As such, it does not perform competitively in practice with the other step size strategies, although it does achieve the same theoretical rate of convergence. Lemma 1.2.3 in Nesterov's Introductory lectures on convex optimization. \end{equation} Within your Python environment, the Wolfram Client Library for Python lets you: Access the power of Wolfram algorithms: Get immediate access to the world's largest integrated algorithmbase: >>> limit = wlexpr ('Limit[x Log[x^2], x -> 0]') >>> session. The effectiveness of this approach is then tightly linked to the ability to quickly solve the linear subproblems. Example: using Frank-Wolfe to solve a Lasso problem Generalized Matching Pursuit and Frank-Wolfe, "Duality between subgradient and conditional gradient methods. and so can be used as a function suboptimality certificate. This step size was used for example by Lacoste-Julien 2016.Lacoste-Julien, Simon. "An algorithm for quadratic programming." &\qquad \text{ (by optimality of $\gamma_t^\star$)}\nonumber\\ \def\yy{\boldsymbol y} e_{t} \leq t C \implies f(\xx_{t}) - f(\xx^\star) \leq \frac{2 C}{t+1}~, We will now minimize the right hand side with respect to $\gamma \in [0, 1]$. f(\xx_{t+1})&\leq f(\xx_t) - \xi g_t + \frac{\xi^2 L }{2}\diam(\mathcal{C})^2\\ g_t = \langle \nabla f(\xx_t), \xx_t - \ss_t \rangle November 14th, 2019. must read. -h_0 \leq f(\xx_{t+1}) - f(\xx_0) &\leq - \sum_{i=0}^t \frac{g_i}{2}\min\left\{\frac{g_i}{C}, 1\right\} \\ Lemma 1: Key recursive inequality. \end{equation} The rest of the algorithm mostly concerns finding the appropriate step size to move in the direction dictated by the linear minimization oracle $\dd_t = \ss_t - \xx_t$, where $\ss_t$ is the result of the linear minimization oracle. &\quad\boldsymbol{x}_{t+1} = \boldsymbol{x}_t + \gamma_t \boldsymbol{d}_t~.\label{eq:update_rule}\\ "Greedy Algorithms, Frank-Wolfe and Friends", If $g_t \leq C$, then $\xi^* = g_t / C$ and using this value in the previous inequality we obtain the bound ICML 2013 for a more recent exposition. with $C = L \diam(\mathcal{C})^2$. Generalized Matching Pursuit and Frank-Wolfe \begin{align} Compared to a projection, the use of a linear minimization oracle has other important consequences. \gamma_t = \frac{\boldsymbol{q_t}^T (\boldsymbol{b} - \boldsymbol{A} \xx)}{\|\boldsymbol{q}_t\|^2}~, Create Free Account. \eqref{eq:lmo}). \begin{equation} &\quad= A_{t+1}(f(\xx_{t+1}) - f(\xx^\star)) - A_t(f(\xx_t) - f(\xx^\star))\\ &f((1 - \gamma_t^\star)\xx_{t} + \gamma_t^\star \ss_t) \\ The remainder of the section is structured as follows: I first introduce two key definition and technical Lemma, and finally prove the convergence results. A consequence of the Lipschitz gradient assumption on $f$ is that we can upper bound the function $f$ at every point $\yy \in \mathcal{C}$ by the following quadratic:The Lipschitz gradient assumption in fact implies the stronger inequality $|f(\yy) - f(\xx) - \langle \nabla f(\xx), \yy-\xx\rangle|$ $\leq$ $\frac{L}{2}\|\xx - \yy\|^2$, see e.g. The Frank-Wolfe algorithm converges under very mild assumptions. "Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization." \gamma_t^\star = \min\Big\{\frac{ g_t}{L\|\xx_t - \ss_t\|^2}, 1 \Big\}~. ", "Step-size adaptivity in Projection-Free Optimization", Introductory lectures on convex optimization, Complexity bounds for primal-dual methods minimizing the model of objective function, A Unified Optimization View on It instead relies on a routine that solves a linear problem over the domain (Eq. For simplicity I assume that the linear subproblems are solved exactly, but these proofs can easily be extended to consider approximate linear minimization oracles. where in the second inequality we have used Young's inequality $ab \leq \frac{a^2}{2} + \frac{b^2}{2}$ with $a = \sqrt{2 h_0}, b = \sqrt{C}$. f(\xx) - f(\xx^\star) \leq \langle\nabla f(\xx), \xx - \xx^\star\rangle~. \gamma_t = \min\Big\{\frac{g_t}{L\,\diam(\mathcal{C})^2}, 1 \Big\}~,\label{eq:step_size_diam} For simplicity we will stick to the euclidean norm. &\quad {\textbf{Variant 1}}: \text{set step size as} \nonumber\\ In upcoming posts, I will discuss other variants with better convergence properties, such as the Away-steps or pairwise FW algorithm and randomized variants. \eqref{eq:recusive_rhs_final} yields the claimed inequality. where $g_t^* = \min_{0\leq i\leq t} g_i$. Given a data matrix $\boldsymbol{A} \in \RR^{n \times p}$, a target variable $\boldsymbol{b} \in \RR^n$, and a regularization parameter $\alpha$, this is a problem of the form \eqref{eq:fw_objective} with where the last inequality follows from the definition of convexityA differentiable function is said to be convex if $f(\yy) \geq f(\xx) + \langle \nabla f(\xx), \yy - \xx\rangle$ for all $\xx, \yy$ in the domain. I tried to mimic as much as possible the python implementation made in this github repo. 741. Official Blog. \end{align} Then we have the following sequence of inequalities
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