Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear gradient mappings and stochastic optimization: A general
  framework with applications to heavy-tail noise   We introduce a general framework for nonlinear stochastic gradient descent
(SGD) for the scenarios when gradient noise exhibits heavy tails. The proposed
framework subsumes several popular nonlinearity choices, like clipped,
normalized, signed or quantized gradient, but we also consider novel
nonlinearity choices. We establish for the considered class of methods strong
convergence guarantees assuming a strongly convex cost function with Lipschitz
continuous gradients under very general assumptions on the gradient noise. Most
notably, we show that, for a nonlinearity with bounded outputs and for the
gradient noise that may not have finite moments of order greater than one, the
nonlinear SGD's mean squared error (MSE), or equivalently, the expected cost
function's optimality gap, converges to zero at rate~$O(1/t^\zeta)$, $\zeta \in
(0,1)$. In contrast, for the same noise setting, the linear SGD generates a
sequence with unbounded variances. Furthermore, for the nonlinearities that can
be decoupled component wise, like, e.g., sign gradient or component-wise
clipping, we show that the nonlinear SGD asymptotically (locally) achieves a
$O(1/t)$ rate in the weak convergence sense and explicitly quantify the
corresponding asymptotic variance. Experiments show that, while our framework
is more general than existing studies of SGD under heavy-tail noise, several
easy-to-implement nonlinearities from our framework are competitive with state
of the art alternatives on real data sets with heavy tail noises.

14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat