SMO算法，platt论文的原始算法及优化算法

 platt论文：[PDF] Sequential Minimal Optimization : A Fast Algorithm for Training Support Vector Machines | Semantic Scholar

算法优化：[PDF] Improvements to Platt's SMO Algorithm for SVM Classifier Design | Semantic Scholar

包含个人platt版SMO代码实现、SMO 优化算法、libsvm：yanzhi0922/SVM: 2024.07.12 (github.com)

数据集获取：LIBSVM Data: Classification, Regression, and Multi-label (ntu.edu.tw)

platt原论文SMO伪代码：

target = desired output vector
point = training point matrix
procedure takeStep(i1,i2)if (i1 == i2) return 0alph1 = Lagrange multiplier for i1y1 = target[i1]E1 = SVM output on point[i1] – y1 (check in error cache)s = y1*y2Compute L, H via equations (13) and (14)if (L == H)return 0k11 = kernel(point[i1],point[i1])k12 = kernel(point[i1],point[i2])k22 = kernel(point[i2],point[i2])eta = k11+k22-2*k12if (eta > 0){a2 = alph2 + y2*(E1-E2)/etaif (a2 < L) a2 = Lelse if (a2 > H) a2 = H}else{Lobj = objective function at a2=LHobj = objective function at a2=Hif (Lobj < Hobj-eps)a2 = Lelse if (Lobj > Hobj+eps)a2 = Helsea2 = alph2}if (|a2-alph2| < eps*(a2+alph2+eps))return 0a1 = alph1+s*(alph2-a2)Update threshold to reflect change in Lagrange multipliersUpdate weight vector to reflect change in a1 & a2, if SVM is linearUpdate error cache using new Lagrange multipliersStore a1 in the alpha arrayStore a2 in the alpha arrayreturn 1
endprocedureprocedure examineExample(i2)y2 = target[i2]alph2 = Lagrange multiplier for i2E2 = SVM output on point[i2] – y2 (check in error cache)r2 = E2*y2if ((r2 < -tol && alph2 < C) || (r2 > tol && alph2 > 0)){if (number of non-zero & non-C alpha > 1){i1 = result of second choice heuristic (section 2.2)if takeStep(i1,i2)return 1}loop over all non-zero and non-C alpha, starting at a random point{i1 = identity of current alphaif takeStep(i1,i2)return 1}loop over all possible i1, starting at a random point{i1 = loop variableif (takeStep(i1,i2)return 1}}return 0
endproceduremain routine:numChanged = 0;examineAll = 1;while (numChanged > 0 | examineAll){numChanged = 0;if (examineAll)loop I over all training examplesnumChanged += examineExample(I)elseloop I over examples where alpha is not 0 & not CnumChanged += examineExample(I)if (examineAll == 1)examineAll = 0else if (numChanged == 0)examineAll = 1
}

以上算法基本原理相同，结果相同，优化是时间复杂度上的优化 ，libsvm时间复杂度最优