arXiv subjects

Wenzhi Gao

Publications and source records attributed to Wenzhi Gao.

27 records · Page 2Linked to original sources

arXiv2024

Stochastic Weakly Convex Optimization Beyond Lipschitz Continuity

This paper considers stochastic weakly convex optimization without the standard Lipschitz continuity assumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of stochastic algorithms, including the stochastic subgradient method, preserve the $\mathcal{O} ( 1 / \sqrt{K})$ convergence rate with constant failure rate. Our analyses rest on rather weak assumptions: the Lipschitz parameter can be either bounded by a general growth function of $\|x\|$ or locally estimated through independent random samples.

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arXiv2024

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.

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arXiv2024

An Enhanced ADMM-based Interior Point Method for Linear and Conic Optimization

The ADMM-based interior point (ABIP, Lin et al. 2021) method is a hybrid algorithm that effectively combines interior point method (IPM) and first-order methods to achieve a performance boost in large-scale linear optimization. Different from traditional IPM that relies on computationally intensive Newton steps, the ABIP method applies the alternating direction method of multipliers (ADMM) to approximately solve the barrier penalized problem. However, similar to other first-order methods, this technique remains sensitive to condition number and inverse precision. In this paper, we provide an enhanced ABIP method with multiple improvements. Firstly, we develop an ABIP method to solve the general linear conic optimization and establish the associated iteration complexity. Secondly, inspired by some existing methods, we develop different implementation strategies for ABIP method, which substantially improve its performance in linear optimization. Finally, we conduct extensive numerical experiments in both synthetic and real-world datasets to demonstrate the empirical advantage of our developments. In particular, the enhanced ABIP method achieves a 5.8x reduction in the geometric mean of run time on $105$ selected LP instances from Netlib, and it exhibits advantages in certain structured problems such as SVM and PageRank. However, the enhanced ABIP method still falls behind commercial solvers in many benchmarks, especially when high accuracy is desired. We posit that it can serve as a complementary tool alongside well-established solvers.

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arXiv2024

OptiMUS: Scalable Optimization Modeling with (MI)LP Solvers and Large Language Models

Optimization problems are pervasive in sectors from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers because the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. This paper introduces OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve (mixed integer) linear programming problems from their natural language descriptions. OptiMUS can develop mathematical models, write and debug solver code, evaluate the generated solutions, and improve its model and code based on these evaluations. OptiMUS utilizes a modular structure to process problems, allowing it to handle problems with long descriptions and complex data without long prompts. Experiments demonstrate that OptiMUS outperforms existing state-of-the-art methods on easy datasets by more than $20\%$ and on hard datasets (including a new dataset, NLP4LP, released with this paper that features long and complex problems) by more than $30\%$.

cs.AI↗

arXiv2023

HDSDP: Software for Semidefinite Programming

HDSDP is a numerical software solving the semidefinite programming problems. The main framework of HDSDP resembles the dual-scaling interior point solver DSDP [BY2008] and several new features, including a dual method based on the simplified homogeneous self-dual embedding, have been implemented. The embedding technique enhances stability of the dual method and several new heuristics and computational techniques are designed to accelerate its convergence. HDSDP aims to show how dual-scaling algorithm benefits from the self-dual embedding and it is developed in parallel to DSDP5.8. Numerical experiments over several classical benchmark datasets exhibit its robustness and efficiency, and particularly its advantages on SDP instances featuring low-rank structure and sparsity. HDSDP is open-sourced under MIT license and available at https://github.com/COPT-Public/HDSDP.

cs.MS↗

arXiv2023

Delayed Stochastic Algorithms for Distributed Weakly Convex Optimization

This paper studies delayed stochastic algorithms for weakly convex optimization in a distributed network with workers connected to a master node. Recently, Xu et al. 2022 showed that an inertial stochastic subgradient method converges at a rate of $\mathcal{O}(τ_{\text{max}}/\sqrt{K})$ which depends on the maximum information delay $τ_{\text{max}}$. In this work, we show that the delayed stochastic subgradient method ($\texttt{DSGD}$) obtains a tighter convergence rate which depends on the expected delay $\barτ$. Furthermore, for an important class of composition weakly convex problems, we develop a new delayed stochastic prox-linear ($\texttt{DSPL}$) method in which the delays only affect the high-order term in the complexity rate and hence, are negligible after a certain number of $\texttt{DSPL}$ iterations. In addition, we demonstrate the robustness of our proposed algorithms against arbitrary delays. By incorporating a simple safeguarding step in both methods, we achieve convergence rates that depend solely on the number of workers, eliminating the effect of the delay. Our numerical experiments further confirm the empirical superiority of our proposed methods.

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arXiv2023

OptiMUS: Optimization Modeling Using MIP Solvers and large language models

Optimization problems are pervasive across various sectors, from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers, as the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve MILP problems from their natural language descriptions. OptiMUS is capable of developing mathematical models, writing and debugging solver code, developing tests, and checking the validity of generated solutions. To benchmark our agent, we present NLP4LP, a novel dataset of linear programming (LP) and mixed integer linear programming (MILP) problems. Our experiments demonstrate that OptiMUS solves nearly twice as many problems as a basic LLM prompting strategy. OptiMUS code and NLP4LP dataset are available at \href{https://github.com/teshnizi/OptiMUS}{https://github.com/teshnizi/OptiMUS}

cs.AI↗

arXiv2022

Optimal Diagonal Preconditioning

Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice, most lack guarantees on reductions in condition number. Moreover, the degree to which we can improve over existing heuristic preconditioners remains an important practical question. In this paper, we study the problem of optimal diagonal preconditioning that achieves maximal reduction in the condition number of any full-rank matrix by scaling its rows and/or columns. We first reformulate the problem as a quasi-convex problem and provide a simple algorithm based on bisection. Then we develop an interior point algorithm with $O(\log(1/ε))$ iteration complexity, where each iteration consists of a Newton update based on the Nesterov-Todd direction. Next, we specialize to one-sided optimal diagonal preconditioning problems, and demonstrate that they can be formulated as standard dual SDP problems. We then develop efficient customized solvers and study the empirical performance of our optimal diagonal preconditioning procedures through extensive experiments on large matrices. Our findings suggest that optimal diagonal preconditioners can significantly improve upon existing heuristics-based diagonal preconditioners at reducing condition numbers and speeding up iterative methods. Moreover, our implementation of customized solvers, combined with a random row/column sampling step, can find near-optimal diagonal preconditioners for matrices up to size 200,000 in reasonable time, demonstrating their practical appeal.

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arXiv2021

Minibatch and Momentum Model-based Methods for Stochastic Weakly Convex Optimization

Stochastic model-based methods have received increasing attention lately due to their appealing robustness to the stepsize selection and provable efficiency guarantee. We make two important extensions for improving model-based methods on stochastic weakly convex optimization. First, we propose new minibatch model-based methods by involving a set of samples to approximate the model function in each iteration. For the first time, we show that stochastic algorithms achieve linear speedup over the batch size even for non-smooth and non-convex (particularly, weakly convex) problems. To this end, we develop a novel sensitivity analysis of the proximal mapping involved in each algorithm iteration. Our analysis appears to be of independent interests in more general settings. Second, motivated by the success of momentum stochastic gradient descent, we propose a new stochastic extrapolated model-based method, greatly extending the classic Polyak momentum technique to a wider class of stochastic algorithms for weakly convex optimization. The rate of convergence to some natural stationarity condition is established over a fairly flexible range of extrapolation terms. While mainly focusing on weakly convex optimization, we also extend our work to convex optimization. We apply the minibatch and extrapolated model-based methods to stochastic convex optimization, for which we provide a new complexity bound and promising linear speedup in batch size. Moreover, an accelerated model-based method based on Nesterov's momentum is presented, for which we establish an optimal complexity bound for reaching optimality.

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