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 minimize 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 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 ). 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 .
L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication is carried out, where is the approximate Newton's direction, is the current gradient, 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 take as given , the position at the k-th iteration, and where 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
We 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 descent direction.
Thus we can compute the descent direction as follows:
This formulation is valid whether we are minimizing or maximizing. Note that if we are maximizing, the search direction would be the negative of z (since z is "downhill"), and should be negative definite rather than positive definite. 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.
The scaling of the initial matrix 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 will be satisfied by the chosen step since a step length greater than may be needed to satisfy this condition. Some implementations address this by skipping the BFGS update when 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 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 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.
Implementation of variants
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