7,267 research outputs found
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
Capacity Bounds for Broadcast Channels with Confidential Messages
In this paper, we study capacity bounds for discrete memoryless broadcast
channels with confidential messages. Two private messages as well as a common
message are transmitted; the common message is to be decoded by both receivers,
while each private message is only for its intended receiver. In addition, each
private message is to be kept secret from the unintended receiver where secrecy
is measured by equivocation. We propose both inner and outer bounds to the rate
equivocation region for broadcast channels with confidential messages. The
proposed inner bound generalizes Csisz\'{a}r and K\"{o}rner's rate equivocation
region for broadcast channels with a single confidential message, Liu {\em et
al}'s achievable rate region for broadcast channels with perfect secrecy,
Marton's and Gel'fand and Pinsker's achievable rate region for general
broadcast channels. Our proposed outer bounds, together with the inner bound,
helps establish the rate equivocation region of several classes of discrete
memoryless broadcast channels with confidential messages, including less noisy,
deterministic, and semi-deterministic channels. Furthermore, specializing to
the general broadcast channel by removing the confidentiality constraint, our
proposed outer bounds reduce to new capacity outer bounds for the discrete
memory broadcast channel.Comment: 27 pages, 1 figure, submitted to IEEE Transaction on Information
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