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I am interested in a class of optimization problems of which we know that the input variable is first subjected to noise $\xi$ before entering the data-producing process $f$.

I write the objective in probability, e.g. $x^* = \underset{x \in X}{\arg } \{ P[\partial f(x + \xi)^\intercal w \leq \epsilon_1 ] \geq 1 - \epsilon_2\}$ where $X \subset \mathbb{R}^d$ and $\epsilon_i$ and $w$ are constant w.r.t $x$.

Notes :

  • $f$ is not known explicitly but is continuous and quite regular, so we are able to compute subgradients for any realization of $\xi$. I am pretty sure it is not convex over all $X$.

  • We do not know the distribution of $\xi$ and it might have a dependency on $x$.

Question :

  • I've seen many works in stochastic approximation dealing with additive noise (noisy zeroth and first- order oracles); is there any relevant literature on "noisy control variables" ? I would be very grateful for any pointers, especially to approximation algorithms.

Thank you in advance

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