Limited-memory BFGS

The algorithm starts with an initial estimate of the optimal value, ${\displaystyle \mathbf {x} _{0}}$, and proceeds iteratively to refine that estimate with a sequence of better estimates ${\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\ldots }$. The derivatives of the function ${\displaystyle g_{k}:=\nabla f(\mathbf {x} _{k})}$ 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 ${\displaystyle f(\mathbf {x} )}$.

L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication ${\displaystyle d_{k}=-H_{k}g_{k}}$ is carried out, where ${\displaystyle d_{k}}$ is the approximate Newton's direction, ${\displaystyle g_{k}}$ is the current gradient, and ${\displaystyle H_{k}}$ 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."[4][5]

We take as given ${\displaystyle x_{k}}$, the position at the k-th iteration, and ${\displaystyle g_{k}\equiv \nabla f(x_{k})}$ where ${\displaystyle f}$ 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

${\displaystyle s_{k}=x_{k+1}-x_{k}}$

${\displaystyle y_{k}=g_{k+1}-g_{k}}$.

We define ${\displaystyle \rho _{k}={\frac {1}{y_{k}^{\top }s_{k}}}}$, and ${\displaystyle H_{k}^{0}}$ 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

${\displaystyle H_{k+1}=(I-\rho _{k}s_{k}y_{k}^{\top })H_{k}(I-\rho _{k}y_{k}s_{k}^{\top })+\rho _{k}s_{k}s_{k}^{\top }.}$

For a fixed k we define a sequence of vectors ${\displaystyle q_{k-m},\ldots ,q_{k}}$ as ${\displaystyle q_{k}:=g_{k}}$ and ${\displaystyle q_{i}:=(I-\rho _{i}y_{i}s_{i}^{\top })q_{i+1}}$. Then a recursive algorithm for calculating ${\displaystyle q_{i}}$ from ${\displaystyle q_{i+1}}$ is to define ${\displaystyle \alpha _{i}:=\rho _{i}s_{i}^{\top }q_{i+1}}$ and ${\displaystyle q_{i}=q_{i+1}-\alpha _{i}y_{i}}$. We also define another sequence of vectors ${\displaystyle z_{k-m},\ldots ,z_{k}}$ as ${\displaystyle z_{i}:=H_{i}q_{i}}$. There is another recursive algorithm for calculating these vectors which is to define ${\displaystyle z_{k-m}=H_{k}^{0}q_{k-m}}$ and then recursively define ${\displaystyle \beta _{i}:=\rho _{i}y_{i}^{\top }z_{i}}$ and ${\displaystyle z_{i+1}=z_{i}+(\alpha _{i}-\beta _{i})s_{i}}$. The value of ${\displaystyle z_{k}}$ is then our ascent direction.

Thus we can compute the descent direction as follows:

${\displaystyle {\begin{array}{l}q=g_{k}\\{\mathtt {For}}\ i=k-1,k-2,\ldots ,k-m\\\qquad \alpha _{i}=\rho _{i}s_{i}^{\top }q\\\qquad q=q-\alpha _{i}y_{i}\\\gamma _{k}={\frac {s_{k-1}^{\top }y_{k-1}}{y_{k-1}^{\top }y_{k-1}}}\\H_{k}^{0}=\gamma _{k}I\\z=H_{k}^{0}q\\{\mathtt {For}}\ i=k-m,k-m+1,\ldots ,k-1\\\qquad \beta _{i}=\rho _{i}y_{i}^{\top }z\\\qquad z=z+s_{i}(\alpha _{i}-\beta _{i})\\z=-z\end{array}}}$

This formulation gives the search direction for the minimization problem, i.e., ${\displaystyle z=-H_{k}g_{k}}$. For maximization problems, one should thus take -z instead. Note that the initial approximate inverse Hessian ${\displaystyle H_{k}^{0}}$ is chosen as a diagonal matrix or even a multiple of the identity matrix since this is numerically efficient.

The scaling of the initial matrix ${\displaystyle \gamma _{k}}$ ensures that the search direction is well scaled and therefore the unit step length is accepted in most iterations. A Wolfe line search is used to ensure that the curvature condition is satisfied and the BFGS updating is stable. Note that some software implementations use an Armijo backtracking line search, but cannot guarantee that the curvature condition ${\displaystyle y_{k}^{\top }s_{k}>0}$ will be satisfied by the chosen step since a step length greater than ${\displaystyle 1}$ may be needed to satisfy this condition. Some implementations address this by skipping the BFGS update when ${\displaystyle y_{k}^{\top }s_{k}}$ is negative or too close to zero, but this approach is not generally recommended since the updates may be skipped too often to allow the Hessian approximation ${\displaystyle H_{k}}$ to capture important curvature information.

This two loop update only works for the inverse Hessian. Approaches to implementing L-BFGS using the direct approximate Hessian ${\displaystyle B_{k}}$ have also been developed, as have other means of approximating the inverse Hessian.[6]

L-BFGS has been called "the algorithm of choice" for fitting log-linear (MaxEnt) models and conditional random fields with ${\displaystyle \ell _{2}}$-regularization.[2][3]

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.

Notable open source implementations include:

• ALGLIB implements L-BFGS in C++ and C# as well as a separate box/linearly constrained version, BLEIC.
• R's optim general-purpose optimizer routine uses the L-BFGS-B method.
• SciPy's optimization module's minimize method also includes an option to use L-BFGS-B.

Notable non open source implementations include:

• The L-BFGS-B variant also exists as ACM TOMS algorithm 778.[8][12] In February 2011, some of the authors of the original L-BFGS-B code posted a major update (version 3.0).
• A reference implementation in Fortran 77 (and with a Fortran 90 interface).[13][14] This version, as well as older versions, has been converted to many other languages.
• An OWL-QN C++ implementation by its designers.[3][15]

• Liu, D. C.; Nocedal, J. (1989). "On the Limited Memory Method for Large Scale Optimization". Mathematical Programming B. 45 (3): 503–528. CiteSeerX 10.1.1.110.6443. doi:10.1007/BF01589116. S2CID 5681609.
• Haghighi, Aria (2 Dec 2014). "Numerical Optimization: Understanding L-BFGS".
• Pytlak, Radoslaw (2009). "Limited Memory Quasi-Newton Algorithms". Conjugate Gradient Algorithms in Nonconvex Optimization. Springer. pp. 159–190. ISBN 978-3-540-85633-7.