分布式梯度下降法：多智能体系统优化

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"这篇文档主要讨论了在多智能体系统中优化凸聚合函数的问题，其中每个智能体只能根据自身及邻居的信息计算其局部梯度。文章指出，在离散时间的梯度下降法实施中，如何选择合适的智能体步长以确保目标函数的单调演化成为基本挑战。为此，作者提出了一种基于自适应凸组合的分布式算法解决方案，并通过模拟验证了结果。" 正文: 在分布式搜索和优化领域，多智能体系统常常被用来解决复杂的全局问题。在这个系统中，每个智能体是一个独立的决策单元，它们协作寻找全局最优解。"分步实现搜索"这个主题聚焦于如何在多智能体环境中有效执行优化过程，特别是当目标函数是凸的并且其梯度分布时。论文中提到的"convex aggregate function"是指由多个智能体的局部贡献构成的整体优化目标，这些局部贡献可能源于不同的空间或任务区域。在这样的系统中，每个智能体只能访问关于自身和相邻智能体的信息，无法获取全局视图。因此，它们必须通过局部梯度信息来估计全局梯度的方向。传统的梯度下降法在离散时间步骤中可能遇到步长选择的问题。如果步长选取不当，可能会导致目标函数值的不单调递减，甚至可能陷入振荡或不收敛的状态。论文提出了一种新颖的策略，即通过"adaptive convex combinations"来聚合智能体的步长，这允许系统动态调整步长以适应环境变化和网络拓扑结构。这种分布式算法的核心思想在于，它不需要集中式的协调者来确定全局步长，而是让智能体间通过通信和协作来形成一个动态的、自适应的步长组合策略。这既保持了系统的分布式特性，又解决了步长选择的难题，从而保证了目标函数的单调演化，有利于系统收敛到全局最优解。模拟实验部分提供了对这一算法实际效果的验证，展示了在不同场景下算法的性能和稳定性。这些实验结果不仅证实了理论分析的有效性，也为实际应用提供了参考。这篇论文对多智能体系统中的分布式搜索问题进行了深入探讨，提出的算法为解决分布式优化中的步长选择问题提供了一个创新的思路，对于分布式控制、网络优化等领域具有重要的理论与实践价值。

4 Jorge Cort

es and Sonia Mart

ınez

2.2 Graph-theoretic notions

We present some basic notions from algebraic graph theory following the exposition

in [2]. An undirected graph, or simply graph, is a pair G = (V,E), where V is a

ﬁnite set called the vertex set and E is the edge set consisting of unordered pairs

of vertices. For i, j ∈ V and i 6= j, the set {i, j} denotes an undirected edge, and

i and j are neighbors. We let N

(i) denote the set of neighbors of u

in G. The

graph G is connected if for any pair of nodes i, j there exists a sequence of edges

{i,i

},{i

},...,{i

, j} connecting i with j. The adjacency matrix of a graph G is

a non-negative symmetric matrix A = (a

i j

) ∈ R

n×n

≥0

such that a

i j

6= 0 if and only if

{i, j} is an edge of the graph. Here, we consider a

i j

= 1, when {i, j} ∈ E. Consider

the diagonal matrix D = diag(A1

). The Laplacian matrix of G is deﬁned as L =

D − A, which is a symmetric and positive semi-deﬁnite matrix. Note that L has an

eigenvalue at 0 and 1

is the corresponding eigenvector. A graph G is connected if

and only if L is irreducible and 0 is a simple eigenvalue. Finally, a map g : R

→ R

is distributed over G if, for all j ∈ {1, . . ., n}, the component g

can be expressed as

(x) = g

,...,x

), where N

( j) = {i

,...,i

}, for all x ∈ R

2.3 Directions of descent and line search

Given a continuously differentiable function f : R

→ R, we let ∇ f : R

→ R

denote its gradient

∇ f (x) =



∂ f

∂ x

(x),...,

∂ f

∂ x

(x)



Throughout the paper, we use the notation ∇

f to refer to the ith component of ∇ f .

Given a function f : R

→ R and x ∈ R

, v ∈ R

is a direction of descent of f at x if

there exists T > 0 such that

f (x + δ v) < f (x), δ ∈ (0, T ).

If f is continuously differentiable at x, this is equivalent to saying that ∇ f (x)

v < 0.

The procedure of calculating the actual step δ to be taken in the direction v is called

line search. The choice of a stepsize and a direction that guarantees the reduction of

the function at each iterate leads to various gradient algorithms. In particular, one

could aim to ﬁnd the best stepsize that optimizes the decrease in the value of f along

a direction v, i.e.,

= argmin

δ ∈[0,∞)

f (x + δ v). (1)

Let h

(δ ) = f (x + δv). For a continuously differentiable function, it is not difﬁcult

to see that the stepsize (1) is characterized by the equation

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分布式梯度下降法：多智能体系统优化

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