2,741 research outputs found
On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation
In this paper we address the stable numerical solution of nonlinear ill-posed
systems by a trust-region method. We show that an appropriate choice of the
trust-region radius gives rise to a procedure that has the potential to
approach a solution of the unperturbed system. This regularizing property is
shown theoretically and validated numerically.Comment: arXiv admin note: text overlap with arXiv:1410.278
Adaptive hybrid optimization strategy for calibration and parameter estimation of physical models
A new adaptive hybrid optimization strategy, entitled squads, is proposed for
complex inverse analysis of computationally intensive physical models. The new
strategy is designed to be computationally efficient and robust in
identification of the global optimum (e.g. maximum or minimum value of an
objective function). It integrates a global Adaptive Particle Swarm
Optimization (APSO) strategy with a local Levenberg-Marquardt (LM) optimization
strategy using adaptive rules based on runtime performance. The global strategy
optimizes the location of a set of solutions (particles) in the parameter
space. The LM strategy is applied only to a subset of the particles at
different stages of the optimization based on the adaptive rules. After the LM
adjustment of the subset of particle positions, the updated particles are
returned to the APSO strategy. The advantages of coupling APSO and LM in the
manner implemented in squads is demonstrated by comparisons of squads
performance against Levenberg-Marquardt (LM), Particle Swarm Optimization
(PSO), Adaptive Particle Swarm Optimization (APSO; the TRIBES strategy), and an
existing hybrid optimization strategy (hPSO). All the strategies are tested on
2D, 5D and 10D Rosenbrock and Griewank polynomial test functions and a
synthetic hydrogeologic application to identify the source of a contaminant
plume in an aquifer. Tests are performed using a series of runs with random
initial guesses for the estimated (function/model) parameters. Squads is
observed to have the best performance when both robustness and efficiency are
taken into consideration than the other strategies for all test functions and
the hydrogeologic application
The geometry of nonlinear least squares with applications to sloppy models and optimization
Parameter estimation by nonlinear least squares minimization is a common
problem with an elegant geometric interpretation: the possible parameter values
of a model induce a manifold in the space of data predictions. The minimization
problem is then to find the point on the manifold closest to the data. We show
that the model manifolds of a large class of models, known as sloppy models,
have many universal features; they are characterized by a geometric series of
widths, extrinsic curvatures, and parameter-effects curvatures. A number of
common difficulties in optimizing least squares problems are due to this common
structure. First, algorithms tend to run into the boundaries of the model
manifold, causing parameters to diverge or become unphysical. We introduce the
model graph as an extension of the model manifold to remedy this problem. We
argue that appropriate priors can remove the boundaries and improve convergence
rates. We show that typical fits will have many evaporated parameters. Second,
bare model parameters are usually ill-suited to describing model behavior; cost
contours in parameter space tend to form hierarchies of plateaus and canyons.
Geometrically, we understand this inconvenient parametrization as an extremely
skewed coordinate basis and show that it induces a large parameter-effects
curvature on the manifold. Using coordinates based on geodesic motion, these
narrow canyons are transformed in many cases into a single quadratic, isotropic
basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting
algorithms as an Euler approximation to geodesic motion in these natural
coordinates on the model manifold and the model graph respectively. By adding a
geodesic acceleration adjustment to these algorithms, we alleviate the
difficulties from parameter-effects curvature, improving both efficiency and
success rates at finding good fits.Comment: 40 pages, 29 Figure
Do optimization methods in deep learning applications matter?
With advances in deep learning, exponential data growth and increasing model
complexity, developing efficient optimization methods are attracting much
research attention. Several implementations favor the use of Conjugate Gradient
(CG) and Stochastic Gradient Descent (SGD) as being practical and elegant
solutions to achieve quick convergence, however, these optimization processes
also present many limitations in learning across deep learning applications.
Recent research is exploring higher-order optimization functions as better
approaches, but these present very complex computational challenges for
practical use. Comparing first and higher-order optimization functions, in this
paper, our experiments reveal that Levemberg-Marquardt (LM) significantly
supersedes optimal convergence but suffers from very large processing time
increasing the training complexity of both, classification and reinforcement
learning problems. Our experiments compare off-the-shelf optimization
functions(CG, SGD, LM and L-BFGS) in standard CIFAR, MNIST, CartPole and
FlappyBird experiments.The paper presents arguments on which optimization
functions to use and further, which functions would benefit from
parallelization efforts to improve pretraining time and learning rate
convergence
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