skip to main content


Title: Algorithm-hardware co-optimization of the memristor-based framework for solving SOCP and homogeneous QCQP problems
A memristor crossbar, which is constructed with memristor devices, has the unique ability to change and memorize the state of each of its memristor elements. It also has other highly desirable features such as high density, low power operation and excellent scalability. Hence the memristor crossbar technology can potentially be utilized for developing low-complexity and high-scalability solution frameworks for solving a large class of convex optimization problems, which involve extensive matrix operations and have critical applications in multiple disciplines. This paper, as the first attempt towards this direction, proposes a novel memristor crossbar-based framework for solving two important convex optimization problems, i.e., second-order cone programming (SOCP) and homogeneous quadratically constrained quadratic programming (QCQP) problems. In this paper, the alternating direction method of multipliers (ADMM) is adopted. It splits the SOCP and homogeneous QCQP problems into sub-problems that involve the solution of linear systems, which could be effectively solved using the memristor crossbar in O(1) time complexity. The proposed algorithm is an iterative procedure that iterates a constant number of times. Therefore, algorithms to solve SOCP and homogeneous QCQP problems have pseudo-O(N) complexity, which is a significant reduction compared to the state-of-the-art software solvers (O(N3.5)-O(N4)).  more » « less
Award ID(s):
1637559
NSF-PAR ID:
10110068
Author(s) / Creator(s):
; ; ; ; ;
Date Published:
Journal Name:
2017 22nd Asia and South Pacific Design Automation Conference (ASP-DAC)
Page Range / eLocation ID:
788 to 793
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Memristors have recently received significant attention as device-level components for building a novel generation of computing systems. These devices have many promising features, such as non-volatility, low power consumption, high density, and excellent scalability. The ability to control and modify biasing voltages at memristor terminals make them promising candidates to efficiently perform matrix-vector multiplications and solve systems of linear equations. In this article, we discuss how networks of memristors arranged in crossbar arrays can be used for efficiently solving optimization and machine learning problems. We introduce a new memristor-based optimization framework that combines the computational merits of memristor crossbars with the advantages of an operator splitting method, the alternating direction method of multipliers (ADMM). Here, ADMM helps in splitting a complex optimization problem into subproblems that involve the solution of systems of linear equations. The strength of this framework is shown by applying it to linear programming, quadratic programming, and sparse optimization. In addition to ADMM, implementation of a customized power iteration method for eigenvalue/eigenvector computation using memristor crossbars is discussed. The memristor-based power iteration method can further be applied to principal component analysis. The use of memristor crossbars yields a significant speed-up in computation, and thus, we believe, has the potential to advance optimization and machine learning research in artificial intelligence. 
    more » « less
  2. In this paper, we propose a new approach for robust compressive sensing (CS) using memristor crossbars that are constructed by recently invented memristor devices. The exciting features of a memristor crossbar, such as high density, low power and great scalability, make it a promising candidate to perform large-scale matrix operations. To apply memristor crossbars to solve a robust CS problem, the alternating directions method of multipliers (ADMM) is employed to split the original problem into subproblems that involve the solution of systems of linear equations. A system of linear equations can then be solved using memristor crossbars with astonishing O(1) time complexity. We also study the impact of hardware variations on the memristor crossbar based CS solver from both theoretical and practical points of view. The resulting overall complexity is given by O(n), which achieves O(n2.5) speed-up compared to the state-of-the-art software approach. Numerical results are provided to illustrate the effectiveness of the proposed CS solver. 
    more » « less
  3. We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors’ corrected analysis of a method of Skajaa and Ye. It enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, algorithms for nonsymmetric conic optimization also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for nonsymmetric cone optimization: [Formula: see text] iterations to reach an ε-optimal solution, where ν is the barrier parameter of the barrier function used in the optimization. Alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. A simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using an optimal design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared with using state-of-the-art, off-the-shelf conic optimization software. Summary of Contribution: The paper describes an open-source Matlab package for optimization over nonsymmetric cones. A particularly important feature of this software is that, unlike other conic optimization software, it enables optimization over any convex cone as long as a suitable barrier function is available for the cone or its dual, not limiting the user to a small number of specific cones. Nonsymmetric cones for which such barriers are already known include, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Thus, the scope of this software is far larger than most current conic optimization software. This does not come at the price of efficiency, as the worst-case iteration complexity of our algorithm matches the iteration complexity of the most successful interior-point methods for symmetric cones. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, our software can also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. This is also demonstrated in this paper via an example in which our code significantly outperforms Mosek 9 and SCS 2. 
    more » « less
  4. Abstract

    In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$(T+1)-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$ε-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$O((2LTε)d), and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$O((LT4ε)d)for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$O((LT8ε)d/2-1)for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$(Tε)-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

     
    more » « less
  5. The high computation and memory storage of large deep neural networks (DNNs) models pose intensive challenges to the conventional Von-Neumann architecture, incurring substantial data movements in the memory hierarchy. The memristor crossbar array has emerged as a promising solution to mitigate the challenges and enable low-power acceleration of DNNs. Memristor-based weight pruning and weight quantization have been separately investigated and proven effectiveness in reducing area and power consumption compared to the original DNN model. However, there has been no systematic investigation of memristor-based neuromorphic computing (NC) systems considering both weight pruning and weight quantization. In this paper, we propose an unified and systematic memristor-based framework considering both structured weight pruning and weight quantization by incorporating alternating direction method of multipliers (ADMM) into DNNs training. We consider hardware constraints such as crossbar blocks pruning, conductance range, and mismatch between weight value and real devices, to achieve high accuracy and low power and small area footprint. Our framework is mainly integrated by three steps, i.e., memristor- based ADMM regularized optimization, masked mapping and retraining. Experimental results show that our proposed frame- work achieves 29.81× (20.88×) weight compression ratio, with 98.38% (96.96%) and 98.29% (97.47%) power and area reduction on VGG-16 (ResNet-18) network where only have 0.5% (0.76%) accuracy loss, compared to the original DNN models. We share our models at anonymous link http://bit.ly/2Jp5LHJ . 
    more » « less