Frank-Wolfe - Revision history

Tbb5: /* Numerical Example */

2021-12-16T02:49:44Z

Numerical Example

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	& = -2 \cdot 4 + -2 \cdot 4 = -16		& = -2 \cdot 4 + -2 \cdot 4 = -16
	\end{align}</math>		\end{align}</math>
	Since the solution is not zero, the extreme point <math>x_0</math> is not optimal and ~~next~~ iteration is required.		Since the solution is not zero, the extreme point <math>x_0</math> is not optimal and another iteration is required.

Tbb5: /* Numerical Example */

2021-12-16T02:49:07Z

Numerical Example

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	<math display="block">\begin{align} x_1 &= x_0 + \alpha_0p_0 \\		<math display="block">\begin{align} x_1 &= x_0 + \alpha_0p_0 \\
	&= [0,0] + 16/64 \cdot [4,4] \\		&= [0,0] + 1/4 \cdot [4,4] \\
	\therefore \; x_1 &= [1,1]		\therefore \; x_1 &= [1,1]
	\end{align} </math>		\end{align} </math>

Tbb5: /* Numerical Example */

2021-12-16T02:47:53Z

Numerical Example

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	&= 32\alpha_0^2 -16\alpha_0 +2 \\		&= 32\alpha_0^2 -16\alpha_0 +2 \\
	\nabla f(\alpha_0) &= 64\alpha_0 - 16 = 0 \\		\nabla f(\alpha_0) &= 64\alpha_0 - 16 = 0 \\
	\therefore \qquad \alpha_0 &= 16/64 &= 1/4		\therefore \qquad \alpha_0 &= 16/64 = 1/4
	\end{align}</math>		\end{align}</math>

Tbb5: Simplify fraction

2021-12-16T02:46:51Z

Simplify fraction

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	&= 32\alpha_0^2 -16\alpha_0 +2 \\		&= 32\alpha_0^2 -16\alpha_0 +2 \\
	\nabla f(\alpha_0) &= 64\alpha_0 - 16 = 0 \\		\nabla f(\alpha_0) &= 64\alpha_0 - 16 = 0 \\
	\therefore \qquad \alpha_0 &= 16/64		\therefore \qquad \alpha_0 &= 16/64 &= 1/4
	\end{align}</math>		\end{align}</math>

Tbb5: generalize solver

2021-12-16T02:45:22Z

generalize solver

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	<math display="block"> \nabla f(0,0) \cdot (x,y)^T </math>		<math display="block"> \nabla f(0,0) \cdot (x,y)^T </math>

	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 ~~the software GAMS~~, the solution to linear problem is:		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:
	<math display="block"> \nabla f(0,0) \cdot (x,y)^T = (2,2)</math>		<math display="block"> \nabla f(0,0) \cdot (x,y)^T = (2,2)</math>

Tbb5: Editorials from our Google doc

2021-12-16T02:42:23Z

Editorials from our Google doc

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	== Introduction ==		== 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.<ref name="Frank">Frank, M. and Wolfe, P. (1956). [https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800030109 An algorithm for quadratic programming.] ''Naval Research Logistics Quarterly'', 3(1-2):95–110.</ref> 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.		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.<ref name="Frank">Frank, M. and Wolfe, P. (1956). [https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800030109 An algorithm for quadratic programming.] ''Naval Research Logistics Quarterly'', 3(1-2):95–110.</ref> 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 ~~the~~ practical ~~convergence~~, which can potentially be ~~an extremely slow computation~~ process. ~~Nonetheless,~~ new ~~sub-method has been proposed~~ to accelerate the algorithm by better aligning the descent direction while still preserving the projection-free property. <ref name="Combettes">Combettes, C.W.. and Pokutta, S. (2020). [https://arxiv.org/abs/2003.06369 Boosting Frank-Wolfe by chasing gradients.] ''International Conference on Machine Learning'' (pp. 2111-2121). PMLR. </ref> 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.		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. <ref name="Combettes">Combettes, C.W.. and Pokutta, S. (2020). [https://arxiv.org/abs/2003.06369 Boosting Frank-Wolfe by chasing gradients.] ''International Conference on Machine Learning'' (pp. 2111-2121). PMLR. </ref> 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 ==		== Theory & Methodology ==

Alt65: /* Step 3. Determine step length */

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Step 3. Determine step length

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	<math display="block">f(x_k + \alpha_k p_k) < f(x_k)</math>		<math display="block">f(x_k + \alpha_k p_k) < f(x_k)</math>
	<math display="block">min_{\alpha \in [0,1]} \, f(x_k + \~~alpha~~ p_k)</math>		<math display="block">min_{\alpha \in [0,1]} \, f(x_k + \alpha_k p_k)</math>

	====Step 4. Set new iteration point====		====Step 4. Set new iteration point====

Alt65 at 08:59, 11 December 2021

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	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.<ref name="LeBlanc">LeBlanc, L., Morlok, E.K., & Pierskalla, W.P. (1975). [https://www.sciencedirect.com/science/article/abs/pii/0041164775900301 An efficient approach to solving the road network equilibrium traffic assignment problem.] ''Transportation Research'', 9, 309-318.</ref> 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.		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.<ref name="LeBlanc">LeBlanc, L., Morlok, E.K., & Pierskalla, W.P. (1975). [https://www.sciencedirect.com/science/article/abs/pii/0041164775900301 An efficient approach to solving the road network equilibrium traffic assignment problem.] ''Transportation Research'', 9, 309-318.</ref> 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.
	~~== 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.

	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.		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.

Alt65 at 08:58, 11 December 2021

2021-12-11T08:58:48Z

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	Thus, the extreme point <math>(1,1)</math> is the optimum solution for this non-linear problem.		Thus, the extreme point <math>(1,1)</math> is the optimum solution for this non-linear problem.

			== Applications ==

			=== Image and Video Co-localization ===
			[[File:Co-localization with Frank-Wolfe Algorithm.png\|thumb\|586x586px\|'''Figure 2.''' The process of co-localization that captures common objects by using bounding boxes.<ref name="Joulin">Joulin, A., Tang, K., & Li, F.F. (2014). [https://link.springer.com/chapter/10.1007/978-3-319-10599-4_17 Efficient image and video co-localization with frank-wolfe algorithm.] ''European Conference on Computer Vision'' (pp. 253-268). Springer, Cham.</ref>\|alt=]]

			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.<ref name="Harzallah">Harzallah, H., Jurie, F., & Schmid, C. (2009). [https://ieeexplore.ieee.org/document/5459257 Combining efficient object localization and image classification.] ''2009 IEEE 12th international conference on computer vision'' (pp. 237-244). IEEE.</ref> 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.<ref name="Joulin" />

			=== 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.<ref name="LeBlanc">LeBlanc, L., Morlok, E.K., & Pierskalla, W.P. (1975). [https://www.sciencedirect.com/science/article/abs/pii/0041164775900301 An efficient approach to solving the road network equilibrium traffic assignment problem.] ''Transportation Research'', 9, 309-318.</ref> 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.
			== 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.

			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\|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 <ref name="Pokutta">Pokutta, S., Spiegel, C., & Zimmer, M. (2020). [https://arxiv.org/abs/2010.07243 Deep neural network training with Frank-Wolfe.] ''arXiv preprint arXiv:2010.07243''.</ref>:

			<math display=block>\theta_{t+1} = \theta_t + \alpha(\nu_t - \theta_t)</math>

			with <math>\alpha \in</math> [0,1] and <math>\nu_t</math> being

			<math display=block>\nu_t = argmin_{\nu \in C} <\nabla L (\theta_t) , \nu>.</math>

			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.<ref name="Huang">Huang, Z., and Wang, N. (2018). [https://doi.org/10.1007/978-3-030-01270-0_19 Data-driven sparse structure selection for deep neural networks.] ''Proceedings of the European conference on computer vision'' (pp.304-320). ECCV. </ref> 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 ==		== Conclusion ==

Alt65: /* Numerical Example */

2021-12-11T08:43:49Z

Numerical Example

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	'''Step 1. Utilize new iteration point as the starting point '''		'''Step 1. Utilize new iteration point as the starting point '''

	From ''Step 4'', <math> x_1 = [1,1]</math>.		From ''Step 4'' in ''Iteration 1'', <math> x_1 = [1,1]</math>.
	Therefore, substituting <math>x_1</math> into <math>\delta f_x(x,y)</math> and <math>\delta f_y(x,y)</math> from ''Step 1'',		Therefore, substituting new iteration point <math>x_1</math> into <math>\delta f_x(x,y)</math> and <math>\delta f_y(x,y)</math> from ''Step 1'',
	<math display="block">\nabla f(1,1) = z_1 = [0, 0] </math>		<math display="block">\nabla f(1,1) = z_1 = [0, 0] </math>