Blast wave

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.

Like the original BFGS, L-BFGS uses an approximation to the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense n×n approximation to the 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 H_k, 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 m<10). These updates are used to implicitly do operations requiring the H_k-vector product.

Algorithm

L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication for finding the search direction is carried out $d_{k}=-H_{k}g_{k}\,\!$. There are multiple published approaches to using a history of updates to form this direction vector. Here, we give a common approach, the so-called "two loop recursion."

We'll take as given $x_{k}\,\!$, the position at the $k\,\!$-th iteration, and $g_{k}\equiv \nabla f(x_{k})$ where $f\,\!$ is the function being minimized, and all vectors are column vectors. Then we keep the updates $s_{k}=x_{k+1}-x_{k}\,\!$ and $y_{k}=g_{k+1}-g_{k}\,\!$. We'll define $\rho _{k}={\frac {1}{y_{k}^{\rm {T}}s_{k}}}$, and $H_{k}^{0}\,\!$ will be the 'initial' approximate of the inverse Hessian that our estimate at iteration $k\,\!$ begins with. Then we can compute the (uphill) direction as follows:

$q=g_{k}\,\!$

For $i=k-1,k-2,\ldots ,k-m$

$\alpha _{i}=\rho _{i}s_{i}^{\rm {T}}q\,\!$

$q=q-\alpha _{i}y_{i}\,\!$

$H_{k}=y_{k}^{\rm {T}}s_{k}/y_{k}^{\rm {T}}y_{k}$

$z=H_{k}q$

For $i=k-m,k-m+1,\ldots ,k-1$

$\beta _{i}=\rho _{i}y_{i}^{\rm {T}}z\,\!$

$z=z+s_{i}(\alpha _{i}-\beta _{i})\,\!$

Stop with $H_{k}g_{k}=z\,\!$

This formulation is valid whether we are minimizing or maximizing. Note that if we are minimizing, the search direction would be the negative of z (since z is "uphill"), and if we are maximizing, $H_{k}^{0}$ should be negative definite rather than positive definite. We would typically do a backtracking line search in the search direction (any line search would be valid, but L-BFGS does not require exact line searches in order to converge).

Commonly, the inverse Hessian $H_{k}^{0}\,\!$ is represented as a diagonal matrix, so that initially setting $z\,\!$ requires only an element-by-element multiplication.

This two loop update only works for the inverse Hessian. Approaches to implementing L-BFGS using the direct approximate Hessian $B_{k}\,\!$ have also been developed, as have other means of approximating the inverse Hessian.

Applications

L-BFGS has been called "the algorithm of choice" for fitting log-linear (MaxEnt) models and conditional random fields with $\ell _{2}$-regularization.

Variants

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.

L-BFGS-B

The L-BFGS-B algorithm extends L-BFGS to handle simple box constraints (aka bound constraints) on variables; that is, constraints of the form $l_{i}\leq x_{i}\leq u_{i}$ where l_i and u_i are per-variable constant lower and upper bounds, respectively (for each x_i, 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.

OWL-QN

Orthant-wise limited-memory quasi-Newton (OWL-QN) is an L-BFGS variant for fitting $\ell _{1}$-regularized models, exploiting the inherent sparsity of such models. It minimizes functions of the form

$f({\vec {x}})=g({\vec {x}})+C\|{\vec {x}}\|_{1}$

where $g$ 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 $\|{\vec {x}}\|_{1}$ term becomes a smooth linear term which can be handled by L-BFGS. After a L-BFGS step, the method allows some variables to change sign, and repeats the process.

O-LBFGS

Schraudolf et al. present an online approximation to both BFGS and L-BFGS.

Implementations

An early, open source implementation of L-BFGS in Fortran exists in Netlib as a shar archive [1]. Multiple other open source implementations have been produced as translations of this Fortran code (e.g. java, and python via SciPy). Other implementations exist (e.g. Matlab (optimization toolbox), Matlab (BSD)), frequently as part of generic optimization libraries (e.g. Mathematica, FuncLib C# library, and dlib C++ library). The libLBFGS is a C implementation.

Implementations of variants

The L-BFGS-B variant also exists as ACM TOMS algorithm 778. In February 2011, some of the authors of the original L-BFGS-B code posted a major update (version 3.0).

A reference implementation is available in Fortran 77 (and with a Fortran 90 interface) at the author's website. This version, as well as older versions, has been converted to many other languages, including a Java wrapper for v3.0; Matlab interfaces for v3.0, v2.4, and v2.1; a C++ interface for v2.1; a Python interface for v3.0 as part of scipy.optimize.minimize; an OCaml interface for v2.1 and v3.0; version 2.3 has been converted to C by f2c and is available at this website; and R's optim general-purpose optimizer routine includes L-BFGS-B by using method="L-BFGS-B".

OWL-QN implementations are available in:

• C++ implementation by its designers, includes the original ICML paper on the algorithm
• Python implementation by Michael Subotin, intended for use with SciPy
• The CRF toolkit Wapiti includes a C implementation

Works cited

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Further reading

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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang