  We consider a stochastic gradient descent (SGD) algorithm for solving linear
inverse problems (e.g., CT image reconstruction) in the Banach space framework
of variable exponent Lebesgue spaces $\ell^{(p_n)}(\mathbb{R})$. Such
non-standard spaces have been recently proved to be the appropriate functional
framework to enforce pixel-adaptive regularisation in signal and image
processing applications. Compared to its use in Hilbert settings, however, the
application of SGD in the Banach setting of $\ell^{(p_n)}(\mathbb{R})$ is not
straightforward, due, in particular to the lack of a closed-form expression and
the non-separability property of the underlying norm. In this manuscript, we
show that SGD iterations can effectively be performed using the associated
modular function. Numerical validation on both simulated and real CT data show
significant improvements in comparison to SGD solutions both in Hilbert and
other Banach settings, in particular when non-Gaussian or mixed noise is
observed in the data.
