Vishal Halder

Nibedita Roy, Vishal Halder, Gugan Thoppe et al.

We study mean estimation of a random vector $X$ in a distributed parameter-server-worker setup. Worker $i$ observes samples of $a_i^\top X$, where $a_i^\top$ is the $i$th row of a known sensing matrix $A$. The key challenges are adversarial measurements and asynchrony: a fixed subset of workers may transmit corrupted measurements, and workers are activated asynchronously--only one is active at any time. In our previous work, we proposed a two-timescale $\ell_1$-minimization algorithm and established asymptotic recovery under a null-space-property-like condition on $A$. In this work, we establish tight non-asymptotic convergence rates under the same null-space-property-like condition. We also identify relaxed conditions on $A$ under which exact recovery may fail but recovery of a projected component of $\mathbb{E}[X]$ remains possible. Overall, our results provide a unified finite-time characterization of robustness, identifiability, and statistical efficiency in distributed linear estimation with adversarial workers, with implications for network tomography and related distributed sensing problems.

Vishal Halder, Alexandre Reiffers-Masson, Abdeldjalil Aïssa-El-Bey et al.

Let $A \in \mathbb{R}^{m \times n}$ be an arbitrary, known matrix and $e$ a $q$-sparse adversarial vector. Given $y = A x^\star + e$ and $q$, we seek the smallest set containing $x^\star$ -- hence the one conveying maximal information about $x^\star$ -- that is uniformly recoverable from $y$ without knowing $e$. While exact recovery of $x^\star$ via strong (and often impractical) structural assumptions on $A$ or $x^\star$ (e.g., restricted isometry, sparsity) is well studied, recoverability for arbitrary $A$ and $x^\star$ remains open. Our main result shows that the best that one can hope to recover is $x^\star + \ker(U)$, where $U$ is the unique projection matrix onto the intersection of rowspaces of all possible submatrices of $A$ obtained by deleting $2q$ rows. Moreover, we prove that every $x$ that minimizes the $\ell_0$-norm of $y - A x$ lies in $x^\star + \ker(U)$, which then gives a constructive approach to recover this set.