LU102143B1 - Conditional gradient based method for accelerated distributed online optimization - Google Patents
Conditional gradient based method for accelerated distributed online optimization Download PDFInfo
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Abstract
An accelerated distributed online conditional gradient optimization method based on conditional gradient is provided, so that the problem of high time complexity of the distributed online optimization algorithm can be effectively solved. According to the method, a network optimization objective function is decomposed into the sum of local objective functions of different nodes or agents. Thus, the agents obtain just their objective functions and cooperate to solve the optimization problem through the interaction of information with adjacent agents. The method avoids the deficiency that the traditional conditional gradient algorithm is not sensitive to the gradient size by constructing a new time-varying cost function with a regularization item to the local cost information of each agent. The method further improves the convergence rate of the algorithm exponentially by a local linear optimization setup instead of projection operation. Finally, the experimental results performed on various tasks indicate that the method works also well in practice and compares favorably to other distributed online optimization methods.
Description
TT BL-5165
CONDITIONAL GRADIENT BASED METHOD FOR ACCELERATED DISTRIBUTED ONLINE OPTIMIZATION HUT0zTas
TECHNICAL FIELD The present invention relates to a conditional gradient based method for accelerated distributed online optimization, and belongs to the field of machine learning.
BACKGROUND Distributed convex optimization has drawn great attention from researchers in many fields. Classical problems such as distributed tracking, distributed estimation and distributed detection are essentially optimization problems. The distributed optimization problem is mainly to perform global optimization tasks assigned to each node in the network. Since each node has limited resources or partial information about the task, the nodes, therefore, collaborate to perform data collection and update the local estimation by sharing the collected information. Distributed optimization imposes a low computing burden on each node, and the whole network remains robust even though certain node undergoes a local failure. Therefore, it is possible to effectively overcome deficiencies in a single information processing unit in a centralized scenario.
Distributed optimization has been widely used in the case of time-invariant cost functions. However, distributed network systems are usually in dynamic and uncertain environments in practice. For example, considering the issue of tracking moving targets, it is intended to track the position, velocity and acceleration of the object. Such problems have always been main focus of online learning in the field of machine learning. Thus, the combination of online optimization and distributed optimization and the employment of any variable cost function to represent the uncertainty of a multi-agent network system can be effective for real-time processing of dynamic data streams of network nodes.
With the rapid development of distributed online optimization, many traditional optimization algorithms have been extended to distributed online situations. In recent years, traditional optimization algorithms such as gradient descent and dual averaging have been widely used in distributed online optimization. Conditional gradient algorithm (also known as Frank-Wolfe, FW) is essentially a first-order optimization method, which can theoretically achieve a lower convergence rate than other effective optimization algorithms. In consideration of practical multi-dimensional optimization problems, it is infeasible to use second-order information or other superlinear operation. In addition, the FW method has been proved to be a powerful tool for solving large-scale optimization problems since it can effectively avoid critical issues such as the difficulty of calculating orthogonal projections in the first-order optimization method. Hence, with the introduction of a local linear optimization step in the existing conditional gradient algorithm, an accelerated distributed online conditional gradient algorithm is proposed, and the conditional —————m———————EEE
BL-5165 ’ gradient based online optimization algorithm is extended to the distributed setting. LU102143
SUMMARY The technical problem to be solved by the present invention is to provide a conditional gradient based method for accelerated distributed online optimization, which is aimed at | accelerating the convergence of a model in a distributed network.
To solve the above technical problem, the following technical solution is adopted in the present invention.
In a distributed online convex optimization setup, each node represents an agent, and in each iteration, the agents generate decision-making information, submit the decision-making information independently and obtain corresponding cost functions. Different agents have varying degrees of importance relative to each other in the exchange of information, and with a weighted average, an agent having a higher degree of importance can be assigned a higher weight to provide more valuable information therefrom, thereby reducing the error of the entire distributed system. Besides, a local linear optimization step is introduced in the existing conditional gradient algorithm to accelerate the convergence of an entire network model.
BRIEF DESCRIPTION OF DRAWINGS FIG. 1 illustrates the relative error versus the number of iterations when the present invention is applied to an L1 regularized logistic regression model problem. FIG. 2 illustrates the relative error versus the number of iterations when the present invention is applied to an L2 regularized logistic regression model problem.
DETAILED DESCRIPTION The present invention solves the distributed optimization problem on a connected undirected network and avoids excessive communication cost of a central node due to deficiencies in a single information processing unit in a centralized scenario. The following specific steps are performed. Step 1: a loss function f(t) =f, (1) is revealed. Step 3: a subgradient of information generated by agents is calculated: g, € fi, (x) Regarding each agent: 7, aC +1) Step 4: 4 Pis 7 A (Xu) Xp = Yo Xa =X, +0 (Pis Xi ) In a distributed network, the transfer of information of agents is based on a weighted average —_———————"""
BL-5165 (the third line in Step 4) to ensure that the information of an important agent is fully utilized. CA 02143 local linear optimization setup À (610705 8.) is further introduced in our method for the purpose of accelerating the convergence of an entire network model. p is a parameter for the local linear optimization setup. @ isa learning rate.
The present invention will be further described below with reference to the accompanying drawings.
FIG. 1 shows the convergence result of the proposed method performing on L1 regularized logistic regression model. In case of an online distributed learning setting our objective is to solve the L1 regularized logistic regression problem. For the synthetic dataset, numerical result is shown in FIG. 1. It can be seen that the accelerated distributed online conditional gradient algorithm performs better than other algorithms. FIG. 1 also shows that the convergence of the algorithm 1s obviously faster than that of other algorithms at the beginning.
FIG. 2 shows the convergence result of the proposed method performing on L2 regularized logistic regression model. An experiment was carried out on an actual data set with satisfying results. As shown in FIG. 2, the algorithm provided herein achieves the acceleration effect. As shown in FIG. 2, the loss of the algorithm reaches the minimum rapidly, and it also indicates the algorithm is better than other algorithms in performance and thus may be more suitable for practical use.
Claims (3)
1. A conditional gradient-based method for accelerated distributed online optimization, wherein agents in a distributed network submit local information independently and then obtain local cost functions; the agents communicate with each other by a weighted average method, and find next iteration direction by a local linear optimization step after the communications among the agents.
2. The method according to claim 1, wherein regarding that agents in a distributed network submit local information independently and then obtain local cost functions, in a distributed online convex optimization setup, each node represents an agent, and in each iteration, the agents generate decision-making information, submit the decision-making information independently and obtain corresponding cost functions.
3. The method according to claim 1, wherein regarding that the agents communicate with each other by a weighted average method, different agents have varying degrees of importance relative to each other in the exchange of information, and with a weighted average, an agent having a higher degree of importance is assigned a higher weight to provide more valuable information therefrom, thereby reducing the error of the entire distributed system. | ee
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| CN201911045411.1A CN110768841A (en) | 2019-10-30 | 2019-10-30 | Acceleration distributed online optimization method based on condition gradient |
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