Continuation of Nesterov’s Smoothing for Regression with Structured Sparsity in High-Dimensional Neuroimaging
Résumé
redictive models can be used on high-dimensional brain images to decode cognitive states or diagnosis/prognosis
of a clinical condition/evolution. Spatial regularization through
structured sparsity offers new perspectives in this context and
reduces the risk of overfitting the model while providing in-
terpretable neuroimaging signatures by forcing the solution to
adhere to domain-specific constraints. Total Variation (TV) is
a promising candidate for structured penalization: it enforces
spatial smoothness of the solution while segmenting predictive
regions from the background. We consider the problem of
minimizing the sum of a smooth convex loss, a non-smooth convex
penalty (whose proximal operator is known) and a wide range of
possible complex, non-smooth convex structured penalties such as
TV or overlapping group Lasso. Existing solvers are either limited
in the functions they can minimize or in their practical capacity
to scale to high-dimensional imaging data. Nesterov’s smoothing
technique can be used to minimize a large number of non-
smooth convex structured penalties. However, reasonable preci-
sion requires a small smoothing parameter, which slows down
the convergence speed to unacceptable levels. To benefit from the
versatility of Nesterov’s smoothing technique, we propose a first
order continuation algorithm, CONESTA, which automatically
generates a sequence of decreasing smoothing parameters. The
generated sequence maintains the optimal convergence speed
towards any globally desired precision. Our main contributions
are: To propose an expression of the duality gap to probe the
current distance to the global optimum in order to adapt the
smoothing parameter and the convergence speed. This expression
is applicable to many penalties and can be used with other
solvers than CONESTA. We also propose an expression for the
particular smoothing parameter that minimizes the number of
iterations required to reach a given precision. Further, we provide
a convergence proof and its rate, which is an improvement over
classical proximal gradient smoothing methods. We demonstrate
on both simulated and high-dimensional structural neuroimaging
data that CONESTA significantly outperforms many state-of-the-
art solvers in regard to convergence speed and precision.
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