arXiv subjects

Haihao Lu

Publications and source records attributed to Haihao Lu.

At least 37 records · Page 2Linked to original sources

arXiv2024

PDOT: a Practical Primal-Dual Algorithm and a GPU-Based Solver for Optimal Transport

In this paper, we propose a practical primal-dual algorithm with theoretical guarantees and develop a GPU-based solver, which we dub PDOT, for solving large-scale optimal transport problems. Compared to Sinkhorn algorithm or classic LP algorithms, PDOT can achieve high-accuracy solution while efficiently taking advantage of modern computing architecture, i.e., GPUs. On the theoretical side, we show that PDOT has a data-independent $\widetilde O(mn(m+n)^{1.5}\log(\frac{1}ε))$ local flop complexity where $ε$ is the desired accuracy, and $m$ and $n$ are the dimension of the original and target distribution, respectively. We further present a data-dependent $\widetilde O(mn(m+n)^{3.5}Δ+ mn(m+n)^{1.5}\log(\frac{1}ε))$ global flop complexity of PDOT, where $Δ$ is the precision of the data. On the numerical side, we present PDOT, an open-source GPU solver based on the proposed algorithm. Our extensive numerical experiments consistently demonstrate the well balance of PDOT in computing efficiency and accuracy of the solution, compared to Gurobi and Sinkhorn algorithms.

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arXiv2024

cuPDLP.jl: A GPU Implementation of Restarted Primal-Dual Hybrid Gradient for Linear Programming in Julia

In this paper, we provide an affirmative answer to the long-standing question: Are GPUs useful in solving linear programming? We present cuPDLP.jl, a GPU implementation of restarted primal-dual hybrid gradient (PDHG) for solving linear programming (LP). We show that this prototype implementation in Julia has comparable numerical performance on standard LP benchmark sets to Gurobi, a highly optimized implementation of the simplex and interior-point methods. This demonstrates the power of using GPUs in linear programming, which, for the first time, showcases that GPUs and first-order methods can lead to performance comparable to state-of-the-art commercial optimization LP solvers on standard benchmark sets.

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arXiv2024

First-Order Methods for Linear Programming

Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order methods for solving large-scale linear programming.

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arXiv2024

cuPDLP-C: A Strengthened Implementation of cuPDLP for Linear Programming by C language

A recent GPU implementation of the Restarted Primal-Dual Hybrid Gradient Method for Linear Programming was proposed in Lu and Yang (2023). Its computational results demonstrate the significant computational advantages of the GPU-based first-order algorithm on certain large-scale problems. The average performance also achieves a level close to commercial solvers for the first time in history. However, due to limitations in experimental hardware and the disadvantage of implementing the algorithm in Julia compared to C language, neither the commercial solver nor cuPDLP reached their maximum efficiency. Therefore, in this report, we have re-implemented and optimized cuPDLP in C language. Utilizing state-of-the-art CPU and GPU hardware, we extensively compare cuPDLP with the best commercial solvers. The experiments further highlight its substantial computational advantages and potential for solving large-scale linear programming problems. We also discuss the profound impact this breakthrough may have on mathematical programming research and the entire operations research community.

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arXiv2023

Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming

There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the proposed stochastic method exhibits a linear convergence rate for solving sharp instances with a high probability. In addition, we propose an efficient coordinate-based stochastic oracle for unconstrained bilinear problems, which has $\mathcal O(1)$ per iteration cost and improves the complexity of the existing deterministic and stochastic algorithms. Finally, we show that the obtained linear convergence rate is nearly optimal (upto $\log$ terms) for a wide class of stochastic primal dual methods.

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arXiv2023

On the Geometry and Refined Rate of Primal-Dual Hybrid Gradient for Linear Programming

We study the convergence behaviors of primal-dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

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arXiv2023

Analysis of Dual-Based PID Controllers through Convolutional Mirror Descent

Dual-based proportional-integral-derivative (PID) controllers are often employed in practice to solve online allocation problems with global constraints, such as budget pacing in online advertising. However, controllers are used in a heuristic fashion and come with no provable guarantees on their performance. This paper provides the first regret bounds on the performance of dual-based PID controllers for online allocation problems. We do so by first establishing a fundamental connection between dual-based PID controllers and a new first-order algorithm for online convex optimization called \emph{Convolutional Mirror Descent} (CMD), which updates iterates based on a weighted moving average of past gradients. CMD recovers, in a special case, online mirror descent with momentum and optimistic mirror descent. We establish sufficient conditions under which CMD attains low regret for general online convex optimization problems with adversarial inputs. We leverage this new result to give the first regret bound for dual-based PID controllers for online allocation problems. As a byproduct of our proofs, we provide the first regret bound for CMD for non-smooth convex optimization, which might be of independent interest.

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arXiv2023

Optimizing Scalable Targeted Marketing Policies with Constraints

Targeted marketing policies target different customers with different marketing actions. While most research has focused on training targeting policies without managerial constraints, in practice, many firms face managerial constraints when implementing these policies. For example, firms may face volume constraints on the maximum or minimum number of actions they can take, or on the minimum acceptable outcomes for different customer segments. They may also face similarity (fairness) constraints that require similar actions with different groups of customers. Traditional optimization methods face challenges when solving problems with either many customers or many constraints. We show how recent advances in linear programming can be adapted to the targeting of marketing actions. We provide a theoretical guarantee comparing how the proposed algorithm scales compared to state-of-the-art benchmarks (primal simplex, dual simplex and barrier methods). We also extend existing guarantees on optimality and computation speed, by adapting them to accommodate the characteristics of targeting problems. We implement the proposed algorithm using data from a field experiment with over 2 million customers, and six different marketing actions (including a no action ``Control''). We use this application to evaluate the computation speed and range of problems the algorithm can solve, comparing it to benchmark methods. The findings confirm that the algorithm makes it feasible to train large-scale targeting problems that include volume and similarity constraints.

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arXiv2023

A Field Guide for Pacing Budget and ROS Constraints

Budget pacing is a popular service that has been offered by major internet advertising platforms since their inception. Budget pacing systems seek to optimize advertiser returns subject to budget constraints by smoothly spending advertiser budgets. In the past few years, autobidding products that provide real-time bidding as a service to advertisers have seen a prominent rise in adoption. A popular autobidding strategy is value maximization subject to return-on-spend (ROS) constraints. For historical/business reasons, the systems that govern these two services, namely budget pacing and ROS pacing, are not always a unified and coordinated entity that optimizes a global objective subject to both constraints. The purpose of this work is to theoretically and empirically compare algorithms with different degrees of coordination between these two pacing systems. In particular, we compare (a) a fully-decoupled sequential algorithm that first constructs the advertiser's ROS-pacing bid and then lowers that bid for budget pacing; (b) a minimally-coupled min-pacing algorithm that runs these two services independently, obtains the bid multipliers from both of them and applies the minimum of the two multipliers as the effective multiplier; and (c) a fully-coupled dual-based algorithm that optimally combines the dual variables from both the systems. Our main contribution is to theoretically analyze the min-pacing algorithm and show that it attains similar guarantees to the fully-coupled canonical dual-based algorithm. On the other hand, we show that the sequential algorithm, even though appealing by virtue of being fully decoupled, could badly violate the constraints. We validate our theoretical findings empirically by showing that the min-pacing algorithm performs almost as well as the canonical dual-based algorithm on a semi-synthetic dataset based on a large online advertising platform's data.

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arXiv2023

Achieving Fairness and Accuracy in Regressive Property Taxation

Regressivity in property taxation, or the disproportionate overassessment of lower-valued properties compared to higher-valued ones, results in an unfair taxation burden for Americans living in poverty. To address regressivity and enhance both the accuracy and fairness of property assessments, we introduce a scalable property valuation model called the $K$-segment model. Our study formulates a mathematical framework for the $K$-segment model, which divides a single model into $K$ segments and employs submodels for each segment. Smoothing methods are incorporated to balance and smooth the multiple submodels within the overall model. To assess the fairness of our proposed model, we introduce two innovative fairness measures for property evaluation and taxation, focusing on group-level fairness and extreme sales price portions where unfairness typically arises. Compared to the model employed currently in practice, our study demonstrates that the $K$-segment model effectively improves fairness based on the proposed measures. Furthermore, we investigate the accuracy--fairness trade-off in property assessments and illustrate how the $K$-segment model balances high accuracy with fairness for all properties. Our work uncovers the practical impacts of the $K$-segment models in addressing regressivity in property taxation, offering a tangible solution for policymakers and property owners. By implementing this model, we pave the way for a fairer taxation system, ensuring a more equitable distribution of tax burdens.

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arXiv2023

On the Convergence of L-shaped Algorithms for Two-Stage Stochastic Programming

In this paper, we design, analyze, and implement a variant of the two-loop L-shaped algorithms for solving two-stage stochastic programming problems that arise from important application areas including revenue management and power systems. We consider the setting in which it is intractable to compute exact objective function and (sub)gradient information, and instead, only estimates of objective function and (sub)gradient values are available. Under common assumptions including fixed recourse and bounded (sub)gradients, the algorithm generates a sequence of iterates that converge to a neighborhood of optimality, where the radius of the convergence neighborhood depends on the level of the inexactness of objective function estimates. The number of outer and inner iterations needed to find an approximate optimal iterate is provided. Finally, we show a sample complexity result for the algorithm with a Polyak-type step-size policy that can be extended to analyze other situations. We also present a numerical study that verifies our theoretical results and demonstrates the superior empirical performance of our proposed algorithms over classic solvers.

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arXiv2023

Online Ad Procurement in Non-stationary Autobidding Worlds

Today's online advertisers procure digital ad impressions through interacting with autobidding platforms: advertisers convey high level procurement goals via setting levers such as budget, target return-on-investment, max cost per click, etc.. Then ads platforms subsequently procure impressions on advertisers' behalf, and report final procurement conversions (e.g. click) to advertisers. In practice, advertisers may receive minimal information on platforms' procurement details, and procurement outcomes are subject to non-stationary factors like seasonal patterns, occasional system corruptions, and market trends which make it difficult for advertisers to optimize lever decisions effectively. Motivated by this, we present an online learning framework that helps advertisers dynamically optimize ad platform lever decisions while subject to general long-term constraints in a realistic bandit feedback environment with non-stationary procurement outcomes. In particular, we introduce a primal-dual algorithm for online decision making with multi-dimension decision variables, bandit feedback and long-term uncertain constraints. We show that our algorithm achieves low regret in many worlds when procurement outcomes are generated through procedures that are stochastic, adversarial, adversarially corrupted, periodic, and ergodic, respectively, without having to know which procedure is the ground truth. Finally, we emphasize that our proposed algorithm and theoretical results extend beyond the applications of online advertising.

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arXiv2023

On a Unified and Simplified Proof for the Ergodic Convergence Rates of PPM, PDHG and ADMM

We present a unified viewpoint of proximal point method (PPM), primal-dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) for solving convex-concave primal-dual problems. This viewpoint shows the equivalence of these three algorithms upto a norm change, and it leads to a four-line simple proof of their $\mathcal O(1/k)$ ergodic rates. The simple proof technique is not limited to these three algorithms, but can also be utilized to analyze related algorithms, such as gradient descent, linearized PDHG, inexact algorithms, just to name a few.

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arXiv2023

On the Infimal Sub-differential Size of Primal-Dual Hybrid Gradient Method and Beyond

Primal-dual hybrid gradient method (PDHG, a.k.a. Chambolle and Pock method) is a well-studied algorithm for minimax optimization problems with a bilinear interaction term. Recently, PDHG is used as the base algorithm for a new LP solver PDLP that aims to solve large LP instances by taking advantage of modern computing resources, such as GPU and distributed system. Most of the previous convergence results of PDHG are either on duality gap or on distance to the optimal solution set, which are usually hard to compute during the solving process. In this paper, we propose a new progress metric for analyzing PDHG, which we dub infimal sub-differential size (IDS), by utilizing the geometry of PDHG iterates. IDS is a natural extension of the gradient norm of smooth problems to non-smooth problems, and it is tied with KKT error in the case of LP. Compared to traditional progress metrics for PDHG, IDS always has a finite value and can be computed only using information of the current solution. We show that IDS monotonically decays, and it has an $\mathcal O(\frac{1}{k})$ sublinear rate for solving convex-concave primal-dual problems, and it has a linear convergence rate if the problem further satisfies a regularity condition that is satisfied by applications such as linear programming, quadratic programming, TV-denoising model, etc. The simplicity of our analysis and the monotonic decay of IDS suggest that IDS is a natural progress metric to analyze PDHG. As a by-product of our analysis, we show that the primal-dual gap has $\mathcal O(\frac{1}{\sqrt{k}})$ convergence rate for the last iteration of PDHG for convex-concave problems. The analysis and results on PDHG can be directly generalized to other primal-dual algorithms, for example, proximal point method (PPM), alternating direction method of multipliers (ADMM) and linearized alternating direction method of multipliers (l-ADMM).

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arXiv2023

On the Linear Convergence of Extra-Gradient Methods for Nonconvex-Nonconcave Minimax Problems

Recently, minimax optimization received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical Gradient Descent-Ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the Proximal Point Method (PPM) converges linearly for a family of nonconvex-nonconcave problems. In this paper, we study the convergence of a damped version of Extra-Gradient Method (EGM) which avoids potentially costly proximal computations, only relying on gradient evaluation. We show that EGM converges linearly for smooth minimax optimization problem satisfying the same nonconvex-nonconcave condition needed by PPM.

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arXiv2023

A $J$-Symmetric Quasi-Newton Method for Minimax Problems

Minimax problems have gained tremendous attentions across the optimization and machine learning community recently. In this paper, we introduce a new quasi-Newton method for minimax problems, which we call $J$-symmetric quasi-Newton method. The method is obtained by exploiting the $J$-symmetric structure of the second-order derivative of the objective function in minimax problem. We show that the Hessian estimation (as well as its inverse) can be updated by a rank-2 operation, and it turns out that the update rule is a natural generalization of the classic Powell symmetric Broyden (PSB) method from minimization problems to minimax problems. In theory, we show that our proposed quasi-Newton algorithm enjoys local Q-superlinear convergence to a desirable solution under standard regularity conditions. Furthermore, we introduce a trust-region variant of the algorithm that enjoys global R-superlinear convergence. Finally, we present numerical experiments that verify our theory and show the effectiveness of our proposed algorithms compared to Broyden's method and the extragradient method on three classes of minimax problems.

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arXiv2022

Faster First-Order Primal-Dual Methods for Linear Programming using Restarts and Sharpness

First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in traditional linear programming (LP) applications. This paper exploits the sharpness of primal-dual formulations of LP instances to achieve linear convergence using restarts in a general setting that applies to ADMM (alternating direction method of multipliers), PDHG (primal-dual hybrid gradient method) and EGM (extragradient method). In the special case of PDHG, without restarts we show an iteration count lower bound of $Ω(κ^2 \log(1/ε))$, while with restarts we show an iteration count upper bound of $O(κ\log(1/ε))$, where $κ$ is a condition number and $ε$ is the desired accuracy. Moreover, the upper bound is optimal for a wide class of primal-dual methods, and applies to the strictly more general class of sharp primal-dual problems. We develop an adaptive restart scheme and verify that restarts significantly improve the ability of PDHG, EGM, and ADMM to find high accuracy solutions to LP problems.

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arXiv2022

Practical Large-Scale Linear Programming using Primal-Dual Hybrid Gradient

We present PDLP, a practical first-order method for linear programming (LP) that can solve to the high levels of accuracy that are expected in traditional LP applications. In addition, it can scale to very large problems because its core operation is matrix-vector multiplications. PDLP is derived by applying the primal-dual hybrid gradient (PDHG) method, popularized by Chambolle and Pock (2011), to a saddle-point formulation of LP. PDLP enhances PDHG for LP by combining several new techniques with older tricks from the literature; the enhancements include diagonal preconditioning, presolving, adaptive step sizes, and adaptive restarting. PDLP improves the state of the art for first-order methods applied to LP. We compare PDLP with SCS, an ADMM-based solver, on a set of 383 LP instances derived from MIPLIB 2017. With a target of $10^{-8}$ relative accuracy and 1 hour time limit, PDLP achieves a 6.3x reduction in the geometric mean of solve times and a 4.6x reduction in the number of instances unsolved (from 227 to 49). Furthermore, we highlight standard benchmark instances and a large-scale application (PageRank) where our open-source prototype of PDLP, written in Julia, outperforms a commercial LP solver.

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