# Stochastic Gradient Noise This synthesis example continues {doc}`Analysis with Stochastic Gradient Noise <../examples_analysis/ana_stoch_noise_pgd>`. An algorithm to solve a composite optimization problem $\min_{\beta \in \R^d} f(\beta) + g(\beta)$ must be synthesized with ordering $(\partial g, \nabla f)$. The Operators and Network in the System are ```{math} \begin{align*} \mat{c}{z_k^1 \\ z_k^2 \\ z_{p, k} \\ y_k^1 \\ y_k^2} &= \mat{cc:c:cc}{ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \hdl 0 & 0 & 0 & 0 & I \hdl I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ } \mat{c}{w_k^1 \\ w_k^2 \\ w_{p, k} \\ u_k^1 \\ u_k^2}, & \mat{c}{w_k^1 \\ w_k^2} \in \mat{c}{\partial g_(z_k^1) \\ \nabla f(z_k^2)}. \end{align*} ``` Synthesis is performed with parameters $m=1, L=10, \Omega = d I$. The controller is synthesized with the settings: 1. $\nabla f$ must be evaluated explicitly, 2. The worst-case convergence rate must satisfies $\rho < 0.9$, 3. The stochastic sensitivity should be minimized. The output of Synthesis is a controller with description ```{math} \begin{align*} \mat{c}{x_{k+1}^1 \\ x_{k+1}^2 \hl u_k^1 \\ u_k^2} &= \left[\mat{cc|cc}{ 1 & -0.0865 & -0.1815 & -0.0950 \\ 0 & 0.0001 & -0.9049 & 0.0950 \hl 1& -0.8144 & -0.8144 & 0 \\ 1 & -0.0865 & -0.0865 & 0 } \otimes I_d \right] \mat{c}{x_{k+1}^1 \\ x_{k+1}^2 \hl y_k^1 \\ y_k^2}, & \mat{c}{w_k^1 \\ w_k^2} \in \mat{c}{\partial g_(z_k^1) \\ \nabla f(z_k^2)}. \end{align*} ``` The controller is used to solve a composite optimization problem in which $f$ is a quadratic and $g$ is the indicator function of an $L_1$ ball. The gradient noise is i.i.d. normally distributed with covariance $100 I$. Figure [1](#long-time) plots the squared error $\norm{z_{p, k}}_2^2$ for 2000 time steps. The empirical mean (low dotted red line) with value $150.62$ is computed by averaging the squared error from times 200 to 2000. The mean bound (high dotted gray line) is computed by $d (\text{stdev}^2) \text{sensitivity}^2 = 40 (10^2) (0.3082)^2 = 380.0048$. :::{figure} _static/empirical_mean_h2_dark.png :align: center :class: only-dark :name: long-time *Figure 1* Long-time mean square error bounds ::: :::{figure} _static/empirical_mean_h2_light.png :align: center :class: only-light :name: long-time *Figure 1:* Long-time mean square error bounds ::: Figure [2](#noisy-exec) plots signals of the in the first 100 iterations of algorithm execution. :::{figure} _static/syn_h2_signals_dark.png :align: center :class: only-dark :name: noisy-exec *Figure 2* Signals in the noisy algorithm execution ::: :::{figure} _static/syn_h2_signals_light.png :align: center :class: only-light :name: noisy-exec *Figure 2:* Signals in the noisy algorithm execution ::: Figure [3](#noisy-track) plots the tracking errors. After an initial transient, the tracking errors settle to a noise floor. :::{figure} _static/err_bounded_h2_dark.png :align: center :class: only-dark :name: noisy-track *Figure 3* Tracking bounds in the noisy algorithm execution ::: :::{figure} _static/err_bounded_h2_light.png :align: center :class: only-light :name: noisy-track *Figure 3:* Tracking bounds in the noisy algorithm execution ::: ```{literalinclude} ../../../examples/analysis/syn_quad_h2_single.m :linenos: true :caption: Code for Synthesis with stochastic gradient noise :language: matlab ```