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 estimation 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.
Contents
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 . There are multiple published approaches 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 , the position at the
-th iteration, and
where
is the function being minimized, and all vectors are column vectors. We also assume that we have stored the last
updates of the form
and
. We'll define
, and
will be the 'initial' approximate of the inverse Hessian that our estimate at iteration
begins with. Then we can compute the (uphill) direction as follows:
For
![]()
![]()
![]()
![]()
For
![]()
![]()
Stop with
![]()
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, 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 is represented as a diagonal matrix, so that initially setting
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 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 -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 li ≤ xi ≤ ui 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 -regularized models, exploiting the inherent sparsity of such models.[2] It minimizes functions of the form
where 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
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:
- fmincon (Matlab optimization toolbox)
- FMINLBFGS (for Matlab, BSD license)
- minFunc (also for Matlab)
- LBFGS-D (in the D programming language))
- Frequently as part of generic optimization libraries (e.g. Mathematica, FuncLib C# library, and dlib C++ library)
- The libLBFGS is a C implementation.
- Maximization in Two-Class Logistic Regression (in Microsoft Azure ML)
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:
- C++ implementation by its designers, includes the original ICML paper on the algorithm[2]
- The CRF toolkit Wapiti includes a C implementation
- libLBFGS
Works cited
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 2.0 2.1 2.2 2.3 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 7.0 7.1 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
Further reading
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.