Frank–Wolfe algorithm: Difference between revisions

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{{Short description|Optimization algorithm}}

The '''Frank–Wolfe algorithm''', also known as the ''convex combination algorithm'', is a classic algorithm in [[operations research]] (OR). It was originally proposed by Marguerite Frank and Phil Wolfe in 1956 as a procedure for solving [[quadratic programming]] problems with linear constraints. At each step the objective function is linearized and then a step is taken in a direction that reduces the objective while maintaining feasibility. The algorithm can be seen as a generalization of the primal [[simplex algorithm]] for [[linear programming]]. In recent years it has been widely used for determining the equilibrium flows in transportation networks.

The '''Frank–Wolfe algorithm''' is an [[iterative method|iterative]] [[First-order approximation|first-order]] [[Mathematical optimization|optimization]] [[algorithm]] for [[constrained optimization|constrained]] [[convex optimization]]. Also known as the '''conditional gradient method''',<ref>{{Cite journal | last1 = Levitin | first1 = E. S. | last2 = Polyak | first2 = B. T. | doi = 10.1016/0041-5553(66)90114-5 | title = Constrained minimization methods | journal = USSR Computational Mathematics and Mathematical Physics | volume = 6 | issue = 5 | pages = 1 | year = 1966 }}</ref> '''reduced gradient algorithm''' and the '''convex combination algorithm''', the method was originally proposed by [[Marguerite Frank]] and [[Philip Wolfe (mathematician)|Philip Wolfe]] in&nbsp;1956.<ref>{{Cite journal | last1 = Frank | first1 = M. | last2 = Wolfe | first2 = P. | doi = 10.1002/nav.3800030109 | title = An algorithm for quadratic programming | journal = Naval Research Logistics Quarterly | volume = 3 | issue = 1–2 | pages = 95–110 | year = 1956 }}</ref> In each iteration, the Frank–Wolfe algorithm considers a [[linear approximation]] of the objective function, and moves towards a minimizer of this linear function (taken over the same ___domain).

===Problem statement===

:Minimize <math> f(\mathbf{x}) = \frac{1}{2} \mathbf{x}^{\mathrm{T}} E\mathbf{x} + \mathbf{h}^{\mathrm{T}} \mathbf{x} </math>

:subject to <math> \mathbf{x} \epsilon \mathbf{P}</math>.

Where the n×n matrix E is [[Positive-semidefinite_matrix|positive semidefinite]], h is an n×1 vector, and <math>\mathbf{P}</math> represents a feasible region defined by a mix of linear inequality and equality constraints (for example Ax ≤ b, Cx = d).

Suppose <math>\mathcal{D}</math> is a [[compact space|compact]] [[convex set]] in a [[vector space]] and <math>f \colon \mathcal{D} \to \mathbb{R}</math> is a [[convex function|convex]], [[differentiable function|differentiable]] [[real-valued function]]. The Frank–Wolfe algorithm solves the [[optimization problem]]

===Algorithm===

:Minimize <math> f(\mathbf{x})</math>

:subject to <math> \mathbf{x} \epsilonin \mathbfmathcal{PD}</math>.

===Algorithm===

'''Step 1.''' Initialization. Let <math>k \leftarrow 0</math> and let <math>x_0</math> be any point in <math>\mathbf{P}</math>.

[[File:Frank-Wolfe Algorithm.png|thumbnail|right|A step of the Frank–Wolfe algorithm]]

:'''Step 1.'Initialization:'' Initialization. Let <math>k \leftarrow 0</math>, and let <math>x_0\mathbf{x}_0 \!</math> be any point in <math>\mathbfmathcal{PD}</math>.

'''Step 2.''' Convergence test. If <math> \nabla f(x_k)=0</math> then Stop, we have found the minimum.

:'''Step 31.''' ''Direction-finding subproblem.:'' The approximation of the problem that is obtained by replacing the function f with its first-order Taylor expansion around <math>x_k</math> is found. Solve forFind <math>\barmathbf{xs}_k</math>: solving

::Minimize <math>f(x_k) + \nablamathbf{s}^T \nabla f(x_k) \barmathbf{x}_k)</math>

::Subject to <math> \barmathbf{xs}_k \epsilonin \mathbfmathcal{PD}</math>

:''(noteInterpretation: thatMinimize thisthe islinear aapproximation Linearof Program.the problem <math>x_k</math>given isby fixedthe duringfirst-order Step[[Taylor 3,series|Taylor whileapproximation]] theof minimization takes place by<math>f</math> varyingaround <math>\barmathbf{x}_k \!</math> and is equivalentconstrained to minimizationstay ofwithin <math>\nabla^T f(x_k) \barmathcal{xD}_k</math>.).''

:'''Step 42.''' ''Step size determination.:'' FindSet <math>\lambdaalpha \leftarrow \frac{2}{k+2}</math>, or alternatively find <math>\alpha</math> that minimizes <math> f(x_k\mathbf{x}_k+\lambdaalpha(\barmathbf{xs}_k -x_k\mathbf{x}_k))</math> subject to <math>0 \le \lambdaalpha \le 1</math> . If <math>\lambda = 0</math> then Stop, we have found the minimum.

:'''Step 53.''' ''Update.:'' Let <math>x_\mathbf{x}_{k+1}\leftarrow x_k\mathbf{x}_k+\lambdaalpha(\barmathbf{xs}_k-x_k\mathbf{x}_k)</math>, let <math>k \leftarrow k+1</math> and go back to Step 21.

===Comments=Properties==

While competing methods such as [[gradient descent]] for constrained optimization require a [[Projection (mathematics)|projection step]] back to the feasible set in each iteration, the Frank–Wolfe algorithm only needs the solution of a convex problem over the same set in each iteration, and automatically stays in the feasible set.

The convergence of the Frank–Wolfe algorithm is sublinear in general: the error in the objective function to the optimum is <math>O(1/k)</math> after ''k'' iterations, so long as the gradient is [[Lipschitz continuity|Lipschitz continuous]] with respect to some norm. The same convergence rate can also be shown if the sub-problems are only solved approximately.<ref>{{Cite journal | last1 = Dunn | first1 = J. C. | last2 = Harshbarger | first2 = S. | doi = 10.1016/0022-247X(78)90137-3 | title = Conditional gradient algorithms with open loop step size rules | journal = Journal of Mathematical Analysis and Applications | volume = 62 | issue = 2 | pages = 432 | year = 1978 | doi-access = free }}</ref>

The algorithm generally makes good progress towards the optimum during the first few iterations, but convergence often slows down substantially when close to the minimum point. For this reason the algorithm is perhaps best used to find an approximate solution. It can be shown that the worst case convergence rate is sublinear; however, in practice the convergence rate has been observed to improve in case of many constraints.{{fact}}

The iterations of the algorithm can always be represented as a sparse convex combination of the extreme points of the feasible set, which has helped to the popularity of the algorithm for sparse greedy optimization in [[machine learning]] and [[signal processing]] problems,<ref>{{Cite journal | last1 = Clarkson | first1 = K. L. | title = Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm | doi = 10.1145/1824777.1824783 | journal = ACM Transactions on Algorithms | volume = 6 | issue = 4 | pages = 1–30 | year = 2010 | citeseerx = 10.1.1.145.9299 }}</ref> as well as for example the optimization of [[flow network|minimum–cost flow]]s in [[Transport network|transportation network]]s.<ref>{{Cite journal | last1 = Fukushima | first1 = M. | title = A modified Frank-Wolfe algorithm for solving the traffic assignment problem | doi = 10.1016/0191-2615(84)90029-8 | journal = Transportation Research Part B: Methodological | volume = 18 | issue = 2 | pages = 169–177| year = 1984 }}</ref>

==References==

If the feasible set is given by a set of linear constraints, then the subproblem to be solved in each iteration becomes a [[linear programming|linear program]].

*M. Frank and P. Wolfe, ''An algorithm for quadratic programming'', Naval Research Logistics Quarterly, 3 (1956), pp. 95--110.

While the worst-case convergence rate with <math>O(1/k)</math> can not be improved in general, faster convergence can be obtained for special problem classes, such as some strongly convex problems.<ref>{{Cite book|title=Nonlinear Programming|first= Dimitri |last=Bertsekas|year= 1999|page= 215|publisher= Athena Scientific| isbn =978-1-886529-00-7}}</ref>

*[http://www.math.chalmers.se/Math/Grundutb/CTH/tma946/0203/fw_eng.pdf The Frank-Wolfe algorithm] description

==Lower bounds on the solution value, and primal-dual analysis==

*[http://swutc.tamu.edu/Reports/472840-00074-1.pdf/ Lee and Machemehl: Combined traffic signal control and traffic assignment, Technical Report, March 2005]

Since <math>f</math> is [[Convex function|convex]], for any two points <math>\mathbf{x}, \mathbf{y} \in \mathcal{D}</math> we have:

[[Category:Optimization algorithms]]

:<math>

:Minimize <math> f(\mathbf{xy}) = \frac{1}{2}geq f(\mathbf{x}^{\mathrm{T}}) + E(\mathbf{xy} + - \mathbf{hx})^{\mathrm{T}} \nabla f(\mathbf{x} </math>)

</math>

This also holds for the (unknown) optimal solution <math>\mathbf{x}^*</math>. That is, <math>f(\mathbf{x}^*) \ge f(\mathbf{x}) + (\mathbf{x}^* - \mathbf{x})^T \nabla f(\mathbf{x})</math>. The best lower bound with respect to a given point <math>\mathbf{x}</math> is given by

:<math>

\begin{align}

f(\mathbf{x}^*)

& \ge f(\mathbf{x}) + (\mathbf{x}^* - \mathbf{x})^T \nabla f(\mathbf{x}) \\

&\geq \min_{\mathbf{y} \in D} \left\{ f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T \nabla f(\mathbf{x}) \right\} \\

&= f(\mathbf{x}) - \mathbf{x}^T \nabla f(\mathbf{x}) + \min_{\mathbf{y} \in D} \mathbf{y}^T \nabla f(\mathbf{x})

\end{align}

</math>

The latter optimization problem is solved in every iteration of the Frank–Wolfe algorithm, therefore the solution <math>\mathbf{s}_k</math> of the direction-finding subproblem of the <math>k</math>-th iteration can be used to determine increasing lower bounds <math>l_k</math> during each iteration by setting <math>l_0 = - \infty</math> and

:<math>

l_k := \max (l_{k - 1}, f(\mathbf{x}_k) + (\mathbf{s}_k - \mathbf{x}_k)^T \nabla f(\mathbf{x}_k))

</math>

Such lower bounds on the unknown optimal value are important in practice because they can be used as a stopping criterion, and give an efficient certificate of the approximation quality in every iteration, since always <math>l_k \leq f(\mathbf{x}^*) \leq f(\mathbf{x}_k)</math>.

It has been shown that this corresponding [[duality gap]], that is the difference between <math>f(\mathbf{x}_k)</math> and the lower bound <math>l_k</math>, decreases with the same convergence rate, i.e.

<math>

f(\mathbf{x}_k) - l_k = O(1/k) .

</math>

==Notes==

{{Reflist}}

==Bibliography==

*{{cite journal|last=Jaggi|first=Martin|title=Revisiting Frank–Wolfe: Projection-Free Sparse Convex Optimization|journal=Journal of Machine Learning Research: Workshop and Conference Proceedings |volume=28|issue=1|pages=427–435|year= 2013 |url=http://jmlr.csail.mit.edu/proceedings/papers/v28/jaggi13.html}} (Overview paper)

*[http://www.math.chalmers.se/Math/Grundutb/CTH/tma946/0203/fw_eng.pdf The Frank-WolfeFrank–Wolfe algorithm] description

* {{Cite book | last1=Nocedal | first1=Jorge | last2=Wright | first2=Stephen J. | title=Numerical Optimization | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | edition=2nd | isbn=978-0-387-30303-1 | year=2006 }}.

==External links==

*https://conditional-gradients.org/: a survey of Frank–Wolfe algorithms.

*[https://www.youtube.com/watch?v=24e08AX9Eww Marguerite Frank giving a personal account of the history of the algorithm]

== See also ==

* [[Proximal gradient methods]]

{{Optimization algorithms|convex}}

{{DEFAULTSORT:Frank-Wolfe algorithm}}

[[Category:Optimization algorithms and methods]]

[[Category:Iterative methods]]

[[Category:First order methods]]

[[Category:Gradient methods]]