As society grows more reliant on machine learning, ensuring the security of machine learning systems against sophisticated attacks becomes a pressing concern. A recent result of Goldwasser, Kim, Vaikuntanathan, and Zamir (2022) shows that an adversary can plant undetectable backdoors in machine learning models, allowing the adversary to covertly control the model's behavior. Backdoors can be planted in such a way that the backdoored machine learning model is computationally indistinguishable from an honest model without backdoors. In this paper, we present strategies for defending against backdoors in ML models, even if they are undetectable. The key observation is that it is sometimes possible to provably mitigate or even remove backdoors without needing to detect them, using techniques inspired by the notion of random self-reducibility. This depends on properties of the ground-truth labels (chosen by nature), and not of the proposed ML model (which may be chosen by an attacker). We give formal definitions for secure backdoor mitigation, and proceed to show two types of results. First, we show a "global mitigation" technique, which removes all backdoors from a machine learning model under the assumption that the ground-truth labels are close to a Fourier-heavy function. Second, we consider distributions where the ground-truth labels are close to a linear or polynomial function in $\mathbb{R}^n$. Here, we show "local mitigation" techniques, which remove backdoors with high probability for every inputs of interest, and are computationally cheaper than global mitigation. All of our constructions are black-box, so our techniques work without needing access to the model's representation (i.e., its code or parameters). Along the way we prove a simple result for robust mean estimation.

Sparse linear regression (SLR) is a well-studied problem in statistics where one is given a design matrix $X\in\mathbb{R}^{m\times n}$ and a response vector $y=X\theta^*+w$ for a $k$-sparse vector $\theta^*$ (that is, $\|\theta^*\|_0\leq k$) and small, arbitrary noise $w$, and the goal is to find a $k$-sparse $\widehat{\theta} \in \mathbb{R}^n$ that minimizes the mean squared prediction error $\frac{1}{m}\|X\widehat{\theta}-X\theta^*\|^2_2$. While $\ell_1$-relaxation methods such as basis pursuit, Lasso, and the Dantzig selector solve SLR when the design matrix is well-conditioned, no general algorithm is known, nor is there any formal evidence of hardness in an average-case setting with respect to all efficient algorithms. We give evidence of average-case hardness of SLR w.r.t. all efficient algorithms assuming the worst-case hardness of lattice problems. Specifically, we give an instance-by-instance reduction from a variant of the bounded distance decoding (BDD) problem on lattices to SLR, where the condition number of the lattice basis that defines the BDD instance is directly related to the restricted eigenvalue condition of the design matrix, which characterizes some of the classical statistical-computational gaps for sparse linear regression. Also, by appealing to worst-case to average-case reductions from the world of lattices, this shows hardness for a distribution of SLR instances; while the design matrices are ill-conditioned, the resulting SLR instances are in the identifiable regime. Furthermore, for well-conditioned (essentially) isotropic Gaussian design matrices, where Lasso is known to behave well in the identifiable regime, we show hardness of outputting any good solution in the unidentifiable regime where there are many solutions, assuming the worst-case hardness of standard and well-studied lattice problems.