  In many high-dimensional estimation problems the main task consists in
minimizing a cost function, which is often strongly non-convex when scanned in
the space of parameters to be estimated. A standard solution to flatten the
corresponding rough landscape consists in summing the losses associated to
different data points and obtain a smoother empirical risk. Here we propose a
complementary method that works for a single data point. The main idea is that
a large amount of the roughness is uncorrelated in different parts of the
landscape. One can then substantially reduce the noise by evaluating an
empirical average of the gradient obtained as a sum over many random
independent positions in the space of parameters to be optimized. We present an
algorithm, called Averaged Gradient Descent, based on this idea and we apply it
to tensor PCA, which is a very hard estimation problem. We show that Averaged
Gradient Descent over-performs physical algorithms such as gradient descent and
approximate message passing and matches the best algorithmic thresholds known
so far, obtained by tensor unfolding and methods based on sum-of-squares.
