Using Interior Point Methods for Large-scale Support Vector Machine training

Support Vector Machines (SVMs) are powerful machine learning techniques for classification and regression, but the training stage involves a convex quadratic optimization program that is most often computationally expensive. Traditionally, active-set methods have been used rather than interior point methods, due to the Hessian in the standard dual formulation being completely dense. But as active-set methods are essentially sequential, they may not be adequate for machine learning challenges of the future. Additionally, training time may be limited, or data may grow so large that cluster-computing approaches need to be considered. Interior point methods have the potential to answer these concerns directly. They scale efficiently, they can provide good early approximations, and they are suitable for parallel and multi-core environments. To apply them to SVM training, it is necessary to address directly the most computationally expensive aspect of the algorithm. We therefore present an exact reformulation of the standard linear SVM training optimization problem that exploits separability of terms in the objective. By so doing, per-iteration computational complexity is reduced from O(n3) to O(n). We show how this reformulation can be applied to many machine learning problems in the SVM family. Implementation issues relating to specializing the algorithm are explored through extensive numerical experiments. They show that the performance of our algorithm for large dense or noisy data sets is consistent and highly competitive, and in some cases can out perform all other approaches by a large margin. Unlike active set methods, performance is largely unaffected by noisy data. We also show how, by exploiting the block structure of the augmented system matrix, a hybrid MPI/Open MP implementation of the algorithm enables data and linear algebra computations to be efficiently partitioned amongst parallel processing nodes in a clustered computing environment. The applicability of our technique is extended to nonlinear SVMs by low-rank approximation of the kernel matrix. We develop a heuristic designed to represent clusters using a small number of features. Additionally, an early approximation scheme reduces the number of samples that need to be considered. Both elements improve the computational efficiency of the training phase. Taken as a whole, this thesis shows that with suitable problem formulation and efficient implementation techniques, interior point methods are a viable optimization technology to apply to large-scale SVM training, and are able to provide state-of-the-art performance

High performance interior point methods for three-dimensional finite element limit analysis

The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×

Adapting the interior point method for the solution of linear programs on high performance computers

In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial

Solving large scale linear programming problems

The interior point method (IPM) is now well established as a computationaly com-petitive scheme for solving very large scale linear programming problems. The leading variant of the IPM is the primal dual predictor corrector algorithm due to Mehrotra. The main computational efforts in this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations. We describe an implementation of this algorithm for vector processors. At the heart of the implementation is a vectorized matrix multiplication and Cholesky factorization for sparse matrices. We identify the parts where vectorization can be beneficial and discuss in details the merits of alternative vectorization techniques. We show that the best way to utilize a vector processor is by exploiting dense computation within the sparse framework and by unrolling loop operations. We further present an extended definition of supernodes, and describe an implementation based on this new approach. We show that although this approach requires more memory it can increase the scope of dense computation substantially with out adding extra operations. Performance results on standard industrial test problems and comparison between an algorithm that utilizes the extended supernodes and one that utilizes standard supernodes are presented and discussed

Interference Exploitation-based Hybrid Precoding with Robustness Against Phase Errors

Hybrid analog-digital precoding significantly reduces the hardware costs in massive MIMO transceivers when compared to fully-digital precoding at the expense of increased transmit power. In order to mitigate the above shortfall, we use the concept of constructive interference-based precoding, which has been shown to offer significant transmit power savings when compared with the conventional interference suppression-based precoding in fully-digital multiuser MIMO systems. Moreover, in order to circumvent the potential quality-of-service degradation at the users due to the hardware impairments in the transmitters, we judiciously incorporate robustness against such vulnerabilities in the precoder design. Since the undertaken constructive interference-based robust hybrid precoding problem is nonconvex with infinite constraints and thus difficult to solve optimally, we decompose the problem into two subtasks, namely, analog precoding and digital precoding. In this paper, we propose an algorithm to compute the optimal constructive interference-based robust digital precoders. Furthermore, we devise a scheme to facilitate the implementation of the proposed algorithm in a low-complexity and distributed manner. We also discuss block-level analog precoding techniques. Simulation results demonstrate the superiority of the proposed algorithm and its implementation scheme over the state-of-the-art methods