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Restricted-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the collection of quasi-Newton strategies that approximates the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS) utilizing a restricted quantity of laptop memory. It is a popular algorithm for parameter estimation in machine studying. Hessian (n being the variety of variables in the problem), L-BFGS shops only some vectors that symbolize the approximation implicitly. Attributable to its ensuing linear memory requirement, the L-BFGS methodology is particularly effectively suited to optimization issues with many variables. The 2-loop recursion components is extensively used by unconstrained optimizers as a consequence of its effectivity in multiplying by the inverse Hessian. However, it doesn't permit for the express formation of both the direct or inverse Hessian and is incompatible with non-box constraints. An alternate approach 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 illustration enables the use of L-BFGS in constrained settings, for example, as part of the SQP methodology.

Since BFGS (and hence L-BFGS) is designed to reduce clean capabilities without constraints, the L-BFGS algorithm must be modified to handle functions that include non-differentiable parts or constraints. A well-liked class of modifications are known as energetic-set strategies, primarily based on the idea of the energetic set. The thought is that when restricted to a small neighborhood of the present iterate, the operate and constraints might be simplified. The L-BFGS-B algorithm extends L-BFGS to handle simple box constraints (aka certain constraints) on variables; that's, constraints of the form li ≤ xi ≤ ui the place li and ui are per-variable constant lower and higher bounds, respectively (for every xi, both or both bounds may be omitted). The tactic works by identifying fixed and free variables at every step (utilizing a easy gradient methodology), after which using the L-BFGS method on the free variables solely to get higher accuracy, after which repeating the process. The strategy is an lively-set sort technique: at every iterate, it estimates the signal of every element of the variable, and restricts the following step to have the identical sign.

L-BFGS. After an L-BFGS step, MemoryWave Guide the method allows some variables to vary signal, and repeats the process. Schraudolph et al. present a web based approximation to each BFGS and L-BFGS. Just like stochastic gradient descent, this can be utilized to scale back the computational complexity by evaluating the error function and gradient on a randomly drawn subset of the general dataset in each iteration. BFGS (O-BFGS) isn't necessarily convergent. R's optim common-objective optimizer routine uses the L-BFGS-B methodology. SciPy's optimization module's decrease method also includes an choice to use L-BFGS-B. A reference implementation in Fortran 77 (and with a Fortran 90 interface). This model, as well as older versions, has been transformed to many different languages. Liu, Memory Wave D. C.; Nocedal, J. (1989). "On the Restricted Memory Technique for giant Scale Optimization". Malouf, MemoryWave Guide Robert (2002). "A comparison of algorithms for maximum entropy parameter estimation". Proceedings of the Sixth Convention on Natural Language Learning (CoNLL-2002).

Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear fashions". Proceedings of the twenty fourth Worldwide Convention on Machine Learning. Matthies, H.; Strang, G. (1979). "The answer of non linear finite element equations". International Journal for Numerical Methods in Engineering. 14 (11): 1613-1626. Bibcode:1979IJNME..14.1613M. Nocedal, J. (1980). "Updating Quasi-Newton Matrices with Limited Storage". Byrd, R. H.; Nocedal, J.; Schnabel, R. B. (1994). "Representations of Quasi-Newton Matrices and their use in Restricted Memory Strategies". Mathematical Programming. 63 (4): 129-156. doi:10.1007/BF01582063. Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. (1995). "A Restricted Memory Algorithm for Certain 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 giant scale bound constrained optimization". ACM Transactions on Mathematical Software. Schraudolph, N.; Yu, J.; Günter, S. (2007). A stochastic quasi-Newton methodology for on-line convex optimization. Mokhtari, Memory Wave A.; Ribeiro, A. (2015). "International convergence of online limited memory BFGS" (PDF). Journal of Machine Learning Research. 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, J. (2011). "Remark on "algorithm 778: L-BFGS-B: Fortran subroutines for giant-scale sure constrained optimization"". ACM Transactions on Mathematical Software program. Liu, D. C.; Nocedal, J. (1989). "On the Limited Memory Methodology for big Scale Optimization". Haghighi, Aria (2 Dec 2014). "Numerical Optimization: Understanding L-BFGS". Pytlak, Radoslaw (2009). "Restricted Memory Quasi-Newton Algorithms". Conjugate Gradient Algorithms in Nonconvex Optimization.