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.
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.
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 , where is the current derivative and is the inverse of the Hessian matrix. 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."
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.
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.
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). 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.
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 an L-BFGS step, the method allows some variables to change sign, and repeats the process.
Schraudolph et al. present an online approximation to both BFGS and L-BFGS. 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. It has been shown that O-LBFGS has a global almost sure convergence  while the online approximation of BFGS (O-BFGS) is not necessarily convergent.
An early, open source implementation of L-BFGS in Fortran exists in Netlib as a shar archive . 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
A reference implementation 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
OWL-QN implementations are available in:
- C++ implementation by its designers, includes the original ICML paper on the algorithm
- The CRF toolkit Wapiti includes a C implementation
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