Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projection Efficient Subgradient Method and Optimal Nonsmooth
  Frank-Wolfe Method   We consider the classical setting of optimizing a nonsmooth Lipschitz
continuous convex function over a convex constraint set, when having access to
a (stochastic) first-order oracle (FO) for the function and a projection oracle
(PO) for the constraint set. It is well known that to achieve
$\epsilon$-suboptimality in high-dimensions, $\Theta(\epsilon^{-2})$ FO calls
are necessary. This is achieved by the projected subgradient method (PGD).
However, PGD also entails $O(\epsilon^{-2})$ PO calls, which may be
computationally costlier than FO calls (e.g. nuclear norm constraints).
Improving this PO calls complexity of PGD is largely unexplored, despite the
fundamental nature of this problem and extensive literature. We present first
such improvement. This only requires a mild assumption that the objective
function, when extended to a slightly larger neighborhood of the constraint
set, still remains Lipschitz and accessible via FO. In particular, we introduce
MOPES method, which carefully combines Moreau-Yosida smoothing and accelerated
first-order schemes. This is guaranteed to find a feasible
$\epsilon$-suboptimal solution using only $O(\epsilon^{-1})$ PO calls and
optimal $O(\epsilon^{-2})$ FO calls. Further, instead of a PO if we only have a
linear minimization oracle (LMO, a la Frank-Wolfe) to access the constraint
set, an extension of our method, MOLES, finds a feasible $\epsilon$-suboptimal
solution using $O(\epsilon^{-2})$ LMO calls and FO calls---both match known
lower bounds, resolving a question left open since White (1993). Our
experiments confirm that these methods achieve significant speedups over the
state-of-the-art, for a problem with costly PO and LMO calls.

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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th