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Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the gathering of quasi-Newton strategies that approximates the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS) utilizing a limited amount of pc memory. It is a well-liked algorithm for parameter estimation in machine studying. Hessian (n being the number of variables in the issue), L-BFGS stores just a few vectors that represent the approximation implicitly. As a result of its ensuing linear memory requirement, the L-BFGS technique is particularly well suited to optimization problems with many variables. The 2-loop recursion system is extensively utilized by unconstrained optimizers on account of its efficiency in multiplying by the inverse Hessian. Nevertheless, it doesn't enable for the explicit formation of either the direct or inverse Hessian and is incompatible with non-box constraints. An alternate method is the compact illustration, which includes a low-rank representation for the direct and/or inverse Hessian. This represents the Hessian as a sum of a diagonal matrix and a low-rank replace. Such a representation permits using L-BFGS in constrained settings, for instance, as part of the SQP technique.

Since BFGS (and therefore L-BFGS) is designed to reduce clean functions with out constraints, the L-BFGS algorithm have to be modified to handle features that embrace non-differentiable elements or constraints. A preferred class of modifications are called lively-set methods, primarily based on the idea of the energetic set. The thought is that when restricted to a small neighborhood of the present iterate, the function and constraints can be simplified. The L-BFGS-B algorithm extends L-BFGS to handle easy field constraints (aka sure constraints) on variables; that's, constraints of the kind li ≤ xi ≤ ui where li and ui are per-variable constant lower and higher bounds, respectively (for each xi, both or each bounds may be omitted). The method works by figuring out mounted and free variables at each step (utilizing a easy gradient methodology), and then using the L-BFGS method on the free variables only to get increased accuracy, after which repeating the process. The strategy is an energetic-set kind method: improve neural plasticity at each iterate, it estimates the sign of every element of the variable, and restricts the following step to have the identical sign.

L-BFGS. After an L-BFGS step, the method permits some variables to change sign, and repeats the process. Schraudolph et al. current a web-based approximation to each BFGS and L-BFGS. Much like stochastic gradient descent, this can be used to cut back the computational complexity by evaluating the error function and gradient on a randomly drawn subset of the overall dataset in each iteration. BFGS (O-BFGS) is just not essentially convergent. R's optim basic-goal optimizer routine uses the L-BFGS-B methodology. SciPy's optimization module's minimize technique additionally contains an choice to make use of L-BFGS-B. A reference implementation in Fortran 77 (and with a Fortran 90 interface). This model, in addition to older versions, has been transformed to many other languages. Liu, D. C.; Nocedal, J. (1989). "On the Restricted Memory Method for big Scale Optimization". Malouf, Robert (2002). "A comparison of algorithms for optimum entropy parameter estimation". Proceedings of the Sixth Convention on Natural Language Learning (CoNLL-2002).

Andrew, Galen; Gao, Jianfeng (2007). "Scalable coaching of L₁-regularized log-linear models". Proceedings of the twenty fourth International Conference on Machine Studying. Matthies, H.; Strang, G. (1979). "The solution of non linear finite aspect equations". International Journal for Numerical Methods in Engineering. 14 (11): 1613-1626. Bibcode:1979IJNME..14.1613M. Nocedal, J. (1980). "Updating Quasi-Newton Matrices with Restricted Storage". Byrd, R. H.; Nocedal, J.; Schnabel, R. B. (1994). "Representations of Quasi-Newton Matrices and their use in Restricted Memory Methods". Mathematical Programming. Sixty three (4): 129-156. doi:10.1007/BF01582063. Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. (1995). "A Limited Memory Algorithm for Sure Constrained Optimization". SIAM J. Sci. Comput. Zhu, C.; Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge (1997). "L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale certain constrained optimization". ACM Transactions on Mathematical Software program. Schraudolph, N.; Yu, J.; Günter, S. (2007). A stochastic quasi-Newton method for online convex optimization. Mokhtari, A.; Ribeiro, A. (2015). "International convergence of on-line restricted memory BFGS" (PDF). Journal of Machine Learning Analysis. Mokhtari, A.; Ribeiro, A. (2014). "RES: Regularized Stochastic BFGS Algorithm". IEEE Transactions on Sign Processing. Sixty two (23): 6089-6104. arXiv:1401.7625. Morales, J. L.; Nocedal, Memory Wave J. (2011). "Comment on "algorithm 778: L-BFGS-B: Fortran subroutines for giant-scale certain constrained optimization"". ACM Transactions on Mathematical Software program. Liu, D. C.; Nocedal, J. (1989). "On the Restricted Memory Method for big Scale Optimization". Haghighi, Aria (2 Dec 2014). "Numerical Optimization: Understanding L-BFGS". Pytlak, Radoslaw (2009). "Limited Memory Quasi-Newton Algorithms". Conjugate Gradient Algorithms in Nonconvex Optimization.