加速交替方向优化方法：Fast ADMM算法
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"Fast ADMM Algorithms - 一种加速交替方向优化方法的研究"
这篇论文主要探讨了在数学编程和优化领域广泛应用的交替方向方法，特别是在变分图像处理中的重要性，因为该领域经常需要解决非可微目标函数的最小化问题。论文关注的是两种常见的交替方向方法的加速版本：交替方向乘子法（ADMM）和交替最小化算法（AMA）。
1. **交替方向方法**
这些方法的基本思想是将复杂问题分解为更易于处理的子问题，通过交替迭代求解这些子问题来逐步接近全局最优解。在ADMM中，这种方法通过引入拉格朗日乘子将原问题转化为两个更简单的子问题，分别优化两个变量，然后进行融合。而AMA则通过交替地对各个变量进行最小化操作来求解问题。
2. **加速算法**
论文提出了一种类似于Nesterov梯度下降法的加速机制，旨在提高ADMM和AMA的收敛速度。Nesterov的加速技巧通过预估下一步的方向，使得算法能在更短的时间内达到相同的优化效果，从而提高整体效率。
3. **强凸性与全局收敛性**
当目标函数具有强凸性质时，论文提供了两种方法的古典版和加速版的全局收敛界。强凸函数意味着函数的梯度具有一定的下界，这使得优化过程更容易且有唯一的全局最小值。
4. **数值实验**
为了验证快速方法在各种问题上的优越性能，论文进行了大量的数值实验。这些实验结果表明，加速的ADMM和AMA在实际应用中能更快地达到收敛，提高了算法的实用性和效率。
5. **术语和符号**
论文定义了关键术语和符号，为读者理解算法的细节提供了清晰的框架。这包括对拉格朗日乘子、最优条件、以及子问题解的概念的解释。
6. **结构安排**
论文的结构包括介绍、术语和符号、交替方向方法的概述、优化的最优性条件、具体算法的描述、加速策略的详细解释，以及数值实验和结果分析。
这篇“Fast ADMM Algorithms”论文为优化领域提供了一种新的加速策略，对于处理非可微优化问题，尤其是在图像处理等实际应用中，具有重要的理论和实践价值。通过引入Nesterov的加速思想，它改进了ADMM和AMA的性能，提升了算法的收敛速度和计算效率。
Since the introduction of this optimal method, much work has been done applying Nesterov’s concept to
other first-order splitting methods. Nesterov himself showed that O(1/k
2
) complexity results can be proved
for certain classes of non-differentiable problems [28]. Nesterov’s arguments have also been adapted to the
acceleration of proximal-point descent methods by G¨uler [26]. Like Nesterov’s gradient descent method,
G¨uler’s method achieves O(1/k
2
) global convergence, which is provably optimal.
More recently, there has been reinvigorated interest in optimal first-order schemes in the context of
compressed sensing [29, 30]. One particular algorithm of interest is the optimal forward-backward splitting
of Beck and Teboulle [27], which they have termed FISTA. See Algorithm 6.
Algorithm 6 FISTA
Require: y
1
= x
0
∈ R
N
, α
1
= 1, τ < 1/L(∇G)
1: for k = 1, 2, 3, . . . do
2: x
k
= J
τ ∂G
(y
k
− τ ∇H(y
k
))
3: α
k+1
= (1 +
p
1 + 4α
2
k
)/2
4: y
k+1
= x
k
+
α
k
−1
α
k+1
(x
k
− x
k−1
)
5: end for
FISTA is one the first examples of global acceleration being applied to a splitting scheme. However, the
method cannot directly solve saddle point problems involving Lagrange multipliers (i.e., problems involving
linear constraints).
One approach for solving saddle point problems is the accelerated alternating direction method proposed
by Goldfarb et al. [31]. The authors of [31] propose a Nesterov-type acceleration for problems of the form
(1) in the special case where both A and B are identity matrices, and one of H or G is differentiable. This
scheme is based on the “symmetric” ADMM method, which differs from Algorithm 1 in that it involves two
dual updates per iteration rather than one. The authors handle weakly convex problems by introducing a
step-skipping process that applies the acceleration selectively on certain iterations. These “step-skipping”
methods have a more complicated sequence of steps than conventional ADMM, but are necessary to maintain
stability of the method in the presence of a Nesterov-type acceleration step.
3. Global Convergence Bounds for Unaccelerated ADMM
We now develop global convergence bounds for ADMM. We consider the case where both terms in the
objective are strongly convex:
Assumption 1. Both H and G are strongly convex with moduli σ
H
and σ
G
, respectively.
Note that this assumption is rather strong – in many common problems solved using ADMM, one or both
of the objective terms are not strongly convex. Later, we shall see how convergence can be guaranteed for
general objectives by incorporating a simple restart rule.
As we shall demonstrate below, the conventional (unaccelerated) ADMM converges in the dual objective
with rate O(1/k). Later, we shall introduce an accelerated variant that converges with the optimal bound
O(1/k
2
).
3.1. Preliminary Results. Before we prove global convergence results for ADMM, we require some pre-
liminary definitions and lemmas.
To simplify notation, we define the “pushforward” operators, u
+
, v
+
, and λ
+
for ADMM such that
u
+
= argmin
u
H(u) + hλ, −Aui +
τ
2
kb − Au − Bvk
2
,(12)
v
+
= argmin
z
G(z) + hλ, −Bzi+
τ
2
kb − Au
+
− Bzk
2
,(13)
λ
+
= λ + τ(b − Au
+
− Bv
+
).(14)
In plain words, u
+
, v
+
, and λ
+
represent the results of an ADMM iteration starting with v and λ. Note that
that these pushforward operators are implicitly functions of v and λ, but we have omitted this dependence
for brevity.
6
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