Broyden–Fletcher–Goldfarb–Shanno algorithm

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems.^[1]

The BFGS method belongs to quasi-Newton methods, a class of hill-climbing optimization techniques that seek a stationary point of a (preferably twice continuously differentiable) function. For such problems, a necessary condition for optimality is that the gradient be zero. Newton's method and the BFGS methods are not guaranteed to converge unless the function has a quadratic Taylor expansion near an optimum. However, BFGS has proven to have good performance even for non-smooth optimizations.^[2]

In quasi-Newton methods, the Hessian matrix of second derivatives doesn't need to be evaluated directly. Instead, the Hessian matrix is approximated using updates specified by gradient evaluations (or approximate gradient evaluations). Quasi-Newton methods are generalizations of the secant method to find the root of the first derivative for multidimensional problems. In multi-dimensional problems, the secant equation does not specify a unique solution, and quasi-Newton methods differ in how they constrain the solution. The BFGS method is one of the most popular members of this class.^[3] Also in common use is L-BFGS, which is a limited-memory version of BFGS that is particularly suited to problems with very large numbers of variables (e.g., >1000). The BFGS-B^[4] variant handles simple box constraints.

Rationale

The optimization problem is to minimise $f(\mathbf {x} )$ where $\mathbf {x}$ is a vector in $R^{n}$ , and $f$ is a differentiable scalar function. There are no constraints on the values that $\mathbf {x}$ can take.

The algorithm begins at an initial estimate for the optimal value $\mathbf {x} _{0}$ and proceeds iteratively to get a better estimate at each stage.

The search direction p_k at stage k is given by the solution of the analogue of the Newton equation

B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x} _{k}),

where $B_{k}$ is an approximation to the Hessian matrix, which is updated iteratively at each stage, and $\nabla f(\mathbf {x} _{k})$ is the gradient of the function evaluated at x_k. A line search in the direction p_k is then used to find the next point x_k+1 by minimising $f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$ over the scalar $\alpha >0$ .

The quasi-Newton condition imposed on the update of $B_{k}$ is:

B_{k+1}(\mathbf {x} _{k+1}-\mathbf {x} _{k})=\nabla f(\mathbf {x} _{k+1})-\nabla f(\mathbf {x} _{k}).

Let $\mathbf {y} _{k}=\nabla f(\mathbf {x} _{k+1})-\nabla f(\mathbf {x} _{k})$ and $\mathbf {s} _{k}=\mathbf {x} _{k+1}-\mathbf {x} _{k}$ , then $B_{k+1}$ satisfies $B_{k+1}\mathbf {s} _{k}=\mathbf {y} _{k}$ which is the secant equation. The curvature condition $\mathbf {s} _{k}^{\top }\mathbf {y} _{k}>0$ should be satisfied. If the function is not strongly convex, then the condition has to be enforced explicitly.

Instead of requiring the full Hessian matrix at the point x_k+1 to be computed as B_k+1, the approximate Hessian at stage k is updated by the addition of two matrices:

B_{k+1}=B_{k}+U_{k}+V_{k}.

Both U_k and V_k are symmetric rank-one matrices, but their sum is a rank-two update matrix. BFGS and DFP updating matrix both differ from its predecessor by a rank-two matrix. Another simpler rank-one method is known as Symmetric rank-one method which does not guarantee the positive definitiveness. In order to maintain the symmetry and positive definitiveness of $B_{k+1}$ , the update form can be chosen as: $B_{k+1}=B_{k}+\alpha \mathbf {u} \mathbf {u} ^{\top }+\beta \mathbf {v} \mathbf {v} ^{\top }$ . Imposing the secant condition, $B_{k+1}\mathbf {s} _{k}=\mathbf {y} _{k}$ . Choosing $\mathbf {u} =\mathbf {y} _{k}$ and $\mathbf {v} =B_{k}\mathbf {s} _{k}$ , we can obtain:^[5]

\alpha ={\frac {1}{\mathbf {y} _{k}^{T}\mathbf {s} _{k}}}

\beta ={\frac {1}{\mathbf {s} _{k}^{T}B_{k}\mathbf {s} _{k}}}

Finally, we substitute $\alpha$ and $\beta$ into $B_{k+1}=B_{k}+\alpha \mathbf {u} \mathbf {u} ^{\top }+\beta \mathbf {v} \mathbf {v} ^{\top }$ and get the update equation of $B_{k+1}$ :

B_{k+1}=B_{k}+{\frac {\mathbf {y} _{k}\mathbf {y} _{k}^{\mathrm {T} }}{\mathbf {y} _{k}^{\mathrm {T} }\mathbf {s} _{k}}}-{\frac {B_{k}\mathbf {s} _{k}\mathbf {s} _{k}^{\mathrm {T} }B_{k}^{\mathrm {T} }}{\mathbf {s} _{k}^{\mathrm {T} }B_{k}\mathbf {s} _{k}}}

Algorithm

From an initial guess $\mathbf {x} _{0}$ and an approximate Hessian matrix $B_{0}$ the following steps are repeated as $\mathbf {x} _{k}$ converges to the solution:

Obtain a direction $\mathbf {p} _{k}$ by solving $B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x} _{k})$ .
Perform a one-dimensional optimization (line search) to find an acceptable stepsize $\alpha _{k}$ in the direction found in the first step, so $\alpha _{k}=\arg \min f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$ .
Set $\mathbf {s} _{k}=\alpha _{k}\mathbf {p} _{k}$ and update $\mathbf {x} _{k+1}=\mathbf {x} _{k}+\mathbf {s} _{k}$ .
$\mathbf {y} _{k}={\nabla f(\mathbf {x} _{k+1})-\nabla f(\mathbf {x} _{k})}$ .
$B_{k+1}=B_{k}+{\frac {\mathbf {y} _{k}\mathbf {y} _{k}^{\mathrm {T} }}{\mathbf {y} _{k}^{\mathrm {T} }\mathbf {s} _{k}}}-{\frac {B_{k}\mathbf {s} _{k}\mathbf {s} _{k}^{\mathrm {T} }B_{k}}{\mathbf {s} _{k}^{\mathrm {T} }B_{k}\mathbf {s} _{k}}}$ .

$f(\mathbf {x} )$ denotes the objective function to be minimized. Convergence can be checked by observing the norm of the gradient, $\left|\nabla f(\mathbf {x} _{k})\right|$ . Practically, $B_{0}$ can be initialized with $B_{0}=I$ , so that the first step will be equivalent to a gradient descent, but further steps are more and more refined by $B_{k}$ , the approximation to the Hessian.

The first step of the algorithm is carried out using the inverse of the matrix $B_{k}$ , which can be obtained efficiently by applying the Sherman–Morrison formula to the step 5 of the algorithm, giving

B_{k+1}^{-1}=\left(I-{\frac {s_{k}y_{k}^{T}}{y_{k}^{T}s_{k}}}\right)B_{k}^{-1}\left(I-{\frac {y_{k}s_{k}^{T}}{y_{k}^{T}s_{k}}}\right)+{\frac {s_{k}s_{k}^{T}}{y_{k}^{T}s_{k}}}.

This can be computed efficiently without temporary matrices, recognizing that $B_{k}^{-1}$ is symmetric, and that $\mathbf {y} _{k}^{\mathrm {T} }B_{k}^{-1}\mathbf {y} _{k}$ and $\mathbf {s} _{k}^{\mathrm {T} }\mathbf {y} _{k}$ are scalar, using an expansion such as

B_{k+1}^{-1}=B_{k}^{-1}+{\frac {(\mathbf {s} _{k}^{\mathrm {T} }\mathbf {y} _{k}+\mathbf {y} _{k}^{\mathrm {T} }B_{k}^{-1}\mathbf {y} _{k})(\mathbf {s} _{k}\mathbf {s} _{k}^{\mathrm {T} })}{(\mathbf {s} _{k}^{\mathrm {T} }\mathbf {y} _{k})^{2}}}-{\frac {B_{k}^{-1}\mathbf {y} _{k}\mathbf {s} _{k}^{\mathrm {T} }+\mathbf {s} _{k}\mathbf {y} _{k}^{\mathrm {T} }B_{k}^{-1}}{\mathbf {s} _{k}^{\mathrm {T} }\mathbf {y} _{k}}}.

In statistical estimation problems (such as maximum likelihood or Bayesian inference), credible intervals or confidence intervals for the solution can be estimated from the inverse of the final Hessian matrix. However, these quantities are technically defined by the true Hessian matrix, and the BFGS approximation may not converge to the true Hessian matrix.

Implementations

The GSL implements BFGS as gsl_multimin_fdfminimizer_vector_bfgs2. Ceres Solver implements both BFGS and L-BFGS.

In SciPy, the scipy.optimize.fmin_bfgs function implements BFGS. It is also possible to run BFGS using any of the L-BFGS algorithms by setting the parameter L to a very large number.

Octave uses BFGS with a double-dogleg approximation to the cubic line search.

In R, the BFGS algorithm (and the L-BFGS-B version that allows box constraints) is implemented as an option of the base function optim().

In the MATLAB Optimization Toolbox, the fminunc function uses BFGS with cubic line search when the problem size is set to "medium scale."

A high-precision arithmetic version of BFGS (pBFGS), implemented in C++ and integrated with the high-precision arithmetic package ARPREC is robust against numerical instability (e.g. round-off errors).

Another C++ implementation of BFGS (along with L-BFGS, L-BFGS-B, CG, and Newton's method) using Eigen (C++ library) are available on GitHub under the MIT License here.

BFGS and L-BFGS are also implemented in C as part of the open-source Gnu Regression, Econometrics and Time-series Library (gretl).

Notes

↑ Fletcher, Roger (1987), Practical methods of optimization (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-91547-8
↑ Lewis, Adrian S.; Overton, Michael (2009), Nonsmooth optimization via BFGS (PDF)
↑ Nocedal & Wright (2006), page 24
↑ Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge; Zhu, Ciyou (1995), "A Limited Memory Algorithm for Bound Constrained Optimization", SIAM Journal on Scientific Computing, 16 (5): 1190–1208, doi:10.1137/0916069
↑ Fletcher, Roger (1987), Practical methods of optimization (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-91547-8

Bibliography

Avriel, Mordecai (2003), Nonlinear Programming: Analysis and Methods, Dover Publishing, ISBN 0-486-43227-0
Bonnans, J. Frédéric; Gilbert, J. Charles; Lemaréchal, Claude; Sagastizábal, Claudia A. (2006), Numerical optimization: Theoretical and practical aspects, Universitext (Second revised ed. of translation of 1997 French ed.), Berlin: Springer-Verlag, pp. xiv+490, ISBN 3-540-35445-X, MR 2265882, doi:10.1007/978-3-540-35447-5 French Original: Optimisation numérique: Aspects théoriques et pratiques ISBN 3-540-63183-6
Broyden, C. G. (1970), "The convergence of a class of double-rank minimization algorithms", Journal of the Institute of Mathematics and Its Applications, 6: 76–90, doi:10.1093/imamat/6.1.76
Fletcher, R. (1970), "A New Approach to Variable Metric Algorithms", Computer Journal, 13 (3): 317–322, doi:10.1093/comjnl/13.3.317
Fletcher, Roger (1987), Practical methods of optimization (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-91547-8
Goldfarb, D. (1970), "A Family of Variable Metric Updates Derived by Variational Means", Mathematics of Computation, 24 (109): 23–26, doi:10.1090/S0025-5718-1970-0258249-6
Luenberger, David G.; Ye, Yinyu (2008), Linear and nonlinear programming, International Series in Operations Research & Management Science, 116 (Third ed.), New York: Springer, pp. xiv+546, ISBN 978-0-387-74502-2, MR 2423726
Nocedal, Jorge; Wright, Stephen J. (2006), Numerical Optimization (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-30303-1
Shanno, David F. (July 1970), "Conditioning of quasi-Newton methods for function minimization", Mathematics of Computation, 24 (111): 647–656, MR 42:8905, doi:10.1090/S0025-5718-1970-0274029-X
Shanno, David F.; Kettler, Paul C. (July 1970), "Optimal conditioning of quasi-Newton methods", Mathematics of Computation, 24 (111): 657–664, MR 42:8906, doi:10.1090/S0025-5718-1970-0274030-6

External links

Source code of high-precision BFGS A C++ source code of BFGS with high-precision arithmetic

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