BFGS O-BFGS Shouldn t Be Essentially Convergent

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Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the gathering of quasi-Newton methods that approximates the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS) using a restricted quantity of pc memory. It is a popular algorithm for parameter estimation in machine studying. Hessian (n being the variety of variables in the issue), L-BFGS shops just a few vectors that symbolize the approximation implicitly. As a result of its resulting linear memory requirement, the L-BFGS technique is especially effectively suited for optimization problems with many variables. The 2-loop recursion formulation is broadly used by unconstrained optimizers as a consequence of its effectivity in multiplying by the inverse Hessian. However, it does not allow for the specific formation of both the direct or Memory Wave inverse Hessian and is incompatible with non-box constraints. Another method 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 use of L-BFGS in constrained settings, for instance, as a part of the SQP method.



Since BFGS (and hence L-BFGS) is designed to minimize clean features with out constraints, the L-BFGS algorithm have to be modified to handle functions that embody non-differentiable parts or constraints. A popular class of modifications are known as energetic-set methods, based on the concept of the lively set. The thought 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 easy field constraints (aka bound constraints) on variables; that's, constraints of the type li ≤ xi ≤ ui where li and ui are per-variable fixed lower and upper bounds, respectively (for each xi, both or both bounds could also be omitted). The tactic works by figuring out fixed and free variables at every step (using a simple gradient methodology), and then utilizing the L-BFGS method on the free variables solely to get increased accuracy, and then repeating the method. The strategy is an active-set type methodology: at each iterate, it estimates the sign of every part of the variable, and restricts the next step to have the same signal.



L-BFGS. After an L-BFGS step, the tactic permits some variables to change sign, and repeats the method. Schraudolph et al. present an internet approximation to each BFGS and L-BFGS. Just like stochastic gradient descent, this can be used to scale back the computational complexity by evaluating the error perform and gradient on a randomly drawn subset of the overall dataset in every iteration. BFGS (O-BFGS) will not be necessarily convergent. R's optim general-goal optimizer routine makes use of the L-BFGS-B method. SciPy's optimization module's decrease method also includes an option to make use of L-BFGS-B. A reference implementation in Fortran 77 (and with a Fortran 90 interface). This version, in addition to older versions, has been transformed to many different languages. Liu, D. C.; Nocedal, J. (1989). "On the Restricted Memory Technique for large Scale Optimization". Malouf, Robert (2002). "A comparability of algorithms for maximum entropy parameter estimation". Proceedings of the Sixth Convention on Pure Language Studying (CoNLL-2002).



Andrew, Galen; Gao, Jianfeng (2007). "Scalable coaching of L₁-regularized log-linear models". Proceedings of the 24th Worldwide Conference on Machine Learning. Matthies, H.; Strang, G. (1979). "The solution 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 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 Wave Routine Algorithm for Bound 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 online convex optimization. Mokhtari, A.; Ribeiro, A. (2015). "International convergence of online restricted memory BFGS" (PDF). Journal of Machine Learning Analysis. Mokhtari, A.; Ribeiro, A. (2014). "RES: Regularized Stochastic BFGS Algorithm". IEEE Transactions on Sign Processing. 62 (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 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.

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