Log Barriers for Safe Non-convex Black-box Optimization

We address the problem of minimizing a smooth function $f^0(x)$ over a compact set $D$ defined by smooth functional constraints $f^i(x)\leq 0,~ i = 1,\ldots, m$ given noisy value measurements of $f^i(x)$. This problem arises in safety-critical applications, where certain parameters need to be adapted online in a data-driven fashion, such as in personalized medicine, robotics, manufacturing, etc. In such cases, it is important to ensure constraints are not violated while taking measurements and seeking the minimum of the cost function. We propose a new algorithm s0-LBM, which provides provably feasible iterates with high probability and applies to the challenging case of uncertain zero-th order oracle. We also analyze the convergence rate of the algorithm, and empirically demonstrate its effectiveness.