges up to a small bias using trajectories of the $N$-player game with finite-sample guarantees, permitting symmetrized learning without building an explicit MFG model. Finally, for certain games satisfying monotonicity, we prove a sample complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-6})$ for the $N$-agent game to learn an $\varepsilon$-Nash up to symmetrization bias. Our theory is supported by evaluations on MARL benchmarks with thousands of agents.

cs.GT↗

arXiv2024

Multi-level Monte-Carlo Gradient Methods for Stochastic Optimization with Biased Oracles

We consider stochastic optimization when one only has access to biased stochastic oracles of the objective and the gradient, and obtaining stochastic gradients with low biases comes at high costs. This setting captures various optimization paradigms, such as conditional stochastic optimization, distributionally robust optimization, shortfall risk optimization, and machine learning paradigms, such as contrastive learning. We examine a family of multi-level Monte Carlo (MLMC) gradient methods that exploit a delicate tradeoff among bias, variance, and oracle cost. We systematically study their total sample and computational complexities for strongly convex, convex, and nonconvex objectives and demonstrate their superiority over the widely used biased stochastic gradient method. When combined with the variance reduction techniques like SPIDER, these MLMC gradient methods can further reduce the complexity in the nonconvex regime. Our results imply that a series of stochastic optimization problems with biased oracles, previously considered to be more challenging, is fundamentally no harder than the classical stochastic optimization with unbiased oracles. We also delineate the boundary conditions under which these problems become more difficult. Moreover, MLMC gradient methods significantly improve the best-known complexities in the literature for conditional stochastic optimization and shortfall risk optimization. Our extensive numerical experiments on distributionally robust optimization, pricing and staffing scheduling problems, and contrastive learning demonstrate the superior performance of MLMC gradient methods.

math.OC↗

arXiv2024

Independent Policy Mirror Descent for Markov Potential Games: Scaling to Large Number of Players

Markov Potential Games (MPGs) form an important sub-class of Markov games, which are a common framework to model multi-agent reinforcement learning problems. In particular, MPGs include as a special case the identical-interest setting where all the agents share the same reward function. Scaling the performance of Nash equilibrium learning algorithms to a large number of agents is crucial for multi-agent systems. To address this important challenge, we focus on the independent learning setting where agents can only have access to their local information to update their own policy. In prior work on MPGs, the iteration complexity for obtaining $ε$-Nash regret scales linearly with the number of agents $N$. In this work, we investigate the iteration complexity of an independent policy mirror descent (PMD) algorithm for MPGs. We show that PMD with KL regularization, also known as natural policy gradient, enjoys a better $\sqrt{N}$ dependence on the number of agents, improving over PMD with Euclidean regularization and prior work. Furthermore, the iteration complexity is also independent of the sizes of the agents' action spaces.

cs.LG↗

arXiv2024

Truly No-Regret Learning in Constrained MDPs

Constrained Markov decision processes (CMDPs) are a common way to model safety constraints in reinforcement learning. State-of-the-art methods for efficiently solving CMDPs are based on primal-dual algorithms. For these algorithms, all currently known regret bounds allow for error cancellations -- one can compensate for a constraint violation in one round with a strict constraint satisfaction in another. This makes the online learning process unsafe since it only guarantees safety for the final (mixture) policy but not during learning. As Efroni et al. (2020) pointed out, it is an open question whether primal-dual algorithms can provably achieve sublinear regret if we do not allow error cancellations. In this paper, we give the first affirmative answer. We first generalize a result on last-iterate convergence of regularized primal-dual schemes to CMDPs with multiple constraints. Building upon this insight, we propose a model-based primal-dual algorithm to learn in an unknown CMDP. We prove that our algorithm achieves sublinear regret without error cancellations.

cs.LG↗

arXiv2024

Efficient Algorithms for A Class of Stochastic Hidden Convex Optimization and Its Applications in Network Revenue Management

We study a class of stochastic nonconvex optimization in the form of $\min_{x\in\mathcal{X}} F(x):=\mathbb{E}_ξ[f(ϕ(x,ξ))]$, i.e., $F$ is a composition of a convex function $f$ and a random function $ϕ$. Leveraging an (implicit) convex reformulation via a variable transformation $u=\mathbb{E}[ϕ(x,ξ)]$, we develop stochastic gradient-based algorithms and establish their sample and gradient complexities for achieving an $ε$-global optimal solution. Interestingly, our proposed Mirror Stochastic Gradient (MSG) method operates only in the original $x$-space using gradient estimators of the original nonconvex objective $F$ and achieves $\tilde{\mathcal{O}}(ε^{-2})$ complexities, which matches the lower bounds for solving stochastic convex optimization problems. Under booking limits control, we formulate the air-cargo network revenue management (NRM) problem with random two-dimensional capacity, random consumption, and routing flexibility as a special case of the stochastic nonconvex optimization, where the random function $ϕ(x,ξ)=x\wedgeξ$, i.e., the random demand $ξ$ truncates the booking limit decision $x$. Extensive numerical experiments demonstrate the superior performance of our proposed MSG algorithm for booking limit control with higher revenue and lower computation cost than state-of-the-art bid-price-based control policies, especially when the variance of random capacity is large. KEYWORDS: stochastic nonconvex optimization, hidden convexity, gradient methods, passenger network revenue management, air-cargo network revenue management

math.OC↗