  Algorithms to solve variational regularization of ill-posed inverse problems
usually involve operators that depend on a collection of continuous parameters.
When these operators enjoy some (local) regularity, these parameters can be
selected using the so-called Stein Unbiased Risk Estimate (SURE). While this
selection is usually performed by exhaustive search, we address in this work
the problem of using the SURE to efficiently optimize for a collection of
continuous parameters of the model. When considering non-smooth regularizers,
such as the popular l1-norm corresponding to soft-thresholding mapping, the
SURE is a discontinuous function of the parameters preventing the use of
gradient descent optimization techniques. Instead, we focus on an approximation
of the SURE based on finite differences as proposed in (Ramani et al., 2008).
Under mild assumptions on the estimation mapping, we show that this
approximation is a weakly differentiable function of the parameters and its
weak gradient, coined the Stein Unbiased GrAdient estimator of the Risk
(SUGAR), provides an asymptotically (with respect to the data dimension)
unbiased estimate of the gradient of the risk. Moreover, in the particular case
of soft-thresholding, it is proved to be also a consistent estimator. This
gradient estimate can then be used as a basis to perform a quasi-Newton
optimization. The computation of the SUGAR relies on the closed-form (weak)
differentiation of the non-smooth function. We provide its expression for a
large class of iterative methods including proximal splitting ones and apply
our strategy to regularizations involving non-smooth convex structured
penalties. Illustrations on various image restoration and matrix completion
problems are given.
