# Limited-memory BFGS

Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning.[1][2]

Like the original BFGS, L-BFGS uses an approximation to the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense n×n approximation to the inverse Hessian (n being the number of variables in the problem), L-BFGS stores only a few vectors that represent the approximation implicitly. Due to its resulting linear memory requirement, the L-BFGS method is particularly well suited for optimization problems with a large number of variables. Instead of the inverse Hessian Hk, L-BFGS maintains a history of the past m updates of the position x and gradient ∇f(x), where generally the history size m can be small (often m<10). These updates are used to implicitly do operations requiring the Hk-vector product.

## Algorithm

L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication for finding the search direction is carried out $d_k=-H_k g_k\,\!$. There are multiple published approaches to using a history of updates to form this direction vector. Here, we give a common approach, the so-called "two loop recursion."[3][4]

We'll take as given $x_k\,\!$, the position at the $k\,\!$-th iteration, and $g_k\equiv\nabla f(x_k)$ where $f\,\!$ is the function being minimized, and all vectors are column vectors. We also assume that we have stored the last $m$ updates of the form $s_k = x_{k+1} - x_k\,\!$ and $y_k = g_{k+1} - g_k\,\!$. We'll define $\rho_k = \frac{1}{y^{\rm T}_k s_k}$, and $H^0_k\,\!$ will be the 'initial' approximate of the inverse Hessian that our estimate at iteration $k\,\!$ begins with. Then we can compute the (uphill) direction as follows:

$q = g_k\,\!$
For $i=k-1, k-2, \ldots, k-m$
$\alpha_i = \rho_i s^{\rm T}_i q\,\!$
$q = q - \alpha_i y_i\,\!$
$H_k=y^{\rm T}_{k-1} s_{k-1}/y^{\rm T}_{k-1} y_{k-1}$
$z = H_k q$
For $i=k-m, k-m+1, \ldots, k-1$
$\beta_i = \rho_i y^{\rm T}_i z\,\!$
$z = z + s_i (\alpha_i - \beta_i)\,\!$
Stop with $H_k g_k = z\,\!$

This formulation is valid whether we are minimizing or maximizing. Note that if we are minimizing, the search direction would be the negative of z (since z is "uphill"), and if we are maximizing, $H^{0}_k$ should be negative definite rather than positive definite. We would typically do a backtracking line search in the search direction (any line search would be valid, but L-BFGS does not require exact line searches in order to converge).

Commonly, the inverse Hessian $H^0_k\,\!$ is represented as a diagonal matrix, so that initially setting $z\,\!$ requires only an element-by-element multiplication.

This two loop update only works for the inverse Hessian. Approaches to implementing L-BFGS using the direct approximate Hessian $B_k\,\!$ have also been developed, as have other means of approximating the inverse Hessian.[5]

## Applications

L-BFGS has been called "the algorithm of choice" for fitting log-linear (MaxEnt) models and conditional random fields with $\ell_2$-regularization.[2]

## Variants

Since BFGS (and hence L-BFGS) is designed to minimize smooth functions without constraints, the L-BFGS algorithm must be modified to handle functions that include non-differentiable components or constraints. A popular class of modifications are called active-set methods, based on the concept of the active set. The idea is that when restricted to a small neighborhood of the current iterate, the function and constraints can be simplified.

### L-BFGS-B

The L-BFGS-B algorithm extends L-BFGS to handle simple box constraints (aka bound constraints) on variables; that is, constraints of the form $l_i \leq x_i \leq u_i$ where li and ui are per-variable constant lower and upper bounds, respectively (for each xi, either or both bounds may be omitted).[6][7] The method works by identifying fixed and free variables at every step (using a simple gradient method), and then using the L-BFGS method on the free variables only to get higher accuracy, and then repeating the process.

### OWL-QN

Orthant-wise limited-memory quasi-Newton (OWL-QN) is an L-BFGS variant for fitting $\ell_1$-regularized models, exploiting the inherent sparsity of such models.[2] It minimizes functions of the form

$f(\vec x) = g(\vec x) + C \|\vec x\|_1$

where $g$ is a differentiable convex loss function. The method is an active-set type method: at each iterate, it estimates the sign of each component of the variable, and restricts the subsequent step to have the same sign. Once the sign is fixed, the non-differentiable $\|\vec x\|_1$ term becomes a smooth linear term which can be handled by L-BFGS. After a L-BFGS step, the method allows some variables to change sign, and repeats the process.

### O-LBFGS

Schraudolph et al. present an online approximation to both BFGS and L-BFGS.[8] Similar to stochastic gradient descent, this can be used to reduce the computational complexity by evaluating the error function and gradient on a randomly drawn subset of the overall dataset in each iteration.

## Implementations

An early, open source implementation of L-BFGS in Fortran exists in Netlib as a shar archive [1]. Multiple other open source implementations have been produced as translations of this Fortran code (e.g. java, and python via SciPy). Other implementations exist (e.g. Matlab (optimization toolbox), Matlab (BSD)), frequently as part of generic optimization libraries (e.g. Mathematica, FuncLib C# library, and dlib C++ library). The libLBFGS is a C implementation.

### Implementations of variants

The L-BFGS-B variant also exists as ACM TOMS algorithm 778.[7] In February 2011, some of the authors of the original L-BFGS-B code posted a major update (version 3.0).

A reference implementation[9] is available in Fortran 77 (and with a Fortran 90 interface) at the author's website. This version, as well as older versions, has been converted to many other languages, including a Java wrapper for v3.0; Matlab interfaces for v3.0, v2.4, and v2.1; a C++ interface for v2.1; a Python interface for v3.0 as part of scipy.optimize.minimize; an OCaml interface for v2.1 and v3.0; version 2.3 has been converted to C by f2c and is available at this website; and R's optim general-purpose optimizer routine includes L-BFGS-B by using method="L-BFGS-B".[10]

There exists a complete C++11 rewrite of the L-BFGS-B solver using Eigen3. OWL-QN implementations are available in:

## Works cited

1. ^ Malouf, Robert (2002). "A comparison of algorithms for maximum entropy parameter estimation". Proc. Sixth Conf. on Natural Language Learning (CoNLL). pp. 49–55.
2. ^ a b c d Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear models". Proceedings of the 24th International Conference on Machine Learning. doi:10.1145/1273496.1273501. ISBN 9781595937933. edit
3. ^ Matthies, H.; Strang, G. (1979). "The solution of non linear finite element equations". International Journal for Numerical Methods in Engineering 14 (11): 1613–1626. doi:10.1002/nme.1620141104.
4. ^ Nocedal, J. (1980). "Updating Quasi-Newton Matrices with Limited Storage". Mathematics of Computation 35 (151): 773–782. doi:10.1090/S0025-5718-1980-0572855-7.
5. ^ Byrd, R. H.; Nocedal, J.; Schnabel, R. B. (1994). "Representations of Quasi-Newton Matrices and their use in Limited Memory Methods". Mathematical Programming 63 (4): 129–156. doi:10.1007/BF01582063.
6. ^ Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. (1995). "A Limited Memory Algorithm for Bound Constrained Optimization". SIAM J. Sci. Comput. 16 (5): 1190–1208. doi:10.1137/0916069. edit
7. ^ a b Zhu, C.; Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge (1997). "L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization". ACM Transactions on Mathematical Software 23 (4): 550–560. doi:10.1145/279232.279236.
8. ^ N. Schraudolph, J. Yu, and S. Günter (2007). "A stochastic quasi-Newton method for online convex optimization". AISTATS.
9. ^ Morales, J. L.; Nocedal, J. (2011). "Remark on "algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization"". ACM Transactions on Mathematical Software 38: 1. doi:10.1145/2049662.2049669. edit
10. ^ "General-purpose Optimization". R documentation. Comprehensive R Archive Network.