36,330 research outputs found
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
In this paper, we propose to adopt the diffusion approximation tools to study
the dynamics of Oja's iteration which is an online stochastic gradient descent
method for the principal component analysis. Oja's iteration maintains a
running estimate of the true principal component from streaming data and enjoys
less temporal and spatial complexities. We show that the Oja's iteration for
the top eigenvector generates a continuous-state discrete-time Markov chain
over the unit sphere. We characterize the Oja's iteration in three phases using
diffusion approximation and weak convergence tools. Our three-phase analysis
further provides a finite-sample error bound for the running estimate, which
matches the minimax information lower bound for principal component analysis
under the additional assumption of bounded samples.Comment: Appeared in NIPS 201
Practical recommendations for gradient-based training of deep architectures
Learning algorithms related to artificial neural networks and in particular
for Deep Learning may seem to involve many bells and whistles, called
hyper-parameters. This chapter is meant as a practical guide with
recommendations for some of the most commonly used hyper-parameters, in
particular in the context of learning algorithms based on back-propagated
gradient and gradient-based optimization. It also discusses how to deal with
the fact that more interesting results can be obtained when allowing one to
adjust many hyper-parameters. Overall, it describes elements of the practice
used to successfully and efficiently train and debug large-scale and often deep
multi-layer neural networks. It closes with open questions about the training
difficulties observed with deeper architectures
Optimal Statistical Rates for Decentralised Non-Parametric Regression with Linear Speed-Up
We analyse the learning performance of Distributed Gradient Descent in the
context of multi-agent decentralised non-parametric regression with the square
loss function when i.i.d. samples are assigned to agents. We show that if
agents hold sufficiently many samples with respect to the network size, then
Distributed Gradient Descent achieves optimal statistical rates with a number
of iterations that scales, up to a threshold, with the inverse of the spectral
gap of the gossip matrix divided by the number of samples owned by each agent
raised to a problem-dependent power. The presence of the threshold comes from
statistics. It encodes the existence of a "big data" regime where the number of
required iterations does not depend on the network topology. In this regime,
Distributed Gradient Descent achieves optimal statistical rates with the same
order of iterations as gradient descent run with all the samples in the
network. Provided the communication delay is sufficiently small, the
distributed protocol yields a linear speed-up in runtime compared to the
single-machine protocol. This is in contrast to decentralised optimisation
algorithms that do not exploit statistics and only yield a linear speed-up in
graphs where the spectral gap is bounded away from zero. Our results exploit
the statistical concentration of quantities held by agents and shed new light
on the interplay between statistics and communication in decentralised methods.
Bounds are given in the standard non-parametric setting with source/capacity
assumptions
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