Online Robust Mean Estimation

We study the problem of high-dimensional robust mean estimation in an online setting. Specifically, we consider a scenario where $n$ sensors are measuring some common, ongoing phenomenon. At each time step $t=1,2,\ldots,T$, the $i^{th}$ sensor reports its readings $x^{(i)}_t$ for that time step. The algorithm must then commit to its estimate $\mu_t$ for the true mean value of the process at time $t$. We assume that most of the sensors observe independent samples from some common distribution $X$, but an $\epsilon$-fraction of them may instead behave maliciously. The algorithm wishes to compute a good approximation $\mu$ to the true mean $\mu^\ast := \mathbf{E}[X]$. We note that if the algorithm is allowed to wait until time $T$ to report its estimate, this reduces to the well-studied problem of robust mean estimation. However, the requirement that our algorithm produces partial estimates as the data is coming in substantially complicates the situation. We prove two main results about online robust mean estimation in this model. First, if the uncorrupted samples satisfy the standard condition of $(\epsilon,\delta)$-stability, we give an efficient online algorithm that outputs estimates $\mu_t$, $t \in [T],$ such that with high probability it holds that $\|\mu-\mu^\ast\|_2 = O(\delta \log(T))$, where $\mu = (\mu_t)_{t \in [T]}$. We note that this error bound is nearly competitive with the best offline algorithms, which would achieve $\ell_2$-error of $O(\delta)$. Our second main result shows that with additional assumptions on the input (most notably that $X$ is a product distribution) there are inefficient algorithms whose error does not depend on $T$ at all.