7,302 research outputs found
Herding as a Learning System with Edge-of-Chaos Dynamics
Herding defines a deterministic dynamical system at the edge of chaos. It
generates a sequence of model states and parameters by alternating parameter
perturbations with state maximizations, where the sequence of states can be
interpreted as "samples" from an associated MRF model. Herding differs from
maximum likelihood estimation in that the sequence of parameters does not
converge to a fixed point and differs from an MCMC posterior sampling approach
in that the sequence of states is generated deterministically. Herding may be
interpreted as a"perturb and map" method where the parameter perturbations are
generated using a deterministic nonlinear dynamical system rather than randomly
from a Gumbel distribution. This chapter studies the distinct statistical
characteristics of the herding algorithm and shows that the fast convergence
rate of the controlled moments may be attributed to edge of chaos dynamics. The
herding algorithm can also be generalized to models with latent variables and
to a discriminative learning setting. The perceptron cycling theorem ensures
that the fast moment matching property is preserved in the more general
framework
Bayesian Structure Learning for Markov Random Fields with a Spike and Slab Prior
In recent years a number of methods have been developed for automatically
learning the (sparse) connectivity structure of Markov Random Fields. These
methods are mostly based on L1-regularized optimization which has a number of
disadvantages such as the inability to assess model uncertainty and expensive
cross-validation to find the optimal regularization parameter. Moreover, the
model's predictive performance may degrade dramatically with a suboptimal value
of the regularization parameter (which is sometimes desirable to induce
sparseness). We propose a fully Bayesian approach based on a "spike and slab"
prior (similar to L0 regularization) that does not suffer from these
shortcomings. We develop an approximate MCMC method combining Langevin dynamics
and reversible jump MCMC to conduct inference in this model. Experiments show
that the proposed model learns a good combination of the structure and
parameter values without the need for separate hyper-parameter tuning.
Moreover, the model's predictive performance is much more robust than L1-based
methods with hyper-parameter settings that induce highly sparse model
structures.Comment: Accepted in the Conference on Uncertainty in Artificial Intelligence
(UAI), 201
Variational cross-validation of slow dynamical modes in molecular kinetics
Markov state models (MSMs) are a widely used method for approximating the
eigenspectrum of the molecular dynamics propagator, yielding insight into the
long-timescale statistical kinetics and slow dynamical modes of biomolecular
systems. However, the lack of a unified theoretical framework for choosing
between alternative models has hampered progress, especially for non-experts
applying these methods to novel biological systems. Here, we consider
cross-validation with a new objective function for estimators of these slow
dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures
the ability of a rank- projection operator to capture the slow subspace of
the system. It is shown that a variational theorem bounds the GMRQ from above
by the sum of the first eigenvalues of the system's propagator, but that
this bound can be violated when the requisite matrix elements are estimated
subject to statistical uncertainty. This overfitting can be detected and
avoided through cross-validation. These result make it possible to construct
Markov state models for protein dynamics in a way that appropriately captures
the tradeoff between systematic and statistical errors
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