Achievable distributional robustness when the robust risk is only partially identified

In safety-critical applications, machine learning models should generalize well under worst-case distribution shifts, that is, have a small robust risk. Invariance-based algorithms can provably take advantage of structural assumptions on the shifts when the training distributions are heterogeneous enough to identify the robust risk. However, in practice, such identifiability conditions are rarely satisfied -- a scenario so far underexplored in the theoretical literature. In this paper, we aim to fill the gap and propose to study the more general setting when the robust risk is only partially identifiable. In particular, we introduce the worst-case robust risk as a new measure of robustness that is always well-defined regardless of identifiability. Its minimum corresponds to an algorithm-independent (population) minimax quantity that measures the best achievable robustness under partial identifiability. While these concepts can be defined more broadly, in this paper we introduce and derive them explicitly for a linear model for concreteness of the presentation. First, we show that existing robustness methods are provably suboptimal in the partially identifiable case. We then evaluate these methods and the minimizer of the (empirical) worst-case robust risk on real-world gene expression data and find a similar trend: the test error of existing robustness methods grows increasingly suboptimal as the fraction of data from unseen environments increases, whereas accounting for partial identifiability allows for better generalization.

How robust is randomized blind deconvolution via nuclear norm minimization against adversarial noise?

In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behavior of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.