Limited-memory BFGS

<h2 id="algorithm">Algorithm</h2>
The algorithm starts with an initial estimate of the optimal value, 
 
 
 
 
 
 x
 
 
 0
 
 
 
 
 {\displaystyle \mathbf {x} _{0}}
 
, and proceeds iteratively to refine that estimate with a sequence of better estimates 
 
 
 
 
 
 x
 
 
 1
 
 
 ,
 
 
 x
 
 
 2
 
 
 ,
 …
 
 
 {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\ldots }
 
. The derivatives of the function 
 
 
 
 
 g
 
 k
 
 
 :=
 ∇
 f
 (
 
 
 x
 
 
 k
 
 
 )
 
 
 {\displaystyle g_{k}:=\nabla f(\mathbf {x} _{k})}
 
 are used as a key driver of the algorithm to identify the direction of steepest descent, and also to form an estimate of the Hessian matrix (second derivative) of 
 
 
 
 f
 (
 
 x
 
 )
 
 
 {\displaystyle f(\mathbf {x} )}
 
.
L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication 
 
 
 
 
 d
 
 k
 
 
 =
 −
 
 H
 
 k
 
 
 
 g
 
 k
 
 
 
 
 {\displaystyle d_{k}=-H_{k}g_{k}}
 
 is carried out, where 
 
 
 
 
 d
 
 k
 
 
 
 
 {\displaystyle d_{k}}
 
 is the approximate Newton's direction, 
 
 
 
 
 g
 
 k
 
 
 
 
 {\displaystyle g_{k}}
 
 is the current gradient, and 
 
 
 
 
 H
 
 k
 
 
 
 
 {\displaystyle H_{k}}
 
 is the inverse of the Hessian matrix. There are multiple published approaches using a history of updates to form this direction vector. Here, we give a common approach, the so-called "two loop recursion."<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
We take as given 
 
 
 
 
 x
 
 k
 
 
 
 
 {\displaystyle x_{k}}
 
, the position at the k-th iteration, and 
 
 
 
 
 g
 
 k
 
 
 ≡
 ∇
 f
 (
 
 x
 
 k
 
 
 )
 
 
 {\displaystyle g_{k}\equiv \nabla f(x_{k})}
 
 where 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is the function being minimized, and all vectors are column vectors. We also assume that we have stored the last m updates of the form

s
          
            k
          
        
        =
        
          x
          
            k
            +
            1
          
        
        −
        
          x
          
            k
          
        
      
    
    {\displaystyle s_{k}=x_{k+1}-x_{k}}

y
 
 k
 
 
 =
 
 g
 
 k
 +
 1
 
 
 −
 
 g
 
 k
 
 
 
 
 {\displaystyle y_{k}=g_{k+1}-g_{k}}
 
.
We define 
 
 
 
 
 ρ
 
 k
 
 
 =
 
 
 1
 
 
 y
 
 k
 
 
 ⊤
 
 
 
 s
 
 k
 
 
 
 
 
 
 
 {\displaystyle \rho _{k}={\frac {1}{y_{k}^{\top }s_{k}}}}
 
, and 
 
 
 
 
 H
 
 k
 
 
 0
 
 
 
 
 {\displaystyle H_{k}^{0}}
 
 will be the 'initial' approximate of the inverse Hessian that our estimate at iteration k begins with.
The algorithm is based on the BFGS recursion for the inverse Hessian as

H
          
            k
            +
            1
          
        
        =
        (
        I
        −
        
          ρ
          
            k
          
        
        
          s
          
            k
          
        
        
          y
          
            k
          
          
            ⊤
          
        
        )
        
          H
          
            k
          
        
        (
        I
        −
        
          ρ
          
            k
          
        
        
          y
          
            k
          
        
        
          s
          
            k
          
          
            ⊤
          
        
        )
        +
        
          ρ
          
            k
          
        
        
          s
          
            k
          
        
        
          s
          
            k
          
          
            ⊤
          
        
        .
      
    
    {\displaystyle H_{k+1}=(I-\rho _{k}s_{k}y_{k}^{\top })H_{k}(I-\rho _{k}y_{k}s_{k}^{\top })+\rho _{k}s_{k}s_{k}^{\top }.}

For a fixed k we define a sequence of vectors 
 
 
 
 
 q
 
 k
 −
 m
 
 
 ,
 …
 ,
 
 q
 
 k
 
 
 
 
 {\displaystyle q_{k-m},\ldots ,q_{k}}
 
 as 
 
 
 
 
 q
 
 k
 
 
 :=
 
 g
 
 k
 
 
 
 
 {\displaystyle q_{k}:=g_{k}}
 
 and 
 
 
 
 
 q
 
 i
 
 
 :=
 (
 I
 −
 
 ρ
 
 i
 
 
 
 y
 
 i
 
 
 
 s
 
 i
 
 
 ⊤
 
 
 )
 
 q
 
 i
 +
 1
 
 
 
 
 {\displaystyle q_{i}:=(I-\rho _{i}y_{i}s_{i}^{\top })q_{i+1}}
 
. Then a recursive algorithm for calculating 
 
 
 
 
 q
 
 i
 
 
 
 
 {\displaystyle q_{i}}
 
 from 
 
 
 
 
 q
 
 i
 +
 1
 
 
 
 
 {\displaystyle q_{i+1}}
 
 is to define 
 
 
 
 
 α
 
 i
 
 
 :=
 
 ρ
 
 i
 
 
 
 s
 
 i
 
 
 ⊤
 
 
 
 q
 
 i
 +
 1
 
 
 
 
 {\displaystyle \alpha _{i}:=\rho _{i}s_{i}^{\top }q_{i+1}}
 
 and 
 
 
 
 
 q
 
 i
 
 
 =
 
 q
 
 i
 +
 1
 
 
 −
 
 α
 
 i
 
 
 
 y
 
 i
 
 
 
 
 {\displaystyle q_{i}=q_{i+1}-\alpha _{i}y_{i}}
 
. We also define another sequence of vectors 
 
 
 
 
 z
 
 k
 −
 m
 
 
 ,
 …
 ,
 
 z
 
 k
 
 
 
 
 {\displaystyle z_{k-m},\ldots ,z_{k}}
 
 as 
 
 
 
 
 z
 
 i
 
 
 :=
 
 H
 
 i
 
 
 
 q
 
 i
 
 
 
 
 {\displaystyle z_{i}:=H_{i}q_{i}}
 
. There is another recursive algorithm for calculating these vectors which is to define 
 
 
 
 
 z
 
 k
 −
 m
 
 
 =
 
 H
 
 k
 
 
 0
 
 
 
 q
 
 k
 −
 m
 
 
 
 
 {\displaystyle z_{k-m}=H_{k}^{0}q_{k-m}}
 
 and then recursively define 
 
 
 
 
 β
 
 i
 
 
 :=
 
 ρ
 
 i
 
 
 
 y
 
 i
 
 
 ⊤
 
 
 
 z
 
 i
 
 
 
 
 {\displaystyle \beta _{i}:=\rho _{i}y_{i}^{\top }z_{i}}
 
 and 
 
 
 
 
 z
 
 i
 +
 1
 
 
 =
 
 z
 
 i
 
 
 +
 (
 
 α
 
 i
 
 
 −
 
 β
 
 i
 
 
 )
 
 s
 
 i
 
 
 
 
 {\displaystyle z_{i+1}=z_{i}+(\alpha _{i}-\beta _{i})s_{i}}
 
. The value of 
 
 
 
 
 z
 
 k
 
 
 
 
 {\displaystyle z_{k}}
 
 is then our ascent direction.
Thus we can compute the descent direction as follows:

q
                =
                
                  g
                  
                    k
                  
                
              
            
            
              
                
                  
                    F
                    o
                    r
                  
                
                 
                i
                =
                k
                −
                1
                ,
                k
                −
                2
                ,
                …
                ,
                k
                −
                m
              
            
            
              
                
                
                  α
                  
                    i
                  
                
                =
                
                  ρ
                  
                    i
                  
                
                
                  s
                  
                    i
                  
                  
                    ⊤
                  
                
                q
              
            
            
              
                
                q
                =
                q
                −
                
                  α
                  
                    i
                  
                
                
                  y
                  
                    i
                  
                
              
            
            
              
                
                  γ
                  
                    k
                  
                
                =
                
                  
                    
                      
                        s
                        
                          k
                          −
                          m
                        
                        
                          ⊤
                        
                      
                      
                        y
                        
                          k
                          −
                          m
                        
                      
                    
                    
                      
                        y
                        
                          k
                          −
                          m
                        
                        
                          ⊤
                        
                      
                      
                        y
                        
                          k
                          −
                          m
                        
                      
                    
                  
                
              
            
            
              
                
                  H
                  
                    k
                  
                  
                    0
                  
                
                =
                
                  γ
                  
                    k
                  
                
                I
              
            
            
              
                z
                =
                
                  H
                  
                    k
                  
                  
                    0
                  
                
                q
              
            
            
              
                
                  
                    F
                    o
                    r
                  
                
                 
                i
                =
                k
                −
                m
                ,
                k
                −
                m
                +
                1
                ,
                …
                ,
                k
                −
                1
              
            
            
              
                
                
                  β
                  
                    i
                  
                
                =
                
                  ρ
                  
                    i
                  
                
                
                  y
                  
                    i
                  
                  
                    ⊤
                  
                
                z
              
            
            
              
                
                z
                =
                z
                +
                
                  s
                  
                    i
                  
                
                (
                
                  α
                  
                    i
                  
                
                −
                
                  β
                  
                    i
                  
                
                )
              
            
            
              
                z
                =
                −
                z
              
            
          
        
      
    
    {\displaystyle {\begin{array}{l}q=g_{k}\\{\mathtt {For}}\ i=k-1,k-2,\ldots ,k-m\\\qquad \alpha _{i}=\rho _{i}s_{i}^{\top }q\\\qquad q=q-\alpha _{i}y_{i}\\\gamma _{k}={\frac {s_{k-m}^{\top }y_{k-m}}{y_{k-m}^{\top }y_{k-m}}}\\H_{k}^{0}=\gamma _{k}I\\z=H_{k}^{0}q\\{\mathtt {For}}\ i=k-m,k-m+1,\ldots ,k-1\\\qquad \beta _{i}=\rho _{i}y_{i}^{\top }z\\\qquad z=z+s_{i}(\alpha _{i}-\beta _{i})\\z=-z\end{array}}}

This formulation gives the search direction for the minimization problem, i.e., 
 
 
 
 z
 =
 −
 
 H
 
 k
 
 
 
 g
 
 k
 
 
 
 
 {\displaystyle z=-H_{k}g_{k}}
 
. For maximization problems, one should thus take -z instead. Note that the initial approximate inverse Hessian 
 
 
 
 
 H
 
 k
 
 
 0
 
 
 
 
 {\displaystyle H_{k}^{0}}
 
 is chosen as a diagonal matrix or even a multiple of the identity matrix since this is numerically efficient.
The scaling of the initial matrix 
 
 
 
 
 γ
 
 k
 
 
 
 
 {\displaystyle \gamma _{k}}
 
 ensures that the search direction is well scaled and therefore the unit step length is accepted in most iterations. A <a href="/facts/Wolfe_conditions/5C8cDvC8">Wolfe line search</a> is used to ensure that the curvature condition is satisfied and the BFGS updating is stable. Note that some software implementations use an Armijo <a href="/facts/Backtracking_line_search/cw1BQdIE">backtracking line search</a>, but cannot guarantee that the curvature condition 
 
 
 
 
 y
 
 k
 
 
 ⊤
 
 
 
 s
 
 k
 
 
 >
 0
 
 
 {\displaystyle y_{k}^{\top }s_{k}>0}
 
 will be satisfied by the chosen step since a step length greater than 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 may be needed to satisfy this condition. Some implementations address this by skipping the BFGS update when 
 
 
 
 
 y
 
 k
 
 
 ⊤
 
 
 
 s
 
 k
 
 
 
 
 {\displaystyle y_{k}^{\top }s_{k}}
 
 is negative or too close to zero, but this approach is not generally recommended since the updates may be skipped too often to allow the Hessian approximation 
 
 
 
 
 H
 
 k
 
 
 
 
 {\displaystyle H_{k}}
 
 to capture important curvature information. Some solvers employ so called damped (L)BFGS update which modifies quantities 
 
 
 
 
 s
 
 k
 
 
 
 
 {\displaystyle s_{k}}
 
 and 
 
 
 
 
 y
 
 k
 
 
 
 
 {\displaystyle y_{k}}
 
 in order to satisfy the curvature condition.
The two-loop recursion formula is widely used by unconstrained optimizers due to its efficiency in multiplying by the inverse Hessian. However, it does not allow for the explicit formation of either the direct or inverse Hessian and is incompatible with non-box constraints. An alternative approach is the <a href="/facts/Compact_quasi-Newton_representation/N8ChSQmf">compact representation</a>, which involves a low-rank representation for the direct and/or inverse Hessian.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 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 example, as part of the SQP method.

<h2 id="applications">Applications</h2>
L-BFGS has been called "the algorithm of choice" for fitting <a href="/facts/Multinomial_logistic_regression/aeF5lqDE">log-linear (MaxEnt) models</a> and <a href="/facts/Conditional_random_field/TNuLaGzM">conditional random fields</a> with <a href="/facts/Regularization_(mathematics)/K601A3y2">
 
 
 
 
 ℓ
 
 2
 
 
 
 
 {\displaystyle \ell _{2}}
 
-regularization</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="variants">Variants</h2>
Since BFGS (and hence L-BFGS) is designed to minimize <a href="/facts/Smooth_function/mffL4ch2">smooth</a> functions without <a href="/facts/Constraint_(mathematics)/QLFIFlFX">constraints</a>, the L-BFGS algorithm must be modified to handle functions that include non-<a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> components or constraints. A popular class of modifications are called active-set methods, based on the concept of the <a href="/facts/Active_set/WLtXKtq9">active set</a>. The idea is that when restricted to a small neighborhood of the current iterate, the function and constraints can be simplified.

<h3>L-BFGS-B</h3>
The L-BFGS-B algorithm extends L-BFGS to handle simple box constraints (aka bound constraints) on variables; that is, constraints of the form li ≤ xi ≤ ui where li and ui are per-variable constant lower and upper bounds, respectively (for each xi, either or both bounds may be omitted).<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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.

<h3>OWL-QN</h3>
Orthant-wise limited-memory quasi-Newton (OWL-QN) is an L-BFGS variant for fitting <a href="/facts/Taxicab_geometry/zU7gGUPv">
 
 
 
 
 ℓ
 
 1
 
 
 
 
 {\displaystyle \ell _{1}}
 
</a>-<a href="/facts/Regularization_(mathematics)/K601A3y2">regularized</a> models, exploiting the inherent <a href="/facts/Sparse_matrix/FFtN0XjR">sparsity</a> of such models.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
It minimizes functions of the form

f
        (
        
          
            
              x
              →
            
          
        
        )
        =
        g
        (
        
          
            
              x
              →
            
          
        
        )
        +
        C
        ‖
        
          
            
              x
              →
            
          
        
        
          ‖
          
            1
          
        
      
    
    {\displaystyle f({\vec {x}})=g({\vec {x}})+C\|{\vec {x}}\|_{1}}

where 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is a <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> <a href="/facts/Convex_function/IbzG5SLF">convex</a> <a href="/facts/Loss_function/xv5ozuhl">loss function</a>. The method is an active-set type method: at each iterate, it estimates the <a href="/facts/Sign_(mathematics)/z0xollg1">sign</a> of each component of the variable, and restricts the subsequent step to have the same sign. Once the sign is fixed, the non-differentiable 
 
 
 
 ‖
 
 
 
 x
 →
 
 
 
 
 ‖
 
 1
 
 
 
 
 {\displaystyle \|{\vec {x}}\|_{1}}
 
 term becomes a smooth linear term which can be handled by L-BFGS. After an L-BFGS step, the method allows some variables to change sign, and repeats the process.

<h3>O-LBFGS</h3>
Schraudolph et al. present an <a href="/facts/Online_machine_learning/4BoJlDfg">online</a> approximation to both BFGS and L-BFGS.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> Similar to <a href="/facts/Stochastic_gradient_descent/HbcaYqQP">stochastic gradient descent</a>, this can be used to reduce the computational complexity by evaluating the error function and gradient on a randomly drawn subset of the overall dataset in each iteration. It has been shown that O-LBFGS has a global almost sure convergence <a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> while the online approximation of BFGS (O-BFGS) is not necessarily convergent.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h2 id="implementation-of-variants">Implementation of variants</h2>
Notable open source implementations include:

<ul><li><a href="/facts/ALGLIB/fSduLPd2">ALGLIB</a> implements L-BFGS in C++ and C# as well as a separate box/linearly constrained version, BLEIC.</li>
<li><a href="/facts/R_(programming_language)/LSrkr8K8">R</a>'s optim general-purpose optimizer routine uses the L-BFGS-B method.</li>
<li><a href="/facts/SciPy/bNcMQGc0">SciPy</a>'s optimization module's minimize method also includes an option to use L-BFGS-B.</li></ul>
Notable non open source implementations include:

<ul><li>The L-BFGS-B variant also exists as ACM TOMS algorithm 778.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> In February 2011, some of the authors of the original L-BFGS-B code posted a major update (version 3.0).</li>
<li>A reference implementation in <a href="/facts/Fortran_77/m1KqjcMU">Fortran 77</a> (and with a <a href="/facts/Fortran/m1KqjcMU">Fortran 90</a> interface).<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> This version, as well as older versions, has been converted to many other languages.</li>
<li>An OWL-QN C++ implementation by its designers.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li></ul>
<h2 id="works-cited">Works cited</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li>Liu, D. C.; Nocedal, J. (1989). <a href="http://www.ece.northwestern.edu/~nocedal/PSfiles/limited-memory.ps.gz">"On the Limited Memory Method for Large Scale Optimization"</a>. Mathematical Programming B. 45 (3): 503–528. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.6443">10.1.1.110.6443</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01589116">10.1007/BF01589116</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:5681609">5681609</a>.</li>
<li>Haghighi, Aria (2 Dec 2014). <a href="http://aria42.com/blog/2014/12/understanding-lbfgs">"Numerical Optimization: Understanding L-BFGS"</a>.</li>
<li>Pytlak, Radoslaw (2009). <a href="https://books.google.com/books?id=RhRkaDPmwVoC&pg=PA159">"Limited Memory Quasi-Newton Algorithms"</a>. Conjugate Gradient Algorithms in Nonconvex Optimization. Springer. pp. 159–190. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-85633-7.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Liu, D. C.; Nocedal, J. (1989). "On the Limited Memory Method for Large Scale Optimization". Mathematical Programming B. 45 (3): 503–528. CiteSeerX 10.1.1.110.6443. doi:10.1007/BF01589116. S2CID 5681609. <a href="http://www.ece.northwestern.edu/~nocedal/PSfiles/limited-memory.ps.gz" target="_blank">http://www.ece.northwestern.edu/~nocedal/PSfiles/limited-memory.ps.gz</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Malouf, Robert (2002). "A comparison of algorithms for maximum entropy parameter estimation". Proceedings of the Sixth Conference on Natural Language Learning (CoNLL-2002). pp. 49–55. doi:10.3115/1118853.1118871. <a href="https://dl.acm.org/citation.cfm?id=1118871" target="_blank">https://dl.acm.org/citation.cfm?id=1118871</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear models". Proceedings of the 24th International Conference on Machine Learning. doi:10.1145/1273496.1273501. ISBN 9781595937933. S2CID 5853259. <a href="9781595937933" target="_blank">9781595937933</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. doi:10.1002/nme.1620141104. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Nocedal, J. (1980). "Updating Quasi-Newton Matrices with Limited Storage". Mathematics of Computation. 35 (151): 773–782. doi:10.1090/S0025-5718-1980-0572855-7. <a href="https://doi.org/10.1090%2FS0025-5718-1980-0572855-7" target="_blank">https://doi.org/10.1090%2FS0025-5718-1980-0572855-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Byrd, R. H.; Nocedal, J.; Schnabel, R. B. (1994). "Representations of Quasi-Newton Matrices and their use in Limited Memory Methods". Mathematical Programming. 63 (4): 129–156. doi:10.1007/BF01582063. S2CID 5581219. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Malouf, Robert (2002). "A comparison of algorithms for maximum entropy parameter estimation". Proceedings of the Sixth Conference on Natural Language Learning (CoNLL-2002). pp. 49–55. doi:10.3115/1118853.1118871. <a href="https://dl.acm.org/citation.cfm?id=1118871" target="_blank">https://dl.acm.org/citation.cfm?id=1118871</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear models". Proceedings of the 24th International Conference on Machine Learning. doi:10.1145/1273496.1273501. ISBN 9781595937933. S2CID 5853259. <a href="9781595937933" target="_blank">9781595937933</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. (1995). "A Limited Memory Algorithm for Bound Constrained Optimization". SIAM J. Sci. Comput. 16 (5): 1190–1208. Bibcode:1995SJSC...16.1190B. doi:10.1137/0916069. S2CID 6398414. <a href="https://digital.library.unt.edu/ark:/67531/metadc666315/" target="_blank">https://digital.library.unt.edu/ark:/67531/metadc666315/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Zhu, C.; Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge (1997). "L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization". ACM Transactions on Mathematical Software. 23 (4): 550–560. doi:10.1145/279232.279236. S2CID 207228122. <a href="https://doi.org/10.1145%2F279232.279236" target="_blank">https://doi.org/10.1145%2F279232.279236</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Andrew, Galen; Gao, Jianfeng (2007). "Scalable training of L₁-regularized log-linear models". Proceedings of the 24th International Conference on Machine Learning. doi:10.1145/1273496.1273501. ISBN 9781595937933. S2CID 5853259. <a href="9781595937933" target="_blank">9781595937933</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Schraudolph, N.; Yu, J.; Günter, S. (2007). A stochastic quasi-Newton method for online convex optimization. AISTATS. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Mokhtari, A.; Ribeiro, A. (2015). "Global convergence of online limited memory BFGS" (PDF). Journal of Machine Learning Research. 16: 3151–3181. arXiv:1409.2045. <a href="http://www.jmlr.org/papers/volume16/mokhtari15a/mokhtari15a.pdf" target="_blank">http://www.jmlr.org/papers/volume16/mokhtari15a/mokhtari15a.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Mokhtari, A.; Ribeiro, A. (2014). "RES: Regularized Stochastic BFGS Algorithm". IEEE Transactions on Signal Processing. 62 (23): 6089–6104. arXiv:1401.7625. Bibcode:2014ITSP...62.6089M. CiteSeerX 10.1.1.756.3003. doi:10.1109/TSP.2014.2357775. S2CID 15214938. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
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</ol>

Limited-memory BFGS open-in-new

Limited-memory BFGS