Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterizing Parametric and Convergence Stability in Nonconvex and
  Nonsmooth Optimizations: A Geometric Approach   We consider stability issues in minimizing a continuous (probably
parameterized, nonconvex and nonsmooth) real-valued function $f$. We call a
point stationary if all its possible directional derivatives are nonnegative.
In this work, we focus on two notions of stability on stationary points of $f$:
parametric stability and convergence stability. Parametric considerations are
widely studied in various fields, including smoothed analysis, numerical
stability, condition numbers and sensitivity analysis for linear programming.
Parametric stability asks whether minor perturbations on parameters lead to
dramatic changes in the position and $f$ value of a stationary point.
Meanwhile, convergence stability indicates a non-escapable solution: Any point
sequence iteratively produced by an optimization algorithm cannot escape from a
neighborhood of a stationary point but gets close to it in the sense that such
stationary points are stable to the precision parameter and algorithmic
numerical errors. It turns out that these notions have deep connections to
geometry theory. We show that parametric stability is linked to deformations of
graphs of functions. On the other hand, convergence stability is concerned with
area partitioning of the function domain. Utilizing these connections, we prove
quite tight conditions of these two stability notions for a wide range of
functions and optimization algorithms with small enough step sizes and
precision parameters. These conditions are subtle in the sense that a slightly
weaker function requirement goes to the opposite of primitive intuitions and
leads to wrong conclusions. We present three applications of this theory. These
applications reveal some understanding on Nash equilibrium computation,
nonconvex and nonsmooth optimization, as well as the new optimization
methodology of deep neural networks.

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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
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The categories of this paper are: stat
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th