NESVM: A Fast Gradient Method for Support Vector Machines. Support vector machines (SVMs) are invaluable tools for many practical applications in artificial intelligence, e.g., classification and event recognition. However, popular SVM solvers are not sufficiently efficient for applications with a great deal of samples as well as a large number of features. In this paper, thus, we present NESVM, a fast gradient SVM solver that can optimize various SVM models, e.g., classical SVM, linear programming SVM and least square SVM. Compared against SVM-Perf (whose convergence rate in solving the dual SVM is upper bounded by O(1/√k) where k is the number of iterations) and Pegasos (online SVM that converges at rate O(1/k) for the primal SVM), NESVM achieves the optimal convergence rate at O(1/k2) and a linear time complexity. In particular, NESVM smoothes the nondifferentiable hinge loss and ℓ1-norm in the primal SVM. Then the optimal gradient method without any line search is adopted to solve the optimization. In each iteration round, the current gradient and historical gradients are combined to determine the descent direction, while the Lipschitz constant determines the step size. Only two matrix-vector multiplications are required in each iteration round. Therefore, NESVM is more efficient than existing SVM solvers. In addition, NESVM is available for both linear and nonlinear kernels. We also propose ”homotopy NESVM” to accelerate NESVM by dynamically decreasing the smooth parameter and using the continuation method. Our experiments on census income categorization, indoor/outdoor scene classification event recognition and scene recognition suggest the efficiency and the effectiveness of NESVM. The MATLAB code of NESVM will be available on our website for further assessment.