Second-Order Forward-Mode Automatic Differentiation for Optimization

Adam D. Cobb, Atilim Gunes Baydin, Barak A. Pearlmutter, Susmit Jha

Keywords: Optimization, Automatic Differentiation

TL;DR: This paper introduces a new second-order hyperplane search that uses forward-mode automatic differentiation. We include empirical and theoretical results.

Abstract: Forward gradient methods offer a promising alternative to backpropagation. Optimization that only requires forward passes could simplify hardware implementation, improve parallelism, lower memory cost, and allow for more biologically plausible learning models. This has motivated recent forward-mode automated differentiation (AD) methods. This paper presents a novel second-order forward-mode AD method for optimization that generalizes a second-order line search to a $K$-dimensional hyperplane. Unlike recent work that relies on directional derivatives (or Jacobian–Vector Products, JVPs), we use hyper-dual numbers to jointly evaluate both directional derivatives and their second-order quadratic terms. As a result, we introduce forward-mode weight perturbation with Hessian information for K-dimensional hyper-plane search (FoMoH-$K$D). We derive the convergence properties of FoMoH-$K$D and show how it generalizes to Newton’s method for $K = D$. We demonstrate this generalization empirically, and compare the performance of FoMoH-$K$D to forward gradient descent (FGD) on three case studies: Rosenbrock function used widely for evaluating optimization methods, logistic regression with 7,850 parameters, and learning a CNN classifier with 431,080 parameters. Our experiments show that FoMoH-$K$D not only achieves better performance and accuracy, but also converges faster, thus, empirically verifying our theoretical results.

Primary Area: optimization

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Submission Number: 443