This proof roughly follows that of Lacoste-Julien (2016), with minor differences on how the case $g_t \geq C$ is handled and with a slightly different step size rule: while I consider the step sizes of Variant 1 and 2, Lacoste-Julien considers step sizes of the form of Variant 2 and \eqref{eq:step_size_curvature} The curvature constant is closely related to our Lipschitz assumption on the gradient. Karlijn Willems. \end{equation} Complexity bounds for primal-dual methods minimizing the model of objective function (2015). Again, we make a distinction of cases, this time on $g_t^*$: Hence, in both cases we have The objective function of the line search \eqref{eq:line_search} is of the form The first variant is easy to compute and only relies on knowledge of (a lower bound on) the Lipschitz constant $L$. We will now make a distinction of cases based on the value of $\xi^*$: Combining both cases we have $$. In this part I will present two main convergence results: one for general objectives and one for convex objectives. As it turns out, for a large class of problems, of which the $\ell_1$ or nuclear (also known as trace) norm ball are the most widely known examples, the linear subproblems have either a closed form solution or efficient algorithms exist.For an extensive discussion of the cost the linear minimization oracle, see Jaggi, Martin. Wolfram Alpha is an API which can compute expert-level answers using Wolfram’s algorithms, knowledgebase and AI technology. This will only be the case for other variants such as the Away-steps Frank-Wolfe that we will discuss in upcoming posts. and the domain $\mathcal{C}$ is a convex and compact set.A convex set is one for which any segment between two points lies within the set. In particular, by the definition above we always have $C_f \leq \diam(\mathcal{C})^2 L$, which given \eqref{eq:step_size_diam} suggests the following rule for the step size: This article tells how to create a simple assistant application in Python which can answer simple questions like the ones listed below. \begin{align} community. Note that all the results in this post are in terms of the Lipschitz constant $L$ but analogous results exist in terms of this curvature constant. where $\diam$ denotes the diameter with respect to the euclidean norm.It is possible to use a non-euclidean norm too, as long as the Lipschitz constant $L$ is computed with respect to the same norm. &\quad \boldsymbol{d}_t = \ss_t - \xx_t\\ \begin{equation} The Wolfram Client Library for Python lets Python programs directly integrate Wolfram Language capabilities. This proof uses the same proof technique as that of Nesterov Y. \begin{align} For Variant 2 we have $f(\xx_{t+1}) \leq f((1 - \gamma_t^\star)\xx_{t} + \gamma_t^\star \ss_t)$ since by definition of line search $f(\xx_{t+1})$ is the point that minimizes the objective value in the segment $(1 - \gamma)\xx_{t} + \gamma \ss_t$. \end{equation}. Evaluate any Wolfram Language code from Python: Immediately call all 6000+ built-in Wolfram Language functions in Python: Build up Wolfram Language code directly in Python: Direct support for PIL, Pandas and NumPy libraries: Represent Wolfram Language expressions as Python objects: Within your Python environment, the Wolfram Client Library for Python lets you: Use Wolfram's natural language understanding in Python: © 2019 Wolfram Research. Implementation-of-the-Frank-Wolfe-Algorithm The purpose this project is to implement the Frank-Wolfe Algorithm for transportation network analysis. &f((1 - \gamma_t^\star)\xx_{t} + \gamma_t^\star \ss_t) \\ &\quad\qquad\hfill\text{// exit if gap is below tolerance }\nonumber\\ Which shows the decrease in the Frank-Wolfe gap as a function of the number of iterations. Lemma 1.2.3 in Nesterov's. with $\boldsymbol{q}_t = \boldsymbol{A}(\boldsymbol{s}_t - \xx_t)$. Demyanov, Vladimir and Rubinov, Aleksandr. &\qquad+ \frac{L \gamma^2}{2}\|\ss_t - \xx_t\|^2~.\label{eq:l_smooth_xt} \end{equation} Said otherwise, $\xx^\star$ is a stationary point if there are no feasible descent directions with origin at $\xx^\star$. There are other step size strategies that I did not mention. Because of convexity we can obtain a tighter bound using the following simple inequality, mentioned earlier \eqref{eq:convexity_fw_gap}: datacamp. ", "Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization. f(\xx_{t+1}) \leq f(\xx_t) - \xi g_t + \frac{1}{2}\xi^2 L \diam(\mathcal{C})^2 \end{align}, Contrary to other constrained optimization algorithms like projected gradient descent, the Frank-Wolfe algorithm does not require access to a projection, hence why it is sometimes referred to as a projection-free algorithm. Log in. is positively correlated with the gradient. \begin{align}\label{eq:convexity_fw_gap} We will now minimize the right hand side with respect to $\gamma \in [0, 1]$. A proof with similar convergence rate but different proof techniquescan be found in other papers such as Martin Jaggi's Revisiting Frank-Wolfe or Francesco Locatello's A Unified Optimization View on &\quad g_t = -\langle \nabla f(\xx_t), \dd_t \rangle\\ The claimed bound now follows from definition of $C$. Lemma 1.2.3 in Nesterov's Introductory lectures on convex optimization. Chaining this last inequality with Eq. \end{equation} This feature can be very advantageous in situations with a huge or even infinite number of features, such as architecture optimization in neural networksPing W, Liu Q, Ihler AT. A_{t+1}\frac{\xi^2}{2} &= \frac{t+1}{t+2} \leq 1~, &\quad\qquad\gamma_t = \vphantom{\sum_i}\min\Big\{\frac{g_t}{L\|\dd_t\|^2}, 1 \Big\}\label{eq:step_size}\\ Definition 2: Frank-Wolfe gap. Stay tuned. f(\xx_t) - f(\xx^\star) \leq \frac{2 L \diam(\mathcal{C})^2}{t+1} "Convergence rate of Frank-Wolfe for non-convex objectives." "Step-size adaptivity in Projection-Free Optimization" ArXiv:1806.05123 (2018).. \end{align} Yet another step size strategy that has been proposed is based on the notion of curvature constant. arXiv preprint arXiv:1607.00345 (2016). Cheat Sheets.