Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical applications.
However, there are two problems with this analysis. First, the linear convergence parameter, @math , may be very close to zero and the initial unspecified number of iterations might be very large. In fact, while the result of @cite_2 does not explicitly specify @math , an examine of their proof shows that @math is proportional to the smallest nonzero eigenvalue of @math , where @math is the @math data matrix with its @math -th row be the @math -th data point @math . For example if two data points @math becomes closer and closer, then @math . This dependency is problematic in the data laden domain, and we note that such a dependency does not occur in the analysis of SGD.
@cite_2
2277973662 2099119387 2343345052 1486249389
This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently.