We present a new finite-sample analysis of M-estimators of locations in a Hilbert space using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function (or its score function) and then, we use concentration inequality on M-estimators to investigate the robust estimation of the mean in high dimension in a corrupted setting (adversarial corruption setting) for bounded and unbounded score functions. For a sample of size n and covariance matrix Σ, we attain the minimax speed T r(Σ)/n + Σ op log(1/δ)/n with probability larger than 1 − δ in a heavy-tailed setting. One of the major advantages of our approach compared to others recently proposed is that our estimator is tractable and fast to compute even in very high dimension with a complexity of O(nd log(T r(Σ))) where n is the sample size and Σ is the covariance matrix of the inliers and in the code that we make available for this article is tested to be very fast.
In this paper, we study the stochastic bandits problem with k unknown heavy-tailed and corrupted reward distributions or arms with time-invariant corruption distributions. At each iteration, the player chooses an arm. Given the arm, the environment returns an uncorrupted reward with probability 1−ε and an arbitrarily corrupted reward with probability ε. In our setting, the uncorrupted reward might be heavy-tailed and the corrupted reward might be unbounded. We prove a lower bound on the regret indicating that the corrupted and heavy-tailed bandits are strictly harder than uncorrupted or light-tailed bandits. We observe that the environments can be categorised into hardness regimes depending on the suboptimality gap ∆, variance σ, and corruption proportion . Following this, we design a UCB-type algorithm, namely HuberUCB, that leverages Huber's estimator for robust mean estimation. HuberUCB leads to tight upper bounds on regret in the proposed corrupted and heavy-tailed setting. To derive the upper bound, we prove a novel concentration inequality for Huber's estimator, which might be of independent interest.
This paper investigates robust versions of the general empirical risk minimization algorithm, one of the core techniques underlying modern statistical methods. Success of the empirical risk minimization is based on the fact that for a ‘well-behaved’ stochastic process $\left \{ f(X), \ f\in \mathscr F\right \}$ indexed by a class of functions $f\in \mathscr F$, averages $\frac{1}{N}\sum _{j=1}^N f(X_j)$ evaluated over a sample $X_1,\ldots ,X_N$ of i.i.d. copies of $X$ provide good approximation to the expectations $\mathbb E f(X)$, uniformly over large classes $f\in \mathscr F$. However, this might no longer be true if the marginal distributions of the process are heavy tailed or if the sample contains outliers. We propose a version of empirical risk minimization based on the idea of replacing sample averages by robust proxies of the expectations and obtain high-confidence bounds for the excess risk of resulting estimators. In particular, we show that the excess risk of robust estimators can converge to $0$ at fast rates with respect to the sample size $N$, referring to the rates faster than $N^{-1/2}$. We discuss implications of the main results to the linear and logistic regression problems and evaluate the numerical performance of proposed methods on simulated and real data.
There is a mistake in one of the authors' names (in both online and print versions of the article): it should be Timothée Mathieu instead of Timlothée Mathieu.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.