In this paper, we propose a new stochastic alternating direction method of
multipliers (ADMM) algorithm, which incrementally approximates the full
gradient in the linearized ADMM formulation. Besides having a low per-iteration
complexity as existing stochastic ADMM algorithms, the proposed algorithm
improves the convergence rate on convex problems from $O(\frac 1 {\sqrt{T}})$
to $O(\frac 1 T)$, where $T$ is the number of iterations. This matches the
convergence rate of the batch ADMM algorithm, but without the need to visit all
the samples in each iteration. Experiments on the graph-guided fused lasso
demonstrate that the new algorithm is significantly faster than
state-of-the-art stochastic and batch ADMM algorithms.

本文提出一种新的随机交替方向乘子法（ADMM）算法，其在线性化 ADMM 公式上逐步逼近全梯度。实验证明，该算法在凸优化问题上的收敛速度得到提高，速度显著快于现有的随机和批量 ADMM 算法。

快速随机交替方向乘子法

Fast Stochastic Alternating Direction Method of Multipliers

We consider the problem of learning a structured multi-task regression, where
the output consists of multiple responses that are related by a graph and the
correlated response variables are dependent on the common inputs in a sparse
but synergistic manner. Previous methods such as l1/l2-regularized multi-task
regression assume that all of the output variables are equally related to the
inputs, although in many real-world problems, outputs are related in a complex
manner. In this paper, we propose graph-guided fused lasso (GFlasso) for
structured multi-task regression that exploits the graph structure over the
output variables. We introduce a novel penalty function based on fusion penalty
to encourage highly correlated outputs to share a common set of relevant
inputs. In addition, we propose a simple yet efficient proximal-gradient method
for optimizing GFlasso that can also be applied to any optimization problems
with a convex smooth loss and the general class of fusion penalty defined on
arbitrary graph structures. By exploiting the structure of the non-smooth
''fusion penalty'', our method achieves a faster convergence rate than the
standard first-order method, sub-gradient method, and is significantly more
scalable than the widely adopted second-order cone-programming and
quadratic-programming formulations. In addition, we provide an analysis of the
consistency property of the GFlasso model. Experimental results not only
demonstrate the superiority of GFlasso over the standard lasso but also show
the efficiency and scalability of our proximal-gradient method.

本文提出了一种基于图的融合套索方法（GFlasso）用于结构化多任务回归问题，该方法利用了输出变量之间的图结构，通过融合惩罚函数鼓励高度相关的输出共享一组相关的输入，并提出了一种简单而有效的近端 - 梯度方法用于优化 GFlasso。

一种图结构的多任务回归和通用融合 Lasso 的高效优化方法

Graph-Structured Multi-task Regression and an Efficient Optimization  Method for General Fused Lasso

Many complex disease syndromes such as asthma consist of a large number of
highly related, rather than independent, clinical phenotypes, raising a new
technical challenge in identifying genetic variations associated simultaneously
with correlated traits. In this study, we propose a new statistical framework
called graph-guided fused lasso (GFlasso) to address this issue in a principled
way. Our approach explicitly represents the dependency structure among the
quantitative traits as a network, and leverages this trait network to encode
structured regularizations in a multivariate regression model over the
genotypes and traits, so that the genetic markers that jointly influence
subgroups of highly correlated traits can be detected with high sensitivity and
specificity. While most of the traditional methods examined each phenotype
independently and combined the results afterwards, our approach analyzes all of
the traits jointly in a single statistical method, and borrow information
across correlated phenotypes to discover the genetic markers that perturbe a
subset of correlated triats jointly rather than a single trait. Using simulated
datasets based on the HapMap consortium data and an asthma dataset, we compare
the performance of our method with the single-marker analysis, and other sparse
regression methods such as the ridge regression and the lasso that do not use
any structural information in the traits. Our results show that there is a
significant advantage in detecting the true causal SNPs when we incorporate the
correlation pattern in traits using our proposed methods.

该研究提出了一种名为图引导融合套索（GFlasso）的新的统计框架，通过将量化特征的相关性结构表示为网络，并利用该特征网络在基因型和特征上编码结构正则化，以便高灵敏度和特异性地检测共同影响高度相关特征子组的遗传标记。