# Coordinate Descent This examples Synthesizes cyclic coordinate descent algorithms, continuing from the previous {doc}`simulation <../examples_simulation/sim_coord_descent>` and {doc}`Analysis <../examples_analysis/ana_coord_descent>` examples. Two types of cyclic coordinate descent algorithms are the considered in this demonstration: 1. Full Knowledge: the entire vector $\nabla f(\beta_k)$ is known, 2. Partial Knowledge: only the actively updated portion $[\nabla f(\beta_k)]^i$. The reference coordinate descent algorithm in the prior examples is ```{math} \begin{align} \beta^{i(k)}_{k+1} = \beta^{i(k)}_{k} -\gamma \ [\nabla f(\beta_k)]^i. \end{align} ``` This reference algorithm has Partial Knowledge. Synthesis is performed to design a $c$-block cyclic coordinate descent algorithm with respect to a function $f \in S_{1, 2}$. Figure [1](#coord-6) plots the upper bounds on the convergence rate $\rho$ as $c$ increases. Algorithms with Partial Knowledge (orange, top) have a higher worst-case convergence rate when compared against algorithms with Full Knowledge (blue, bottom) at $c>1$. The two synthesized algorithms are equal in the trivial case of $c=1$. :::{figure} _static/coord_synth_1_2_dark.png :align: center :class: only-dark :name: coord-6 *Figure 1:* Convergence rate of coordinate descent algorithm v.s. number of blocks ::: :::{figure} _static/coord_synth_1_2_light.png :align: center :class: only-light :name: coord-6 *Figure 1:* Convergence rate of coordinate descent algorithm v.s. number of blocks. ::: ```{literalinclude} ../../../examples/synthesis/syn_coord_descent.m :linenos: true :caption: Code for coordinate descent synthesis at $m=1, L=2, c=6$ :language: matlab :lines: 1-26 ```