Frank-Wolfe

Author: Adeline Tse, Jeff Cheung, Tucker Barrett (SYSEN 5800 Fall 2021)

Introduction

The Frank-Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization, first proposed by Marguerite Frank and Philip Wolfe from Princeton University in 1956.[1] It is also known as the “gradient and interpolation” algorithm or sometimes the “conditional gradient method” as it utilizes a simplex method of comparing an initial feasible point with a secondary basic feasible point to maximize the linear constraints in each iteration, to find or approximate the optimal solution.

Advantages of the Frank-Wolfe algorithm include that it is simple to implement; it results in a projection-free computation (that is, it does not require projections back to the constraint set to ensure feasibility); and it generates solution iterations that are sparse with respect to the constraint set. However, one of the major drawbacks of Frank-Wolfe algorithm is its convergence in practical applications, which can potentially be a computationally intensive process. This has enabled research into new revisions and heuristics to accelerate the algorithm by better aligning the descent direction while still preserving the projection-free property. [2] Therefore, despite the downside of Frank-Wolfe algorithm, the numerous benefits of this algorithm allow it to be utilized in artificial intelligence, machine learning, traffic assignment, signal processing, and many more applications.

Theory & Methodology

Figure 1. The Frank-Wolfe method optimizes by considering the linearization of the objective function f and moving the initial position x towards the minimizer of the linear function.[3]

The Frank-Wolfe algorithm uses step size and postulated convexity, which formulates a matrix of positive semidefinite quadratic form. Just like a convex function yields a global minimum at any local minimum on a convex set, by the definition of nonlinear programming, the concave quadratic function would yield a global maximum point at any local maximum on a convex constraint set.

The methodology of the Frank-Wolfe algorithm starts with obtaining the generalized Lagrange multipliers for a quadratic problem, PI. The found multipliers are considered solutions to a new quadratic problem,  PII. The simplex method is then applied to the new problem, PII, and the gradient and interpolation method is utilized to make incremental steps towards the solution of the original quadratic problem, PI. This is performed by taking an initial feasible point, obtaining a second basic feasible point along the gradient of the objective function at the initial point, and maximizing the objective function along the segment that joins these two points.  The found maximum along the segment is utilized as the starting point for the next iteration.[1] Using this method, the values of the objective function converge to zero and in one of the iterations, a secondary point will yield a solution.

Proof

• ${\displaystyle X}$ is a ${\displaystyle [n\times 1]}$ matrix
• ${\displaystyle A}$ is a ${\displaystyle [m\times n]}$ matrix
• ${\displaystyle C}$ is a ${\displaystyle [n\times n]}$ matrix
• ${\displaystyle p}$ and ${\displaystyle b}$ are ${\displaystyle [1\times n]}$ and ${\displaystyle [1\times m]}$ matrices respectively
• ${\displaystyle \nabla f(x)}$ is the gradient

${\displaystyle f(x)=\sum _{j=1}^{n}p_{j}x_{j}-\sum _{j,k=1}^{n,n}X_{j}C_{jk}X_{k}}$
${\displaystyle where\;\sum _{j=1}^{n}A_{ij}X_{j}\leq b_{i}\quad (i=1,...,m)\;and\;X_{j}\geq 0\quad (j=1,...,n)}$

${\displaystyle \nabla f(x)={\frac {df(x)}{dx}}}$

PI is represented by the following equations using matrix notation:

${\displaystyle {\text{Maximize}}\qquad f(x)=px'-x'Cx\quad {\text{subject to}}}$
${\displaystyle (I)={\begin{cases}x\geq 0\\Ax\leq b\end{cases}}}$

Since PI is feasible, any local maximum found that is contained within the convex constraint set is a global maximum. Then by utilizing Kuhn and Tucker's generalizations of the Lagrange Multipliers of the maximization of the objective function ${\displaystyle f(x)}$, ${\displaystyle x_{0}}$ is considered the solution of PI.

${\displaystyle \delta f(x_{0})x_{0}=Max\,[\,\delta f(x_{0})w\,|\,w\geq 0,Aw\leq b],\quad where\;w\;{\text{is an}}\;[n\times 1]\;{\text{matrix}}}$

By the Duality Theorem,

${\displaystyle Min\,[\,ub\,|\,u\geq 0,\,uA\geq \delta f(x_{o})],\quad where\;u\;{\text{is a}}\;[1\times m]\;{\text{matrix}}}$

Therefore, the solution ${\displaystyle x_{0}}$ for PI is only valid if:

${\displaystyle Max\,[\,\delta f(x_{0})x_{0}-ub\,|\,u\geq 0,uA\geq \delta f(x_{0})]=0}$

Since

${\displaystyle \delta f(x)=p-2x'Cx}$
Therefore, by the generalization of Lagrangian, ${\displaystyle g(x,u)}$ is extracted with linear constraints:
${\displaystyle g(x,u)=\delta f(x)x-ub=px-ub-2x'Cx}$
${\displaystyle {\begin{cases}x\geq 0,\quad u\geq 0,\\Ax\leq b,\quad \delta f(x)\leq uA\end{cases}}}$

The completion of the generalized Lagrangian shows that ${\displaystyle m+n}$ positive variables exist for PI, thus a feasible solution exists.

Based on the concavity of ${\displaystyle f}$,

${\displaystyle f(y)-f(x)\leq \delta f(x)(y-x)}$

If ${\displaystyle (x,u)}$ satisfies ${\displaystyle g(x,u)\leq 0}$ (where x is a solution if for some ${\displaystyle u,\;g(x,u)=0}$), then

${\displaystyle f(y)-f(x)\leq \delta f(x)x\leq uAy-\delta f(x)x\leq ub-\delta f(x)x=-g(x,u)}$

By the Boundedness Criterion and the Approximation Criterion,

${\displaystyle f(y)-f(x)\leq -g(x,u)}$

The Frank-Wolfe algorithm then adds a non-negative variable to each of the inequalities to obtain the following:

• ${\displaystyle y}$ and ${\displaystyle \nu }$ are ${\displaystyle [m\times 1]}$ and ${\displaystyle [n\times 1]}$ matrices respectively

${\displaystyle x,\,u,\,y,\,\nu \geq 0}$
${\displaystyle Ax+y=b}$
${\displaystyle 2Cx+A'u'-\nu '=p'}$
${\displaystyle g(x,u)=\delta f(x)x-ub=(uA-\nu )x-u(Ax+y)=-\nu x+uy}$

Thus, the PII problem is to obtain search vectors to satisfy ${\displaystyle \nu x+uy=0}$, and ${\displaystyle x}$ is considered a solution of PI if ${\displaystyle \delta f(x)}$ belongs to the convex cone spanned by the outward normals to the bounding hyperplanes on which ${\displaystyle x}$ lines as shown in Figure 1.

Algorithm

The Frank-Wolfe algorithm can generally be broken down into five steps as described below. Then the loop of iterations continues throughout Step 2 to Step 5 until the minimum extreme point is identified.

Step 1. Define initial solution

If ${\displaystyle x}$ is the extreme point, the initial arbitrary basic feasible solution can be considered as:

${\displaystyle x_{k}\in S\quad where\;k=0}$

The letter ${\displaystyle k}$ denotes the number of iterations.

Step 2. Determine search direction

Search direction, that is the direction vector is:

${\displaystyle p_{k}=y_{k}-x_{k}}$

${\displaystyle x_{k}}$ and ${\displaystyle y_{k}}$ are feasible extreme points and belong to ${\displaystyle S}$ where ${\displaystyle S}$ is convex

First-order Taylor expansion around ${\displaystyle x_{k}}$ and problem is now reformulated to LP:

{\displaystyle {\begin{aligned}min\quad &z_{k}(y)=f(x_{k})+\nabla f(x_{k})^{T}(y-x_{k})\\s.t.\quad &y\in S\\\end{aligned}}}

Step 3. Determine step length

Step size ${\displaystyle \alpha _{k}}$ is defined by the following formula where ${\displaystyle \alpha _{k}}$ must be less than 1 to be feasible:

${\displaystyle f(x_{k}+\alpha _{k}p_{k})
${\displaystyle min_{\alpha \in [0,1]}\,f(x_{k}+\alpha _{k}p_{k})}$

Step 4. Set new iteration point

${\displaystyle x_{k+1}=x_{k}+\alpha _{k}p_{k}}$

Step 5. Stopping criterion

Check if ${\displaystyle x_{k+1}}$ is an approximation of ${\displaystyle x}$, the extreme point. Otherwise, set ${\displaystyle k=k+1}$ and return to Step 2 for the next iteration.

In the Frank-Wolfe algorithm, the value of ${\displaystyle f}$ descends after each iteration and eventually decreases towards ${\displaystyle f(x)}$, the global minimum.

Numerical Example

Consider the non-linear problem below:

{\displaystyle {\begin{aligned}min\quad &(x-1)^{2}+(y-1)^{2}\\s.t.\quad &2x+y\leq 6\\&x+2y\leq 6\\&x\geq 0\\&y\geq 0\end{aligned}}}

ITERATION 1

Step 1. Choose a starting point

Begin by choosing the feasible point ${\displaystyle (0,0)}$ and take the partial derivatives of the objective function ${\displaystyle f}$.

${\displaystyle \delta f_{x}(x,y)=2(x-1)=2x-2}$

${\displaystyle \delta f_{y}(x,y)=2(y-1)=2y-2}$

${\displaystyle \therefore \;\nabla f(0,0)=z_{0}=[-2,-2]}$

Step 2. Determine a search direction

Obtain the search direction by solving the linear problem that is subject to the constraints defined.

${\displaystyle \nabla f(0,0)\cdot (x,y)^{T}}$

The solution of this linear problem is obtained via linear programming algorithm, which is the enumeration with the extreme points in the feasible region. Using the simplex method or an algebraic solver, the solution to linear problem is:

${\displaystyle \nabla f(0,0)\cdot (x,y)^{T}=(2,2)}$

Thus, the direction vector ${\displaystyle p_{0}}$for this iteration ${\displaystyle z_{0}}$ is:

{\displaystyle {\begin{aligned}p_{0}&=(x,y)^{T}-z_{0}\\&=[2,2]-[-2,-2]=[4,4]\end{aligned}}}

Step 3. Determine step length

Evaluate ${\displaystyle z_{\alpha _{0}}=f(4\alpha _{0},4\alpha _{0})}$ to obtain step length ${\displaystyle \alpha _{0}}$.

{\displaystyle {\begin{aligned}\because \;\;\qquad f&=(4\alpha _{0}-1)^{2}+(4\alpha _{0}-1)^{2}\\&=32\alpha _{0}^{2}-16\alpha _{0}+2\\\nabla f(\alpha _{0})&=64\alpha _{0}-16=0\\\therefore \qquad \alpha _{0}&=16/64=1/4\end{aligned}}}

Step 4. Set new iteration point

The next point is determined by the following formula:

{\displaystyle {\begin{aligned}x_{1}&=x_{0}+\alpha _{0}p_{0}\\&=[0,0]+1/4\cdot [4,4]\\\therefore \;x_{1}&=[1,1]\end{aligned}}}

Step 5. Perform optimality test

If the following equation yields a zero, the optimal solution has been found for the problem.

{\displaystyle {\begin{aligned}f_{1}&=\nabla f(x_{0})\cdot p_{0}=\nabla f(0,0)\cdot p_{0}\\&=-2\cdot 4+-2\cdot 4=-16\end{aligned}}}
Since the solution is not zero, the extreme point ${\displaystyle x_{0}}$ is not optimal and another iteration is required.

ITERATION 2

Step 1. Utilize new iteration point as the starting point

From Step 4 in Iteration 1, ${\displaystyle x_{1}=[1,1]}$. Therefore, substituting new iteration point ${\displaystyle x_{1}}$ into ${\displaystyle \delta f_{x}(x,y)}$ and ${\displaystyle \delta f_{y}(x,y)}$ from Step 1,

${\displaystyle \nabla f(1,1)=z_{1}=[0,0]}$

Since the gradient of the new starting point is ${\displaystyle (0,0)}$, the optimality test for the value of ${\displaystyle f}$ will always yield ${\displaystyle 0}$:

${\displaystyle f_{2}=\nabla f(1,1)\cdot p_{1}=[0,0]\cdot p_{1}}$

Thus, the extreme point ${\displaystyle (1,1)}$ is the optimum solution for this non-linear problem.

Applications

Image and Video Co-localization

Figure 2. The process of co-localization that captures common objects by using bounding boxes.[4]

Co-localization is one of the programming methods in the machine learning field used to find a common object within a set of images or video frames. The process of co-localization involves first generating multiple candidate bounding boxes for each image/ video frame, then identifying the specific box in each image/ video frame that jointly contains the common object.[5] This technique is extremely important for analyzing online data. By being able to recognize the common objects, visual concepts can be tagged into specific labels, thus leveraging useful data for other marketing and product design purposes. Such development is especially useful for image and video hosting services such as Flickr and YouTube.

The formulation of co-localization using the Frank-Wolfe algorithm involves minimizing a standard quadratic programming problem and reducing it to a succession of simple linear integer problems. The benefit of the algorithm is that it allows many more images to be analyzed simultaneously and more bounding boxes are generated in each image/ video frame for higher accuracy. By relaxing the discrete non-convex set to its convex hull, the constraints that define the original set can now be separated from one image/ video to another. In the case of co-localizing videos, the constraints use the shortest-path algorithm to reach optimization.[4]

Traffic Assignment Problem

A typical transportation network, also known as flow network, is considered as a convex programming problem with linear constraints. With the vertices called nodes and the edges called arcs, a directed graph called a network is formed and the flow is monitored via the movement from one node, origin (O), to another node, destination (D) along the arc. The objective of this programming problem is to achieve user-equilibrium, which means system flow is set optimal without incurring unnecessarily high cost for users.

The use of the Frank-Wolfe algorithm in this case means that the traffic assignment problem can be applied to nonlinear multicommodity network flow with flow dependent cost.[6] The algorithm is also able to process a larger set of constraints with more O-D pairs. Since the Frank-Wolfe algorithm uses a descent method to search for the direction of extreme points, the technique only recognizes the sequence of the shortest route problems. Therefore, the Frank-Wolfe algorithm is known to solve the traffic assignment problem on networks with several hundred nodes efficiently with the minimal number of iterations.

Due to the algorithm’s effectiveness in optimizing traffic assignment problem, it has been widely applied in many fields other than the estimation of the total demand for travel via road between a set of roads and zones. These applications include data-driven computer networks, fluids in pipes, currents in an electrical circuit and more.

Deep Neural Network Training

Although Stochastic Gradient Descent (SGD) is still the conventional machine learning technique for deep learning, the Frank-Wolfe algorithm has been proven to be applicable for training neural networks as well. Instead of taking projection steps via SGD, the Frank-Wolfe algorithm adopts the simple projection-free first-order function by using the linear minimization oracle (LMO) approach for constrained optimization [7]:

${\displaystyle \theta _{t+1}=\theta _{t}+\alpha (\nu _{t}-\theta _{t})}$

with ${\displaystyle \alpha \in }$ [0,1] and ${\displaystyle \nu _{t}}$ being

${\displaystyle \nu _{t}=argmin_{\nu \in C}<\nabla L(\theta _{t}),\nu >.}$

Through LMO, non-unique solutions to the above equation that were deemed impossible by gradient-descent-based optimization would now be possible. In addition to the linearization approach, sparsity regularization is also applied to the scaling factors in the optimization of deep neural networks.[8] By forcing some factors to zero, the highly complex computational process of training deep neural networks is simplified. Since the other major characteristics of Frank-Wolfe algorithm is describing solutions with sparse structures, deep neural network training benefits greatly from the application of this algorithm.

Conclusion

The Frank-Wolfe algorithm is one of the key techniques in the machine learning field. The algorithm features favor linear minimization, simple implementation, and low complexity. By using the simplex method that requires little alteration to the gradient and interpolation in each iteration, the optimization of the linearized quadratic objective function can be performed in a significantly shorter runtime and larger datasets can be processed more efficiently. Therefore, even though the convergence rate of Frank-Wolfe algorithm is slow due to the search for naive descent directions in each iteration, the benefits that the algorithm brings still outweigh the disadvantages. With the main characteristics of being projection-free and the capability of producing solutions for sparse structures, the algorithm gained popularity with many use cases in machine learning.

References

1. Frank, M. and Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3(1-2):95–110.
2. Combettes, C.W.. and Pokutta, S. (2020). Boosting Frank-Wolfe by chasing gradients. International Conference on Machine Learning (pp. 2111-2121). PMLR.
3. Jaggi, M. (2013). Revisiting Frank-Wolfe: Projection-free sparse convex optimization. Inerational Conference on Machine Learning (pp.427-435). PMLR.
4. Joulin, A., Tang, K., & Li, F.F. (2014). Efficient image and video co-localization with frank-wolfe algorithm. European Conference on Computer Vision (pp. 253-268). Springer, Cham.
5. Harzallah, H., Jurie, F., & Schmid, C. (2009). Combining efficient object localization and image classification. 2009 IEEE 12th international conference on computer vision (pp. 237-244). IEEE.
6. LeBlanc, L., Morlok, E.K., & Pierskalla, W.P. (1975). An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research, 9, 309-318.
7. Pokutta, S., Spiegel, C., & Zimmer, M. (2020). Deep neural network training with Frank-Wolfe. arXiv preprint arXiv:2010.07243.
8. Huang, Z., and Wang, N. (2018). Data-driven sparse structure selection for deep neural networks. Proceedings of the European conference on computer vision (pp.304-320). ECCV.