\section{Experiments}\label{exp}


\begin{figure}[t]
	\centering
	\begin{minipage}{0.32\textwidth}
		\includegraphics[width=1\textwidth]{fig1}
		\caption*{(a)}
	\end{minipage}
	\begin{minipage}{0.32\textwidth}
		\includegraphics[width=1\textwidth]{fig2}
		\caption*{(b)}
	\end{minipage}
	\begin{minipage}{0.32\textwidth}
		\includegraphics[width=1\textwidth]{fig3}
		\caption*{(c)}
	\end{minipage}
	\caption{Safety, convergence, and comparison of Reliable-FW versus s0-LBM ans 0-LBM.}
	\label{t}
\end{figure}

We consider a similar non-convex optimization problem  used in \cite{LB} which models a cutting machine from \cite{num}. Consider the following problem
\begin{align*}
	\min_{(x,y)\in\mathbb{R}^2}\quad \frac{22}{xy}\left(50+\frac{40}{h_1(x,y)}\right),\qquad 
	\text{s.t.} \quad h_2(x,y)\leqslant 0,\,\,100\leqslant x\leqslant 200,\,\,0.08\leqslant y\leqslant 0.16,
\end{align*}
where, $h_1(x,y) = 127.5365-0.84629x-144.21 y+0.001703 x^{2}+0.3656 xy$ and $h_2(x,y)=0.0844-0.010035x+7.0877y$. We set $\sigma_0=0.001$ and $r_0 = 0.01$. For 0-LBM and s0-LBM, we used $\eta_t = 0.1, L = 7, M = 5$, and $\delta = 0.01$. We used $x_0 = [150,\,0.09]^{\top}$ and $\sigma = 0.01$ for Figure~\ref{t} (a) and (b) and $x_0 = [130, 0.09]^{\top}$ and $\sigma=0.06$ for Figure~\ref{t} (c).

Figure~\ref{t} (a) shows function-value in terms of NFO calls for Reliable-FW versus s0-LBM. Note that s0-LBM uses zeroth-order objective function queries opposite to first-order queries in Reliable-FW. Hence, for a fair comparison and to ensure that the performance difference is not affected by the choice of gradient estimator, we apply the (stochastic) gradient estimator of s0L-BM to Reliable-FW and compare only those function calls that have not been used for a gradient approximation.\\
In Figure~\ref{t} (b), function value with respect to objective function calls for Reliable-FW vs 0-LBM have been plotted. Similar to the previous experiment, we use the same gradient estimation for both methods. 
Note that since the Frank-Wolfe step in Reliable-FW is linear optimization of an stochastic vector over a stochastic set, our update is inherently random which result in some oscillation of function-value specially after reaching a local optimum which can also be seen in Figure~\ref{t} (a) and (b).
Figure~\ref{t} (c) illustrates the convergence and safety (feasibility) of iterates of Reliable-FW (marked with "$\times$"), where the original feasible set is depicted in solid and its corresponding estimations during the algorithm are shown in dashed lines.
{"config": "arxiv", "file": "2006.13326/Experiments.tex"}