16,287 research outputs found
Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent
First-order methods play a central role in large-scale machine learning. Even
though many variations exist, each suited to a particular problem, almost all
such methods fundamentally rely on two types of algorithmic steps: gradient
descent, which yields primal progress, and mirror descent, which yields dual
progress.
We observe that the performances of gradient and mirror descent are
complementary, so that faster algorithms can be designed by LINEARLY COUPLING
the two. We show how to reconstruct Nesterov's accelerated gradient methods
using linear coupling, which gives a cleaner interpretation than Nesterov's
original proofs. We also discuss the power of linear coupling by extending it
to many other settings that Nesterov's methods cannot apply to.Comment: A new section added; polished writin
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
Knightian Analysis of the Vickrey Mechanism
We analyze the Vickrey mechanism for auctions of multiple identical goods
when the players have both Knightian uncertainty over their own valuations and
incomplete preferences. In this model, the Vickrey mechanism is no longer
dominant-strategy, and we prove that all dominant-strategy mechanisms are
inadequate. However, we also prove that, in undominated strategies, the social
welfare produced by the Vickrey mechanism in the worst case is not only very
good, but also essentially optimal.Comment: To appear in Econometric
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