Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small random initialization is akin to spectral learning: Optimization
  and generalization guarantees for overparameterized low-rank matrix
  reconstruction   Recently there has been significant theoretical progress on understanding the
convergence and generalization of gradient-based methods on nonconvex losses
with overparameterized models. Nevertheless, many aspects of optimization and
generalization and in particular the critical role of small random
initialization are not fully understood. In this paper, we take a step towards
demystifying this role by proving that small random initialization followed by
a few iterations of gradient descent behaves akin to popular spectral methods.
We also show that this implicit spectral bias from small random initialization,
which is provably more prominent for overparameterized models, also puts the
gradient descent iterations on a particular trajectory towards solutions that
are not only globally optimal but also generalize well. Concretely, we focus on
the problem of reconstructing a low-rank matrix from a few measurements via a
natural nonconvex formulation. In this setting, we show that the trajectory of
the gradient descent iterations from small random initialization can be
approximately decomposed into three phases: (I) a spectral or alignment phase
where we show that that the iterates have an implicit spectral bias akin to
spectral initialization allowing us to show that at the end of this phase the
column space of the iterates and the underlying low-rank matrix are
sufficiently aligned, (II) a saddle avoidance/refinement phase where we show
that the trajectory of the gradient iterates moves away from certain degenerate
saddle points, and (III) a local refinement phase where we show that after
avoiding the saddles the iterates converge quickly to the underlying low-rank
matrix. Underlying our analysis are insights for the analysis of
overparameterized nonconvex optimization schemes that may have implications for
computational problems beyond low-rank reconstruction.

31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg