Limited-memory BFGS(Redirected from L-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.
The algorithm's target problem is to minimise over unconstrained values of the real-vector where is a differentiable scalar function.
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.
The algorithm starts with an initial estimate of the optimal value, , and proceeds iteratively to refine that estimate with a sequence of better estimates . The derivatives of the function are used as a key driver of the algorithm to identify the direction of steepest descent, and also to form an estimate of the Hessian matrix (second derivative) of f.
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 k-th iteration, and 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 and . We'll define , and will be the 'initial' approximate of the inverse Hessian that our estimate at iteration k begins with.
The algorithm is based on the BFGS recursion for the inverse Hessian as
For a fixed k we define a sequence of vectors as and . Then a recursive algorithm for calculating from is to define and .
We also define another sequence of vectors as . There is another recursive algorithm for calculating these vectors which is to define and then recursively define and . The value of is then our approximation for the direction of steepest ascent.
Thus 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).
For minimization, the inverse Hessian must be positive definite. The matrix is thus often represented as a diagonal matrix, with the added benefit that initially setting z only requires 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 variantsEdit
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:
- Malouf, Robert (2002). A comparison of algorithms for maximum entropy parameter estimation (PDF). Proc. Sixth Conf. on Natural Language Learning (CoNLL). pp. 49–55.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- N. Schraudolph, J. Yu, and S. Günter (2007). A stochastic quasi-Newton method for online convex optimization. AISTATS.
- A. Mokhtari and A. Ribeiro (2015). "Global convergence of online limited memory BFGS" (PDF). Journal of Machine Learning Research. 16: 3151–3181.
- A. Mokhtari and A. Ribeiro (2014). "RES: Regularized Stochastic BFGS Algorithm". IEEE Transactions on Signal Processing. 62 (23): 6089–6104. doi:10.1109/TSP.2014.2357775.
- 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.
- "General-purpose Optimization". R documentation. Comprehensive R Archive Network.
- Liu, D. C.; Nocedal, J. (1989). "On the Limited Memory Method for Large Scale Optimization". Mathematical Programming B. 45 (3): 503–528. doi:10.1007/BF01589116.
- Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge; Zhu, Ciyou (1995). "A Limited Memory Algorithm for Bound Constrained Optimization". SIAM Journal on Scientific and Statistical Computing. 16 (5): 1190–1208. doi:10.1137/0916069.
- Haghighi, Aria (2 Dec 2014). "Numerical Optimization: Understanding L-BFGS".