  Achieving accurate approximations to solutions of large linear systems is
crucial, especially when those systems utilize real-world data. A consequence
of using real-world data is that there will inevitably be missingness. Current
approaches for dealing with missing data, such as deletion and imputation, can
introduce bias. Recent studies proposed an adaptation of stochastic gradient
descent (SGD) in specific missing-data models. In this work, we propose a new
algorithm, $\ell$-tuple mSGD, for the setting in which data is missing in a
block-wise, tuple pattern. We prove that our proposed method uses unbiased
estimates of the gradient of the least squares objective in the presence of
tuple missing data. We also draw connections between $\ell$-tuple mSGD and
previously established SGD-type methods for missing data. Furthermore, we prove
our algorithm converges when using updating step sizes and empirically
demonstrate the convergence of $\ell$-tuple mSGD on synthetic data. Lastly, we
evaluate $\ell$-tuple mSGD applied to real-world continuous glucose monitoring
(CGM) device data.
