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"DESCENT"
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Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function
In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an
-accurate solution with probability at least
in at most
iterations, where
is the number of blocks. This extends recent results of Nesterov (SIAM J Optim 22(2): 341–362, 2012), which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing
from the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving first true iteration complexity bounds. For strongly convex functions the method converges linearly. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale
-regularized least squares problems with a billion variables.
Journal Article
Black Indians : a hidden heritage
Traces the history of relations between blacks and American Indians, and the existence of black Indians, from the earliest foreign landings through pioneer days.
Book
Correction: The Single-Breath Diffusing Capacity of CO and NO in Healthy Children of European Descent
[This corrects the article DOI: 10.1371/journal.pone.0113177.].
Journal Article
Moon music
Detective Romulus Poe of Las Vegas investigates the murder of a show girl whose mutilated body was found in the desert. The probe pits Poe against the owner of a casino.
Book
On the complexity analysis of randomized block-coordinate descent methods
In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in Nesterov (SIAM J Optim 22(2):341–362,
2012
), Richtárik and Takáč (Math Program 144(1–2):1–38,
2014
) for minimizing the sum of a smooth convex function and a block-separable convex function, and derive improved bounds on their convergence rates. In particular, we extend Nesterov’s technique developed in Nesterov (SIAM J Optim 22(2):341–362,
2012
) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in Richtárik and Takáč (Math Program 144(1–2):1–38,
2014
). As a result, we also obtain a better high-probability type of iteration complexity. In addition, for unconstrained smooth convex minimization, we develop a new technique called
randomized estimate sequence
to analyze the accelerated RBCD method proposed by Nesterov (SIAM J Optim 22(2):341–362,
2012
) and establish a sharper expected-value type of convergence rate than the one given in Nesterov (SIAM J Optim 22(2):341–362,
2012
).
Journal Article
Queen Elizabeth II : a potrait of her 70 year reign
This is the story of a life dedicated to public service, reported by The Times as it unfolded. From her time as a young princess to that as an internationally admired head of state, Queen Elizabeth has always fascinated and intrigued. Discover insights and memories, and see a changing society reflected in reporting from throughout the 70-year reign of Britain's longest serving monarch.
Book
Tibio-femoral kinematics of the healthy knee joint throughout complete cycles of gait activities
Accurate assessment of 3D tibio-femoral kinematics is essential for understanding knee joint functionality, but also provides a basis for assessing joint pathologies and the efficacy of musculoskeletal interventions. Until now, however, the assessment of functional kinematics in healthy knees has been mostly restricted to the loaded stance phase of gait, and level walking only, but the most critical conditions for the surrounding soft tissues are known to occur during high-flexion activities. This study aimed to determine the ranges of tibio-femoral rotation and condylar translation as well as provide evidence on the location of the centre of rotation during multiple complete cycles of different gait activities.
Based on radiographic images captured using moving fluoroscopy in ten healthy subjects during multiple cycles of level walking, downhill walking and stair descent, 3D femoral and tibial poses were reconstructed to provide a comprehensive description of tibio-femoral kinematics.
Despite a significant increase in joint flexion, the condylar antero-posterior range of motion remained comparable across all activities, with mean translations of 6.3–8.3 mm and 7.3–9.3 mm for the medial and lateral condyles respectively. Only the swing phase of level walking and stair descent exhibited a significantly greater range of motion for the lateral over the medial compartment. Although intra-subject variability was low, considerable differences in joint kinematics were observed between subjects. The observed subject-specific movement patterns indicate that accurate assessment of individual pre-operative kinematics together with individual implant selection and/or surgical implantation decisions might be necessary before further improvement to joint replacement outcome can be achieved.
Journal Article
