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Yinyu Ye

Publications and source records attributed to Yinyu Ye.

At least 73 records · Page 4Linked to original sources

Adam-mini: Use Fewer Learning Rates To Gain More

We propose Adam-mini, an optimizer that achieves on par or better performance than AdamW with 50% less memory footprint. Adam-mini reduces memory by cutting down the learning rate resources in Adam (i.e., $1/\sqrt{v}$). By investigating the Hessian structure of neural nets, we find Adam's $v$ might not function at its full potential as effectively as we expected. We find that $\geq$ 99.9% of these learning rates in $v$ could be harmlessly removed if we (1) carefully partition the parameters into blocks following our new principle on Hessian structure; (2) assign a single but good learning rate to each parameter block. We then provide one simple way to find good learning rates and propose Adam-mini. Empirically, we verify that Adam-mini performs on par or better than AdamW on various language models sized from 39M to 13B for pre-training, supervised fine-tuning, and RLHF. The reduced memory footprint of Adam-mini also alleviates communication overheads among GPUs, thereby increasing throughput. For instance, Adam-mini achieves 49.6% higher throughput than AdamW when pre-training Llama 2-7B on $2\times$ A800-80GB GPUs, which saves 33% wall-clock time for pre-training.

cs.LG↗

Trust Region Methods For Nonconvex Stochastic Optimization Beyond Lipschitz Smoothness

In many important machine learning applications, the standard assumption of having a globally Lipschitz continuous gradient may fail to hold. This paper delves into a more general $(L_0, L_1)$-smoothness setting, which gains particular significance within the realms of deep neural networks and distributionally robust optimization (DRO). We demonstrate the significant advantage of trust region methods for stochastic nonconvex optimization under such generalized smoothness assumption. We show that first-order trust region methods can recover the normalized and clipped stochastic gradient as special cases and then provide a unified analysis to show their convergence to first-order stationary conditions. Motivated by the important application of DRO, we propose a generalized high-order smoothness condition, under which second-order trust region methods can achieve a complexity of $\mathcal{O}(ε^{-3.5})$ for convergence to second-order stationary points. By incorporating variance reduction, the second-order trust region method obtains an even better complexity of $\mathcal{O}(ε^{-3})$, matching the optimal bound for standard smooth optimization. To our best knowledge, this is the first work to show convergence beyond the first-order stationary condition for generalized smooth optimization. Preliminary experiments show that our proposed algorithms perform favorably compared with existing methods.

math.OC↗

Decoupling Learning and Decision-Making: Breaking the $\mathcal{O}(\sqrt{T})$ Barrier in Online Resource Allocation with First-Order Methods

Online linear programming plays an important role in both revenue management and resource allocation, and recent research has focused on developing efficient first-order online learning algorithms. Despite the empirical success of first-order methods, they typically achieve a regret no better than $\mathcal{O}(\sqrt{T})$, which is suboptimal compared to the $\mathcal{O}(\log T)$ bound guaranteed by the state-of-the-art linear programming (LP)-based online algorithms. This paper establishes several important facts about online linear programming, which unveils the challenge for first-order-method-based online algorithms to achieve beyond $\mathcal{O}(\sqrt{T})$ regret. To address the challenge, we introduce a new algorithmic framework that decouples learning from decision-making. For the first time, we show that first-order methods can attain regret $\mathcal{O}(T^{1/3})$ with this new framework.

cs.LG↗

Beyond $\mathcal{O}(\sqrt{T})$ Regret: Decoupling Learning and Decision-making in Online Linear Programming

Online linear programming plays an important role in both revenue management and resource allocation, and recent research has focused on developing efficient first-order online learning algorithms. Despite the empirical success of first-order methods, they typically achieve a regret no better than $\mathcal{O} ( \sqrt{T} )$, which is suboptimal compared to the $\mathcal{O} (\log T)$ bound guaranteed by the state-of-the-art linear programming (LP)-based online algorithms. This paper establishes a general framework that improves upon the $\mathcal{O} ( \sqrt{T} )$ result when the LP dual problem exhibits certain error bound conditions. For the first time, we show that first-order learning algorithms achieve $o( \sqrt{T} )$ regret in the continuous support setting and $\mathcal{O} (\log T)$ regret in the finite support setting beyond the non-degeneracy assumption. Our results significantly improve the state-of-the-art regret results and provide new insights for sequential decision-making.

stat.ML↗

From an Interior Point to a Corner Point: Smart Crossover

Identifying optimal basic feasible solutions to linear programming problems is a critical task for mixed integer programming and other applications. The crossover method, which aims at deriving an optimal extreme point from a suboptimal solution (the output of a starting method such as interior-point methods or first-order methods), is crucial in this process. This method, compared to the starting method, frequently represents the primary computational bottleneck in practical applications. We propose approaches to overcome this bottleneck by exploiting problem characteristics and implementing customized strategies. For problems arising from network applications and exhibiting network structures, we take advantage of the graph structure of the problem and the tree structure of the optimal solutions. Based on these structures, we propose a tree-based crossover method, aiming to recovering basic solutions by identifying nearby spanning tree structures. For general linear programs, we propose recovering an optimal basic solution by identifying the optimal face and employing controlled perturbations based on the suboptimal solution provided by interior-point methods. We prove that an optimal solution for the perturbed problem is an extreme point, and its objective value is at least as good as that of the initial interior point solution. Computational experiments show significant speed-ups achieved by our methods compared to state-of-the-art commercial solvers on classical linear programming problem benchmarks, network flow problem benchmarks, and optimal transport problems.

math.OC↗

When Does Primal Interior Point Method Beat Primal-dual in Linear Optimization?

The primal-dual interior point method (IPM) is widely regarded as the most efficient IPM variant for linear optimization. In this paper, we demonstrate that the improved stability of the pure primal IPM can allow speedups relative to a primal-dual solver, particularly as the IPM approaches convergence. The stability of the primal scaling matrix makes it possible to accelerate each primal IPM step using fast preconditioned iterative solvers for the normal equations. Crucially, we identify properties of the central path that make it possible to stabilize the normal equations. Experiments on benchmark datasets demonstrate the efficiency of primal IPM and showcase its potential for practical applications in linear optimization and beyond.

math.OC↗

Orientation Determination of Cryo-EM Images Using Block Stochastic Riemannian Subgradient Methods

The determination of molecular orientations is crucial for the three-dimensional reconstruction of Cryo-EM images. Traditionally addressed using the common-line method, this challenge is reformulated as a self-consistency error minimization problem constrained to rotation groups. In this paper, we consider the least-squared deviation (LUD) formulation and employ a Riemannian subgradient method to effectively solve the orientation determination problem. To enhance computational efficiency, a block stochastic version of the method is proposed, and its convergence properties are rigorously established. Extensive numerical evaluations reveal that our method not only achieves accuracy comparable to that of state-of-the-art methods but also delivers an average 20-fold speedup. Additionally, we implement a modified formulation and algorithm specifically designed to address scenarios characterized by very low SNR.

math.OC↗

Solving Linear Programs with Fast Online Learning Algorithms

This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a variable-duplication technique that copies each variable $K$ times and reduces the optimality gap and constraint violation by a factor of $\sqrt{K}$. Furthermore, we show how online algorithms can be effectively integrated into sifting, a column generation scheme for large-scale LPs. Numerical experiments demonstrate that our methods can serve as either an approximate direct solver, or an initialization subroutine for exact LP solving.

math.OC↗

Scalable Approximate Optimal Diagonal Preconditioning

We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable to matrices of large dimension, or when access is limited to black-box matrix-vector products, thereby significantly limiting their practical application. In view of these challenges, we propose practical algorithms applicable to finding approximate optimal diagonal preconditioners of large sparse systems. Our approach is based on the idea of dimension reduction, and combines techniques from semi-definite programming (SDP), random projection, semi-infinite programming (SIP), and column generation. Numerical experiments demonstrate that our method scales to sparse matrices of size greater than $10^7$. Notably, our approach is efficient and implementable using only black-box matrix-vector product operations, making it highly practical for a wide variety of applications.

math.NA↗

Gradient Methods with Online Scaling

We introduce a framework to accelerate the convergence of gradient-based methods with online learning. The framework learns to scale the gradient at each iteration through an online learning algorithm and provably accelerates gradient-based methods asymptotically. In contrast with previous literature, where convergence is established based on worst-case analysis, our framework provides a strong convergence guarantee with respect to the optimal scaling matrix for the iteration trajectory. For smooth strongly convex optimization, our results provide an $O(κ^\star \log(1/\varepsilon)$) complexity result, where $κ^\star$ is the condition number achievable by the optimal preconditioner, improving on the previous $O(\sqrt{n}κ^\star \log(1/\varepsilon))$ result. In particular, a variant of our method achieves superlinear convergence on convex quadratics. For smooth convex optimization, we show for the first time that the widely-used hypergradient descent heuristic improves on the convergence of gradient descent.

math.OC↗

A Robust Anchor-based Method for Multi-Camera Pedestrian Localization

This paper addresses the problem of vision-based pedestrian localization, which estimates a pedestrian's location using images and camera parameters. In practice, however, calibrated camera parameters often deviate from the ground truth, leading to inaccuracies in localization. To address this issue, we propose an anchor-based method that leverages fixed-position anchors to reduce the impact of camera parameter errors. We provide a theoretical analysis that demonstrates the robustness of our approach. Experiments conducted on simulated, real-world, and public datasets show that our method significantly improves localization accuracy and remains resilient to noise in camera parameters, compared to methods without anchors.

cs.CV↗

Restarted Primal-Dual Hybrid Conjugate Gradient Method for Large-Scale Quadratic Programming

Convex quadratic programming (QP) is an essential class of optimization problems with broad applications across various fields. Traditional QP solvers, typically based on simplex or barrier methods, face significant scalability challenges. In response to these limitations, recent research has shifted towards matrix-free first-order methods to enhance scalability in QP. Among these, the restarted accelerated primal-dual hybrid gradient (rAPDHG) method, proposed by Lu, has gained notable attention due to its linear convergence rate to an optimal solution and its straightforward implementation on Graphics Processing Units (GPUs). Building on this framework, this paper introduces a restarted primal-dual hybrid conjugate gradient (PDHCG) method, which incorporates conjugate gradient (CG) techniques to address the primal subproblems inexactly. We demonstrate that PDHCG maintains a linear convergence rate with an improved convergence constant and is also straightforward to implement on GPUs. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that our method significantly reduces the number of iterations required to achieve the desired accuracy compared to rAPDHG. Additionally, the GPU implementation of our method achieves state-of-the-art performance on large-scale problems. In most large-scale scenarios, our method is approximately 5 times faster than rAPDHG and about 100 times faster than other existing methods. These results highlight the substantial potential of the proposed PDHCG method to greatly improve both the efficiency and scalability of solving complex quadratic programming challenges.

math.OC↗

Computationally Efficient Estimation of Large Probit Models

Probit models are useful for modeling correlated discrete responses in many disciplines, including consumer choice data in economics and marketing. However, the Gaussian latent variable feature of probit models coupled with identification constraints pose significant computational challenges for its estimation and inference, especially when the dimension of the discrete response variable is large. In this paper, we propose a computationally efficient Expectation-Maximization (EM) algorithm for estimating large probit models. Our work is distinct from existing methods in two important aspects. First, instead of simulation or sampling methods, we apply and customize expectation propagation (EP), a deterministic method originally proposed for approximate Bayesian inference, to estimate moments of the truncated multivariate normal (TMVN) in the E (expectation) step. Second, we take advantage of a symmetric identification condition to transform the constrained optimization problem in the M (maximization) step into a one-dimensional problem, which is solved efficiently using Newton's method instead of off-the-shelf solvers. Our method enables the analysis of correlated choice data in the presence of more than 100 alternatives, which is a reasonable size in modern applications, such as online shopping and booking platforms, but has been difficult in practice with probit models. We apply our probit estimation method to study ordering effects in hotel search results on Expedia's online booking platform.

stat.ME↗

Stochastic Online Fisher Markets: Static Pricing Limits and Adaptive Enhancements

Fisher markets are one of the most fundamental models for resource allocation. However, the problem of computing equilibrium prices in Fisher markets typically relies on complete knowledge of users' budgets and utility functions and requires transactions to happen in a static market where all users are present simultaneously. Motivated by these practical considerations, we study an online variant of Fisher markets, wherein users with privately known utility and budget parameters, drawn i.i.d. from a distribution, arrive sequentially. In this setting, we first study the limitations of static pricing algorithms, which set uniform prices for all users, along two performance metrics: (i) regret, i.e., the optimality gap in the objective of the Eisenberg-Gale program between an online algorithm and an oracle with complete information, and (ii) capacity violations, i.e., the over-consumption of goods relative to their capacities. Given the limitations of static pricing, we design adaptive posted-pricing algorithms, one with knowledge of the distribution of users' budget and utility parameters and another that adjusts prices solely based on past observations of user consumption, i.e., revealed preference feedback, with improved performance guarantees. Finally, we present numerical experiments to compare our revealed preference algorithm's performance to several benchmarks.

cs.GT↗

Accelerating Low-Rank Factorization-Based Semidefinite Programming Algorithms on GPU

In this paper, we address a long-standing challenge: how to achieve both efficiency and scalability in solving semidefinite programming problems. We propose breakthrough acceleration techniques for a wide range of low-rank factorization-based first-order methods using GPUs, making the computation much more efficient and scalable. To illustrate the idea and effectiveness of our approach, we use the low-rank factorization-based SDP solver, LoRADS, as an example, which involves both the classic Burer-Monterio method and a novel splitting scheme with a starting logarithmic rank. Our numerical results demonstrate that the accelerated GPU version of LoRADS, cuLoRADS, can solve huge-scale semidefinite programming problems with remarkable efficiency. By effectively leveraging GPU computational power, cuLoRADS exhibits outstanding performance. Specifically, it can solve a set of MaxCut problems with $10^7 \times 10^7$ matrix variables in 10 seconds to 1 minute each on an NVIDIA H100 GPU with 80GB memory, whereas previous solvers demonstrated the capability of handling problems of this scale, required at least dozens of hours per problem on CPUs. Additionally, cuLoRADS shows exceptional scalability by solving 1) a MaxCut problem with a $170 \text{ million} \times 170 \text{ million}$ matrix variable and 2) a Matrix Completion problem with a 20 million $\times$ 20 million matrix variable and approximately 200 million constraints, both in a matter of minutes.

math.OC↗

A Low-Rank ADMM Splitting Approach for Semidefinite Programming

We introduce a new first-order method for solving general semidefinite programming problems, based on the alternating direction method of multipliers (ADMM) and a matrix-splitting technique. Our algorithm has an advantage over the Burer-Monteiro approach as it only involves much easier quadratically regularized subproblems in each iteration. For a linear objective, the subproblems are well-conditioned quadratic programs that can be efficiently solved by the standard conjugate gradient method. We show that the ADMM algorithm achieves sublinear or linear convergence rates to the KKT solutions under different conditions. Building on this theoretical development, we present LoRADS, a new solver for linear SDP based on the Low-Rank ADMM Splitting approach. LoRADS incorporates several strategies that significantly increase its efficiency. Firstly, it initiates with a warm-start phase that uses the Burer-Monteiro approach. Moreover, motivated by the SDP low-rank theory [So et al. 2008], LoRADS chooses an initial rank of logarithmic order and then employs a dynamic approach to increase the rank. Numerical experiments indicate that LoRADS exhibits promising performance on various SDP problems. A noteworthy achievement of LoRADS is its successful solving of a matrix completion problem with $15,694,167$ constraints and a matrix variable of size $40,000 \times 40,000$ in $351$ seconds.

math.OC↗

A Single-Loop Robust Policy Gradient Method for Robust Markov Decision Processes

Robust Markov Decision Processes (RMDPs) have recently been recognized as a valuable and promising approach to discovering a policy with creditable performance, particularly in the presence of a dynamic environment and estimation errors in the transition matrix due to limited data. Despite extensive exploration of dynamic programming algorithms for solving RMDPs, there has been a notable upswing in interest in developing efficient algorithms using the policy gradient method. In this paper, we propose the first single-loop robust policy gradient (SRPG) method with the global optimality guarantee for solving RMDPs through its minimax formulation. Moreover, we complement the convergence analysis of the nonconvex-nonconcave min-max optimization problem with the objective function's gradient dominance property, which is not explored in the prior literature. Numerical experiments validate the efficacy of SRPG, demonstrating its faster and more robust convergence behavior compared to its nested-loop counterpart.

math.OC↗

A Homogenization Approach for Gradient-Dominated Stochastic Optimization

Gradient dominance property is a condition weaker than strong convexity, yet sufficiently ensures global convergence even in non-convex optimization. This property finds wide applications in machine learning, reinforcement learning (RL), and operations management. In this paper, we propose the stochastic homogeneous second-order descent method (SHSODM) for stochastic functions enjoying gradient dominance property based on a recently proposed homogenization approach. Theoretically, we provide its sample complexity analysis, and further present an enhanced result by incorporating variance reduction techniques. Our findings show that SHSODM matches the best-known sample complexity achieved by other second-order methods for gradient-dominated stochastic optimization but without cubic regularization. Empirically, since the homogenization approach only relies on solving extremal eigenvector problem at each iteration instead of Newton-type system, our methods gain the advantage of cheaper computational cost and robustness in ill-conditioned problems. Numerical experiments on several RL tasks demonstrate the better performance of SHSODM compared to other off-the-shelf methods.

math.OC↗