91,144 research outputs found
Non-stationary Stochastic Optimization
We consider a non-stationary variant of a sequential stochastic optimization
problem, in which the underlying cost functions may change along the horizon.
We propose a measure, termed variation budget, that controls the extent of said
change, and study how restrictions on this budget impact achievable
performance. We identify sharp conditions under which it is possible to achieve
long-run-average optimality and more refined performance measures such as rate
optimality that fully characterize the complexity of such problems. In doing
so, we also establish a strong connection between two rather disparate strands
of literature: adversarial online convex optimization; and the more traditional
stochastic approximation paradigm (couched in a non-stationary setting). This
connection is the key to deriving well performing policies in the latter, by
leveraging structure of optimal policies in the former. Finally, tight bounds
on the minimax regret allow us to quantify the "price of non-stationarity,"
which mathematically captures the added complexity embedded in a temporally
changing environment versus a stationary one
Non-stationary stochastic optimization of an oscillating water column
A non-stationary stochastic optimization methodology
is applied to an OWC (oscillating water column) to find the design
that maximizes the wave energy extraction. Different temporal cycles
are considered to represent the long-term variability of the wave
climate at the site in the optimization problem. The results of the
non-stationary stochastic optimization problem are compared against
those obtained by a stationary stochastic optimization problem. The
comparative analysis reveals that the proposed non-stationary
optimization provides designs with a better fit to reality. However,
the stationarity assumption can be adequate when looking at averaged
system response
On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Optimization
Extrapolation is a well-known technique for solving convex optimization and
variational inequalities and recently attracts some attention for non-convex
optimization. Several recent works have empirically shown its success in some
machine learning tasks. However, it has not been analyzed for non-convex
minimization and there still remains a gap between the theory and the practice.
In this paper, we analyze gradient descent and stochastic gradient descent with
extrapolation for finding an approximate first-order stationary point in smooth
non-convex optimization problems. Our convergence upper bounds show that the
algorithms with extrapolation can be accelerated than without extrapolation
Variance Reduction for Faster Non-Convex Optimization
We consider the fundamental problem in non-convex optimization of efficiently
reaching a stationary point. In contrast to the convex case, in the long
history of this basic problem, the only known theoretical results on
first-order non-convex optimization remain to be full gradient descent that
converges in iterations for smooth objectives, and
stochastic gradient descent that converges in iterations
for objectives that are sum of smooth functions.
We provide the first improvement in this line of research. Our result is
based on the variance reduction trick recently introduced to convex
optimization, as well as a brand new analysis of variance reduction that is
suitable for non-convex optimization. For objectives that are sum of smooth
functions, our first-order minibatch stochastic method converges with an
rate, and is faster than full gradient descent by
.
We demonstrate the effectiveness of our methods on empirical risk
minimizations with non-convex loss functions and training neural nets.Comment: polished writin
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