BFGS O-BFGS Just Isn t Necessarily Convergent

Limited-Memory Wave Routine 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) using a limited amount of pc memory. It is a well-liked algorithm for parameter estimation in machine studying. Hessian (n being the variety of variables in the problem), L-BFGS stores just a few vectors that symbolize the approximation implicitly. As a consequence of its ensuing linear memory requirement, the L-BFGS method is especially effectively fitted to optimization problems with many variables. The two-loop recursion formula is widely utilized by unconstrained optimizers resulting from its effectivity in multiplying by the inverse Hessian. Nevertheless, Memory Wave Routine it does not permit for the explicit formation of either the direct or inverse Hessian and is incompatible with non-box constraints. An alternate strategy is the compact representation, which includes a low-rank illustration for the direct and/or inverse Hessian. This represents the Hessian as a sum of a diagonal matrix and a low-rank update. Such a representation enables the usage of L-BFGS in constrained settings, for example, as part of the SQP technique.



Since BFGS (and therefore L-BFGS) is designed to attenuate easy functions without constraints, the L-BFGS algorithm must be modified to handle features that embrace non-differentiable elements or constraints. A well-liked class of modifications are called lively-set strategies, primarily based on the idea of the energetic set. The concept is that when restricted to a small neighborhood of the current iterate, the function and constraints will be simplified. The L-BFGS-B algorithm extends L-BFGS to handle simple field constraints (aka certain constraints) on variables; that's, constraints of the kind li ≤ xi ≤ ui the place li and ui are per-variable constant lower and upper bounds, respectively (for each xi, both or Memory Wave both bounds could also be omitted). The tactic works by identifying fixed and free variables at each step (using a simple gradient method), and then utilizing the L-BFGS technique on the free variables only to get increased accuracy, and then repeating the process. The strategy is an energetic-set kind method: at every iterate, it estimates the sign of each element of the variable, and restricts the following step to have the same sign.



L-BFGS. After an L-BFGS step, the strategy permits some variables to vary sign, and repeats the process. Schraudolph et al. current an online approximation to each BFGS and L-BFGS. Similar to 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) is not necessarily convergent. R's optim common-function optimizer routine makes use of the L-BFGS-B methodology. SciPy's optimization module's decrease technique also consists of an possibility to use L-BFGS-B. A reference implementation in Fortran 77 (and with a Fortran ninety interface). This model, as well as older versions, has been transformed to many other languages. Liu, D. C.; Nocedal, J. (1989). "On the Restricted Memory Methodology for giant Scale Optimization". Malouf, Robert (2002). "A comparison of algorithms for maximum entropy parameter estimation". Proceedings of the Sixth Conference on Pure Language Learning (CoNLL-2002).



Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear models". Proceedings of the twenty fourth Worldwide Conference on Machine Studying. Matthies, H.; Strang, G. (1979). "The answer of non linear finite factor equations". Worldwide 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 Limited Memory Methods". Mathematical Programming. Sixty three (4): 129-156. doi:10.1007/BF01582063. Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. (1995). "A Restricted 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 big scale bound constrained optimization". ACM Transactions on Mathematical Software program. Schraudolph, N.; Yu, J.; Günter, S. (2007). A stochastic quasi-Newton technique for on-line convex optimization. Mokhtari, A.; Ribeiro, A. (2015). "Global 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 Signal 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 bound constrained optimization"". ACM Transactions on Mathematical Software program. Liu, D. C.; Nocedal, J. (1989). "On the Limited Memory Method for large 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.