GBLassoFV: Generalized Bayesian Lasso with fixed variance
In SafeBayes: Generalized and Safe-Bayesian Ridge and Lasso Regression
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
The function GBLassoFV (Generalized Bayesian Lasso with Fixed Variance) provides a Gibbs sampler to sample the posterior of generalized Bayesian lasso regression models with fixed variance and with learning rate eta.
Usage
1
2
Arguments
y
Vector of outcome variables, numeric, NA allowed, length n.
X
Design matrix, numeric, dimension n x p, n >= 2.
sigma2
Fixed variance parameter sigma^2, numeric. Default NULL, in which case the variance will be estimated from the data.
eta
Learning rate eta, numeric, 0 <= eta <= 1. Default 1.
prior
List containing the following elements
prior$varE: prior for the variance parameter σ^2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chi-square distribution. Default (0,0).
prior$lambda: prior for the penalty parameter lambda with three items.
$value Initial value for lambda. Default 50
$type Can be ‘fixed’: initial value is used as fixed penalty parameter or ‘random’, in which case a prior for lambda is specified. Default ‘random’.
For a Gamma prior on lambda^2: $shape for shape parameter and $rate for the rate parameter; for a Beta prior on lambda: $shape1, $shape2 and $max for lambda proportional to Beta(lambda/max, shape1, shape2). Default: Gamma(0,0).
nIter
Number of iterations, integer. Default 1100.
burnIn
Number of iterations for burn-in, integer. Default 100.
thin
Number of iterations for thinning, integer. Default 10.
minAbsBeta
Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 10^-9.
weights
Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1.
pIter
Print iterations, logical. Default TRUE.
Details
Details on the generalized Bayesian lasso can be found in Chapter 2 of (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009).
Several authors have brought forward the idea of equipping Bayesian updating with a learning rate eta, resulting in an eta-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the eta-generalized posterior, the likelihood is raised to the power eta in order to trade the relative weight of the likelihood and the prior, where eta = 1 corresponds to standard Bayes.
Value
$y
Vector of original outcome variables.
$weights
Vector of weights.
$mu
Posterior mean of the intercept.
$varE
Posterior mean of of the variance.
$yHat
Posterior mean of mu + X*beta + epsilon.
$SD.yHat
Corresponding standard deviation.
$whichNa
Vector with indices of missing values of y.
$fit$pD
Estimated number of effective parameters.
$fit$DIC
Deviance Information Criterion.
$lambda
Posterior mean of lambda.
$bL
Posterior mean of beta.
$SD.bL
Corresponding standard deviation.
$tau2
Posterior mean of tau^2.
$prior
List containing the priors used.
$nIter
Number of iterations.
$burnIn
Number of iterations for burn-in.
$thin
Number of iterations for thinning.
$eta
Learning rate eta.
Author(s)
R. de Heide
References
de Heide, R. 2016.
The Safe-Bayesian Lasso. Master Thesis, Leiden University.
de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009.
Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375-385.
Audibert, J.Y. 2013.
Bayesian generalized double pareto shrinkage. Statistica Sinica 23 119-143.
Catoni, O. 2007.
PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS.
Grunwald, P.D. 2012.
chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings. 169-183. Springer Berlin Heidelberg
McAllester, D. 2003.
PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21
Vovk, V.G. 1990.
Aggregating strategies. In Proc. COLT 90 371-383
Zhang, T. 2004.
Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press 1149-1156
Examples
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rm(list=ls())
library(SafeBayes)
# Simulate data
x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1
y <- NULL
# for each x, a y that is 0 + Gaussian noise
for (i in 1:100) {
y[i] <- 0 + rnorm(1, mean=0, sd=1/4)
}
# Now sample 100 zero's and ones (coin toss)
cointoss <- sample(0:1, 100, replace=TRUE)
# indices of the ones
indices <- which(cointoss==1)
# we replace x and y with (0,0) for the indices the cointoss
# landed tail (1)
x[indices] <- 0
y[indices] <- 0
plot(x,y)
# Determine the generalized posterior for eta = 0.25
obj <- GBLassoFV(y, x, eta=0.25)
# posterior means of the coefficients beta and intercept mu
betafour <- obj$bL
mufour <- obj$mu
Example output
No prior was provided, improper priors are used.
Sigma^2 is not provided and will be estimated from the dataIter: 1 time/iter: 0 varE: 0.022 lambda: 13.456
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Iter: 2 time/iter: 0 varE: 0.022 lambda: 11.269
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Iter: 3 time/iter: 0 varE: 0.022 lambda: 4.97
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Iter: 4 time/iter: 0.001 varE: 0.022 lambda: 2.81
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Iter: 5 time/iter: 0 varE: 0.022 lambda: 4.461
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Iter: 6 time/iter: 0 varE: 0.022 lambda: 2.743
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Iter: 7 time/iter: 0 varE: 0.022 lambda: 1.61
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Iter: 8 time/iter: 0 varE: 0.022 lambda: 1.181
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Iter: 9 time/iter: 0 varE: 0.022 lambda: 1.237
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Iter: 10 time/iter: 0 varE: 0.022 lambda: 12.75
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Iter: 11 time/iter: 0.001 varE: 0.022 lambda: 4.06
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Iter: 12 time/iter: 0 varE: 0.022 lambda: 15.567
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Iter: 13 time/iter: 0 varE: 0.022 lambda: 15.848
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Iter: 14 time/iter: 0 varE: 0.022 lambda: 10.184
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Iter: 15 time/iter: 0 varE: 0.022 lambda: 8.517
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Iter: 16 time/iter: 0 varE: 0.022 lambda: 4.944
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Iter: 17 time/iter: 0 varE: 0.022 lambda: 0.797
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Iter: 18 time/iter: 0.001 varE: 0.022 lambda: 1.601
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Iter: 19 time/iter: 0 varE: 0.022 lambda: 5.469
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Iter: 20 time/iter: 0 varE: 0.022 lambda: 1.614
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Iter: 21 time/iter: 0 varE: 0.022 lambda: 1.886
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Iter: 22 time/iter: 0 varE: 0.022 lambda: 4.242
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Iter: 23 time/iter: 0 varE: 0.022 lambda: 1.38
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Iter: 24 time/iter: 0 varE: 0.022 lambda: 2.967
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Iter: 25 time/iter: 0.003 varE: 0.022 lambda: 3.067
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Iter: 26 time/iter: 0 varE: 0.022 lambda: 2.964
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Iter: 27 time/iter: 0 varE: 0.022 lambda: 2.85
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Iter: 28 time/iter: 0.001 varE: 0.022 lambda: 1.676
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Iter: 29 time/iter: 0 varE: 0.022 lambda: 0.516
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Iter: 30 time/iter: 0 varE: 0.022 lambda: 1.667
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Iter: 31 time/iter: 0 varE: 0.022 lambda: 0.599
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Iter: 32 time/iter: 0 varE: 0.022 lambda: 0.942
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Iter: 33 time/iter: 0 varE: 0.022 lambda: 0.169
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Iter: 34 time/iter: 0 varE: 0.022 lambda: 0.133
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Iter: 35 time/iter: 0 varE: 0.022 lambda: 0.111
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Iter: 36 time/iter: 0.001 varE: 0.022 lambda: 0.104
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Iter: 37 time/iter: 0 varE: 0.022 lambda: 0.078
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Iter: 38 time/iter: 0 varE: 0.022 lambda: 0.032
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Iter: 39 time/iter: 0 varE: 0.022 lambda: 0.133
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Iter: 40 time/iter: 0 varE: 0.022 lambda: 0.579
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Iter: 41 time/iter: 0 varE: 0.022 lambda: 1.47
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Iter: 42 time/iter: 0 varE: 0.022 lambda: 1.62
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Iter: 43 time/iter: 0 varE: 0.022 lambda: 1.351
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Iter: 44 time/iter: 0.001 varE: 0.022 lambda: 11.226
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Iter: 45 time/iter: 0 varE: 0.022 lambda: 13.749
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Iter: 46 time/iter: 0 varE: 0.022 lambda: 3.223
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Iter: 47 time/iter: 0 varE: 0.022 lambda: 4.445
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Iter: 48 time/iter: 0 varE: 0.022 lambda: 6.178
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Iter: 49 time/iter: 0 varE: 0.022 lambda: 8.837
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Iter: 50 time/iter: 0 varE: 0.022 lambda: 31.163
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Iter: 51 time/iter: 0.001 varE: 0.022 lambda: 19.729
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Iter: 52 time/iter: 0 varE: 0.022 lambda: 22.708
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Iter: 53 time/iter: 0 varE: 0.022 lambda: 26.696
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Iter: 54 time/iter: 0 varE: 0.022 lambda: 11.937
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Iter: 55 time/iter: 0 varE: 0.022 lambda: 25.929
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Iter: 56 time/iter: 0 varE: 0.022 lambda: 26.922
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Iter: 57 time/iter: 0.001 varE: 0.022 lambda: 9.284
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Iter: 58 time/iter: 0 varE: 0.022 lambda: 30.096
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Iter: 59 time/iter: 0 varE: 0.022 lambda: 59.941
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Iter: 60 time/iter: 0 varE: 0.022 lambda: 317.413
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Iter: 61 time/iter: 0 varE: 0.022 lambda: 194.221
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Iter: 62 time/iter: 0 varE: 0.022 lambda: 298.815
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Iter: 63 time/iter: 0 varE: 0.022 lambda: 220.312
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Iter: 64 time/iter: 0.001 varE: 0.022 lambda: 74.32
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Iter: 65 time/iter: 0 varE: 0.022 lambda: 101.32
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Iter: 66 time/iter: 0 varE: 0.022 lambda: 55.718
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Iter: 67 time/iter: 0.001 varE: 0.022 lambda: 26.964
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Iter: 68 time/iter: 0 varE: 0.022 lambda: 53.802
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Iter: 69 time/iter: 0 varE: 0.022 lambda: 435.541
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Iter: 70 time/iter: 0 varE: 0.022 lambda: 168.207
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Iter: 71 time/iter: 0 varE: 0.022 lambda: 114.793
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Iter: 72 time/iter: 0 varE: 0.022 lambda: 128.336
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Iter: 73 time/iter: 0 varE: 0.022 lambda: 180.973
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Iter: 74 time/iter: 0 varE: 0.022 lambda: 165.829
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Iter: 75 time/iter: 0.001 varE: 0.022 lambda: 59.301
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Iter: 76 time/iter: 0 varE: 0.022 lambda: 36.012
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Iter: 77 time/iter: 0 varE: 0.022 lambda: 6.383
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Iter: 78 time/iter: 0 varE: 0.022 lambda: 4.596
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Iter: 79 time/iter: 0 varE: 0.022 lambda: 11.74
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Iter: 80 time/iter: 0 varE: 0.022 lambda: 5.349
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Iter: 81 time/iter: 0 varE: 0.022 lambda: 6.167
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Iter: 82 time/iter: 0.001 varE: 0.022 lambda: 10.658
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Iter: 83 time/iter: 0 varE: 0.022 lambda: 15.46
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Iter: 84 time/iter: 0 varE: 0.022 lambda: 32.021
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Iter: 85 time/iter: 0 varE: 0.022 lambda: 50.078
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Iter: 86 time/iter: 0 varE: 0.022 lambda: 40.865
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Iter: 87 time/iter: 0 varE: 0.022 lambda: 88.927
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Iter: 88 time/iter: 0 varE: 0.022 lambda: 84.795
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Iter: 89 time/iter: 0.001 varE: 0.022 lambda: 109.065
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Iter: 90 time/iter: 0 varE: 0.022 lambda: 156.377
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Iter: 91 time/iter: 0 varE: 0.022 lambda: 207.49
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Iter: 92 time/iter: 0.001 varE: 0.022 lambda: 284.797
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Iter: 93 time/iter: 0 varE: 0.022 lambda: 383.244
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Iter: 94 time/iter: 0 varE: 0.022 lambda: 1507.081
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Iter: 95 time/iter: 0 varE: 0.022 lambda: 1443.044
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Iter: 96 time/iter: 0 varE: 0.022 lambda: 888.026
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Iter: 97 time/iter: 0 varE: 0.022 lambda: 1084.895
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Iter: 98 time/iter: 0 varE: 0.022 lambda: 1446.717
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Iter: 99 time/iter: 0 varE: 0.022 lambda: 471.166
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Iter: 100 time/iter: 0.001 varE: 0.022 lambda: 535.332
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Iter: 101 time/iter: 0 varE: 0.022 lambda: 183.422
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Iter: 102 time/iter: 0 varE: 0.022 lambda: 50.056
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Iter: 103 time/iter: 0 varE: 0.022 lambda: 57.275
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Iter: 104 time/iter: 0 varE: 0.022 lambda: 126.194
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Iter: 105 time/iter: 0 varE: 0.022 lambda: 113.796
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Iter: 106 time/iter: 0.001 varE: 0.022 lambda: 120.017
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Iter: 107 time/iter: 0 varE: 0.022 lambda: 42.666
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Iter: 108 time/iter: 0 varE: 0.022 lambda: 208.264
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Iter: 109 time/iter: 0.001 varE: 0.022 lambda: 602.515
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Iter: 110 time/iter: 0 varE: 0.022 lambda: 1921.265
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Iter: 111 time/iter: 0 varE: 0.022 lambda: 2103.287
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Iter: 112 time/iter: 0 varE: 0.022 lambda: 1170.763
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Iter: 113 time/iter: 0 varE: 0.022 lambda: 4685.368
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Iter: 114 time/iter: 0 varE: 0.022 lambda: 5399.16
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Iter: 115 time/iter: 0 varE: 0.022 lambda: 4447.316
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Iter: 116 time/iter: 0.001 varE: 0.022 lambda: 6423.038
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Iter: 117 time/iter: 0 varE: 0.022 lambda: 9076.635
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Iter: 118 time/iter: 0 varE: 0.022 lambda: 253.022
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Iter: 119 time/iter: 0 varE: 0.022 lambda: 322.896
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Iter: 120 time/iter: 0 varE: 0.022 lambda: 1027.708
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Iter: 121 time/iter: 0 varE: 0.022 lambda: 1647.459
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Iter: 122 time/iter: 0 varE: 0.022 lambda: 1687.051
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Iter: 123 time/iter: 0.001 varE: 0.022 lambda: 736.892
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Iter: 124 time/iter: 0 varE: 0.022 lambda: 854.023
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Iter: 125 time/iter: 0 varE: 0.022 lambda: 555.158
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Iter: 126 time/iter: 0.001 varE: 0.022 lambda: 404.102
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Iter: 127 time/iter: 0 varE: 0.022 lambda: 279.216
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Iter: 128 time/iter: 0 varE: 0.022 lambda: 175.487
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Iter: 129 time/iter: 0 varE: 0.022 lambda: 252.52
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Iter: 130 time/iter: 0 varE: 0.022 lambda: 173.788
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Iter: 131 time/iter: 0 varE: 0.022 lambda: 250.226
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Iter: 132 time/iter: 0 varE: 0.022 lambda: 244.941
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Iter: 133 time/iter: 0.001 varE: 0.022 lambda: 376.349
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Iter: 134 time/iter: 0 varE: 0.022 lambda: 181.139
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Iter: 135 time/iter: 0 varE: 0.022 lambda: 385.314
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Iter: 136 time/iter: 0 varE: 0.022 lambda: 750.134
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Iter: 137 time/iter: 0 varE: 0.022 lambda: 915.788
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Iter: 138 time/iter: 0 varE: 0.022 lambda: 765.129
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Iter: 139 time/iter: 0.001 varE: 0.022 lambda: 311.379
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Iter: 140 time/iter: 0 varE: 0.022 lambda: 463.942
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Iter: 141 time/iter: 0 varE: 0.022 lambda: 373.727
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Iter: 142 time/iter: 0 varE: 0.022 lambda: 673.945
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Iter: 143 time/iter: 0.001 varE: 0.022 lambda: 122.193
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Iter: 144 time/iter: 0 varE: 0.022 lambda: 36.414
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Iter: 145 time/iter: 0 varE: 0.022 lambda: 57.65
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Iter: 146 time/iter: 0 varE: 0.022 lambda: 68.418
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Iter: 147 time/iter: 0 varE: 0.022 lambda: 85.69
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Iter: 148 time/iter: 0 varE: 0.022 lambda: 39.633
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Iter: 149 time/iter: 0 varE: 0.022 lambda: 27.73
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Iter: 150 time/iter: 0.001 varE: 0.022 lambda: 13.804
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Iter: 151 time/iter: 0.003 varE: 0.022 lambda: 12.436
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Iter: 152 time/iter: 0 varE: 0.022 lambda: 16.072
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Iter: 153 time/iter: 0 varE: 0.022 lambda: 15.016
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Iter: 154 time/iter: 0 varE: 0.022 lambda: 17.67
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Iter: 155 time/iter: 0 varE: 0.022 lambda: 19.091
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Iter: 156 time/iter: 0.001 varE: 0.022 lambda: 23.258
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Iter: 157 time/iter: 0 varE: 0.022 lambda: 27.779
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Iter: 158 time/iter: 0 varE: 0.022 lambda: 47.661
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Iter: 159 time/iter: 0 varE: 0.022 lambda: 24.498
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Iter: 160 time/iter: 0 varE: 0.022 lambda: 55.796
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Iter: 161 time/iter: 0.001 varE: 0.022 lambda: 26.39
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Iter: 162 time/iter: 0 varE: 0.022 lambda: 31.065
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Iter: 163 time/iter: 0 varE: 0.022 lambda: 13.695
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Iter: 164 time/iter: 0 varE: 0.022 lambda: 25.334
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Iter: 165 time/iter: 0 varE: 0.022 lambda: 25.808
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Iter: 166 time/iter: 0.001 varE: 0.022 lambda: 20.976
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Iter: 167 time/iter: 0 varE: 0.022 lambda: 9.649
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Iter: 168 time/iter: 0 varE: 0.022 lambda: 12.589
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Iter: 169 time/iter: 0 varE: 0.022 lambda: 9.742
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Iter: 170 time/iter: 0 varE: 0.022 lambda: 13.752
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Iter: 171 time/iter: 0 varE: 0.022 lambda: 36.077
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Iter: 172 time/iter: 0.001 varE: 0.022 lambda: 19.563
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Iter: 173 time/iter: 0 varE: 0.022 lambda: 35.96
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Iter: 174 time/iter: 0 varE: 0.022 lambda: 40.309
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Iter: 175 time/iter: 0.001 varE: 0.022 lambda: 49.65
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Iter: 176 time/iter: 0 varE: 0.022 lambda: 15.747
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Iter: 177 time/iter: 0 varE: 0.022 lambda: 15.153
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Iter: 178 time/iter: 0 varE: 0.022 lambda: 48.756
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Iter: 179 time/iter: 0 varE: 0.022 lambda: 69.916
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Iter: 180 time/iter: 0.001 varE: 0.022 lambda: 103.127
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Iter: 181 time/iter: 0 varE: 0.022 lambda: 98.12
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Iter: 182 time/iter: 0 varE: 0.022 lambda: 39.687
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Iter: 183 time/iter: 0 varE: 0.022 lambda: 36.219
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Iter: 184 time/iter: 0 varE: 0.022 lambda: 19.397
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Iter: 185 time/iter: 0 varE: 0.022 lambda: 107.989
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Iter: 186 time/iter: 0.001 varE: 0.022 lambda: 38.974
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Iter: 187 time/iter: 0 varE: 0.022 lambda: 58.731
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Iter: 188 time/iter: 0 varE: 0.022 lambda: 72.948
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Iter: 189 time/iter: 0 varE: 0.022 lambda: 196.729
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Iter: 190 time/iter: 0.001 varE: 0.022 lambda: 167.251
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Iter: 191 time/iter: 0 varE: 0.022 lambda: 465.696
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Iter: 192 time/iter: 0 varE: 0.022 lambda: 166.978
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Iter: 193 time/iter: 0 varE: 0.022 lambda: 290.227
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Iter: 194 time/iter: 0 varE: 0.022 lambda: 362.825
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Iter: 195 time/iter: 0 varE: 0.022 lambda: 283.874
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Iter: 196 time/iter: 0.001 varE: 0.022 lambda: 362.644
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Iter: 197 time/iter: 0 varE: 0.022 lambda: 323.437
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Iter: 198 time/iter: 0 varE: 0.022 lambda: 179.06
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Iter: 199 time/iter: 0 varE: 0.022 lambda: 139.532
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Iter: 200 time/iter: 0 varE: 0.022 lambda: 42.176
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Iter: 201 time/iter: 0 varE: 0.022 lambda: 6.525
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Iter: 202 time/iter: 0.001 varE: 0.022 lambda: 3.213
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Iter: 203 time/iter: 0 varE: 0.022 lambda: 24.349
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Iter: 204 time/iter: 0 varE: 0.022 lambda: 12.1
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Iter: 205 time/iter: 0 varE: 0.022 lambda: 25.685
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Iter: 206 time/iter: 0.001 varE: 0.022 lambda: 20.607
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Iter: 207 time/iter: 0 varE: 0.022 lambda: 52.916
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Iter: 208 time/iter: 0 varE: 0.022 lambda: 30.877
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Iter: 209 time/iter: 0 varE: 0.022 lambda: 18.797
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Iter: 210 time/iter: 0.001 varE: 0.022 lambda: 19.363
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Iter: 211 time/iter: 0 varE: 0.022 lambda: 11.668
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Iter: 212 time/iter: 0 varE: 0.022 lambda: 21.394
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Iter: 213 time/iter: 0 varE: 0.022 lambda: 17.363
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Iter: 214 time/iter: 0 varE: 0.022 lambda: 6.272
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Iter: 215 time/iter: 0 varE: 0.022 lambda: 4.39
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Iter: 216 time/iter: 0.001 varE: 0.022 lambda: 14.455
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Iter: 217 time/iter: 0 varE: 0.022 lambda: 10.684
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Iter: 218 time/iter: 0 varE: 0.022 lambda: 8.715
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Iter: 219 time/iter: 0 varE: 0.022 lambda: 2.193
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Iter: 220 time/iter: 0.001 varE: 0.022 lambda: 3.275
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Iter: 221 time/iter: 0 varE: 0.022 lambda: 4.551
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Iter: 222 time/iter: 0 varE: 0.022 lambda: 0.567
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Iter: 223 time/iter: 0 varE: 0.022 lambda: 1.601
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Iter: 224 time/iter: 0 varE: 0.022 lambda: 0.72
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Iter: 225 time/iter: 0.001 varE: 0.022 lambda: 0.952
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Iter: 226 time/iter: 0 varE: 0.022 lambda: 7.968
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Iter: 227 time/iter: 0 varE: 0.022 lambda: 1.002
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Iter: 228 time/iter: 0 varE: 0.022 lambda: 0.749
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Iter: 229 time/iter: 0 varE: 0.022 lambda: 1.088
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Iter: 230 time/iter: 0.001 varE: 0.022 lambda: 0.571
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Iter: 231 time/iter: 0 varE: 0.022 lambda: 0.99
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Iter: 232 time/iter: 0 varE: 0.022 lambda: 1.019
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Iter: 233 time/iter: 0.001 varE: 0.022 lambda: 0.167
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Iter: 234 time/iter: 0 varE: 0.022 lambda: 0.415
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Iter: 235 time/iter: 0 varE: 0.022 lambda: 0.308
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Iter: 236 time/iter: 0 varE: 0.022 lambda: 0.677
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Iter: 237 time/iter: 0 varE: 0.022 lambda: 0.681
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Iter: 238 time/iter: 0.001 varE: 0.022 lambda: 1.198
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Iter: 239 time/iter: 0 varE: 0.022 lambda: 1.036
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Iter: 240 time/iter: 0 varE: 0.022 lambda: 0.653
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Iter: 241 time/iter: 0 varE: 0.022 lambda: 3.515
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Iter: 242 time/iter: 0 varE: 0.022 lambda: 8.574
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Iter: 243 time/iter: 0.001 varE: 0.022 lambda: 19.037
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Iter: 244 time/iter: 0 varE: 0.022 lambda: 2.939
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Iter: 245 time/iter: 0 varE: 0.022 lambda: 3.787
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Iter: 246 time/iter: 0 varE: 0.022 lambda: 7.823
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Iter: 247 time/iter: 0.001 varE: 0.022 lambda: 4.912
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Iter: 248 time/iter: 0 varE: 0.022 lambda: 2.163
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Iter: 249 time/iter: 0 varE: 0.022 lambda: 5.942
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Iter: 250 time/iter: 0 varE: 0.022 lambda: 4.598
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Iter: 251 time/iter: 0 varE: 0.022 lambda: 16.835
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Iter: 252 time/iter: 0.001 varE: 0.022 lambda: 13.799
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Iter: 253 time/iter: 0 varE: 0.022 lambda: 71.999
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SafeBayes documentation built on May 1, 2019, 9:23 p.m.
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Description Usage Arguments Details Value Author(s) References Examples
Description
The function GBLassoFV (Generalized Bayesian Lasso with Fixed Variance) provides a Gibbs sampler to sample the posterior of generalized Bayesian lasso regression models with fixed variance and with learning rate eta.
Usage
1 2 |
Arguments
y |
Vector of outcome variables, numeric, NA allowed, length n. |
X |
Design matrix, numeric, dimension n x p, n >= 2. |
sigma2 |
Fixed variance parameter sigma^2, numeric. Default NULL, in which case the variance will be estimated from the data. |
eta |
Learning rate eta, numeric, 0 <= eta <= 1. Default 1. |
prior |
List containing the following elements
|
nIter |
Number of iterations, integer. Default 1100. |
burnIn |
Number of iterations for burn-in, integer. Default 100. |
thin |
Number of iterations for thinning, integer. Default 10. |
minAbsBeta |
Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 10^-9. |
weights |
Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. |
pIter |
Print iterations, logical. Default TRUE. |
Details
Details on the generalized Bayesian lasso can be found in Chapter 2 of (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009).
Several authors have brought forward the idea of equipping Bayesian updating with a learning rate eta, resulting in an eta-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the eta-generalized posterior, the likelihood is raised to the power eta in order to trade the relative weight of the likelihood and the prior, where eta = 1 corresponds to standard Bayes.
Value
$y |
Vector of original outcome variables. |
$weights |
Vector of weights. |
$mu |
Posterior mean of the intercept. |
$varE |
Posterior mean of of the variance. |
$yHat |
Posterior mean of mu + X*beta + epsilon. |
$SD.yHat |
Corresponding standard deviation. |
$whichNa |
Vector with indices of missing values of y. |
$fit$pD |
Estimated number of effective parameters. |
$fit$DIC |
Deviance Information Criterion. |
$lambda |
Posterior mean of lambda. |
$bL |
Posterior mean of beta. |
$SD.bL |
Corresponding standard deviation. |
$tau2 |
Posterior mean of tau^2. |
$prior |
List containing the priors used. |
$nIter |
Number of iterations. |
$burnIn |
Number of iterations for burn-in. |
$thin |
Number of iterations for thinning. |
$eta |
Learning rate eta. |
Author(s)
R. de Heide
References
de Heide, R. 2016. The Safe-Bayesian Lasso. Master Thesis, Leiden University.
de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009. Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375-385.
Audibert, J.Y. 2013. Bayesian generalized double pareto shrinkage. Statistica Sinica 23 119-143.
Catoni, O. 2007. PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS.
Grunwald, P.D. 2012. chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings. 169-183. Springer Berlin Heidelberg
McAllester, D. 2003. PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21
Vovk, V.G. 1990. Aggregating strategies. In Proc. COLT 90 371-383
Zhang, T. 2004. Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press 1149-1156
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | rm(list=ls())
library(SafeBayes)
# Simulate data
x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1
y <- NULL
# for each x, a y that is 0 + Gaussian noise
for (i in 1:100) {
y[i] <- 0 + rnorm(1, mean=0, sd=1/4)
}
# Now sample 100 zero's and ones (coin toss)
cointoss <- sample(0:1, 100, replace=TRUE)
# indices of the ones
indices <- which(cointoss==1)
# we replace x and y with (0,0) for the indices the cointoss
# landed tail (1)
x[indices] <- 0
y[indices] <- 0
plot(x,y)
# Determine the generalized posterior for eta = 0.25
obj <- GBLassoFV(y, x, eta=0.25)
# posterior means of the coefficients beta and intercept mu
betafour <- obj$bL
mufour <- obj$mu
|
Example output
No prior was provided, improper priors are used.
Sigma^2 is not provided and will be estimated from the dataIter: 1 time/iter: 0 varE: 0.022 lambda: 13.456
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Iter: 2 time/iter: 0 varE: 0.022 lambda: 11.269
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Iter: 3 time/iter: 0 varE: 0.022 lambda: 4.97
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Iter: 4 time/iter: 0.001 varE: 0.022 lambda: 2.81
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Iter: 5 time/iter: 0 varE: 0.022 lambda: 4.461
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Iter: 6 time/iter: 0 varE: 0.022 lambda: 2.743
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Iter: 7 time/iter: 0 varE: 0.022 lambda: 1.61
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Iter: 8 time/iter: 0 varE: 0.022 lambda: 1.181
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Iter: 9 time/iter: 0 varE: 0.022 lambda: 1.237
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Iter: 10 time/iter: 0 varE: 0.022 lambda: 12.75
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Iter: 11 time/iter: 0.001 varE: 0.022 lambda: 4.06
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Iter: 12 time/iter: 0 varE: 0.022 lambda: 15.567
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Iter: 13 time/iter: 0 varE: 0.022 lambda: 15.848
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Iter: 14 time/iter: 0 varE: 0.022 lambda: 10.184
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Iter: 15 time/iter: 0 varE: 0.022 lambda: 8.517
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Iter: 16 time/iter: 0 varE: 0.022 lambda: 4.944
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Iter: 17 time/iter: 0 varE: 0.022 lambda: 0.797
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Iter: 18 time/iter: 0.001 varE: 0.022 lambda: 1.601
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Iter: 19 time/iter: 0 varE: 0.022 lambda: 5.469
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Iter: 20 time/iter: 0 varE: 0.022 lambda: 1.614
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Iter: 21 time/iter: 0 varE: 0.022 lambda: 1.886
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Iter: 22 time/iter: 0 varE: 0.022 lambda: 4.242
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Iter: 23 time/iter: 0 varE: 0.022 lambda: 1.38
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Iter: 24 time/iter: 0 varE: 0.022 lambda: 2.967
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Iter: 25 time/iter: 0.003 varE: 0.022 lambda: 3.067
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Iter: 26 time/iter: 0 varE: 0.022 lambda: 2.964
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Iter: 27 time/iter: 0 varE: 0.022 lambda: 2.85
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Iter: 28 time/iter: 0.001 varE: 0.022 lambda: 1.676
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Iter: 29 time/iter: 0 varE: 0.022 lambda: 0.516
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Iter: 30 time/iter: 0 varE: 0.022 lambda: 1.667
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Iter: 31 time/iter: 0 varE: 0.022 lambda: 0.599
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Iter: 32 time/iter: 0 varE: 0.022 lambda: 0.942
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Iter: 33 time/iter: 0 varE: 0.022 lambda: 0.169
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Iter: 34 time/iter: 0 varE: 0.022 lambda: 0.133
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Iter: 35 time/iter: 0 varE: 0.022 lambda: 0.111
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Iter: 36 time/iter: 0.001 varE: 0.022 lambda: 0.104
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Iter: 37 time/iter: 0 varE: 0.022 lambda: 0.078
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Iter: 38 time/iter: 0 varE: 0.022 lambda: 0.032
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Iter: 39 time/iter: 0 varE: 0.022 lambda: 0.133
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Iter: 40 time/iter: 0 varE: 0.022 lambda: 0.579
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Iter: 41 time/iter: 0 varE: 0.022 lambda: 1.47
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Iter: 42 time/iter: 0 varE: 0.022 lambda: 1.62
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Iter: 43 time/iter: 0 varE: 0.022 lambda: 1.351
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Iter: 44 time/iter: 0.001 varE: 0.022 lambda: 11.226
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Iter: 45 time/iter: 0 varE: 0.022 lambda: 13.749
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Iter: 46 time/iter: 0 varE: 0.022 lambda: 3.223
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Iter: 47 time/iter: 0 varE: 0.022 lambda: 4.445
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Iter: 48 time/iter: 0 varE: 0.022 lambda: 6.178
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Iter: 49 time/iter: 0 varE: 0.022 lambda: 8.837
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Iter: 50 time/iter: 0 varE: 0.022 lambda: 31.163
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Iter: 51 time/iter: 0.001 varE: 0.022 lambda: 19.729
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Iter: 52 time/iter: 0 varE: 0.022 lambda: 22.708
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Iter: 53 time/iter: 0 varE: 0.022 lambda: 26.696
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Iter: 54 time/iter: 0 varE: 0.022 lambda: 11.937
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Iter: 55 time/iter: 0 varE: 0.022 lambda: 25.929
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Iter: 56 time/iter: 0 varE: 0.022 lambda: 26.922
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Iter: 57 time/iter: 0.001 varE: 0.022 lambda: 9.284
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Iter: 58 time/iter: 0 varE: 0.022 lambda: 30.096
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Iter: 59 time/iter: 0 varE: 0.022 lambda: 59.941
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Iter: 60 time/iter: 0 varE: 0.022 lambda: 317.413
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Iter: 61 time/iter: 0 varE: 0.022 lambda: 194.221
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Iter: 62 time/iter: 0 varE: 0.022 lambda: 298.815
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Iter: 63 time/iter: 0 varE: 0.022 lambda: 220.312
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Iter: 64 time/iter: 0.001 varE: 0.022 lambda: 74.32
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Iter: 65 time/iter: 0 varE: 0.022 lambda: 101.32
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Iter: 66 time/iter: 0 varE: 0.022 lambda: 55.718
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Iter: 67 time/iter: 0.001 varE: 0.022 lambda: 26.964
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Iter: 68 time/iter: 0 varE: 0.022 lambda: 53.802
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Iter: 69 time/iter: 0 varE: 0.022 lambda: 435.541
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Iter: 70 time/iter: 0 varE: 0.022 lambda: 168.207
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Iter: 71 time/iter: 0 varE: 0.022 lambda: 114.793
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Iter: 72 time/iter: 0 varE: 0.022 lambda: 128.336
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Iter: 73 time/iter: 0 varE: 0.022 lambda: 180.973
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Iter: 74 time/iter: 0 varE: 0.022 lambda: 165.829
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Iter: 75 time/iter: 0.001 varE: 0.022 lambda: 59.301
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Iter: 76 time/iter: 0 varE: 0.022 lambda: 36.012
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Iter: 77 time/iter: 0 varE: 0.022 lambda: 6.383
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Iter: 78 time/iter: 0 varE: 0.022 lambda: 4.596
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Iter: 79 time/iter: 0 varE: 0.022 lambda: 11.74
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Iter: 80 time/iter: 0 varE: 0.022 lambda: 5.349
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Iter: 81 time/iter: 0 varE: 0.022 lambda: 6.167
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Iter: 82 time/iter: 0.001 varE: 0.022 lambda: 10.658
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Iter: 83 time/iter: 0 varE: 0.022 lambda: 15.46
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Iter: 84 time/iter: 0 varE: 0.022 lambda: 32.021
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Iter: 85 time/iter: 0 varE: 0.022 lambda: 50.078
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Iter: 86 time/iter: 0 varE: 0.022 lambda: 40.865
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Iter: 87 time/iter: 0 varE: 0.022 lambda: 88.927
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Iter: 88 time/iter: 0 varE: 0.022 lambda: 84.795
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Iter: 89 time/iter: 0.001 varE: 0.022 lambda: 109.065
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Iter: 90 time/iter: 0 varE: 0.022 lambda: 156.377
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Iter: 91 time/iter: 0 varE: 0.022 lambda: 207.49
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Iter: 92 time/iter: 0.001 varE: 0.022 lambda: 284.797
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Iter: 93 time/iter: 0 varE: 0.022 lambda: 383.244
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Iter: 94 time/iter: 0 varE: 0.022 lambda: 1507.081
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Iter: 95 time/iter: 0 varE: 0.022 lambda: 1443.044
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Iter: 96 time/iter: 0 varE: 0.022 lambda: 888.026
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Iter: 97 time/iter: 0 varE: 0.022 lambda: 1084.895
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Iter: 98 time/iter: 0 varE: 0.022 lambda: 1446.717
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Iter: 99 time/iter: 0 varE: 0.022 lambda: 471.166
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Iter: 100 time/iter: 0.001 varE: 0.022 lambda: 535.332
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Iter: 101 time/iter: 0 varE: 0.022 lambda: 183.422
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Iter: 102 time/iter: 0 varE: 0.022 lambda: 50.056
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Iter: 103 time/iter: 0 varE: 0.022 lambda: 57.275
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Iter: 104 time/iter: 0 varE: 0.022 lambda: 126.194
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Iter: 105 time/iter: 0 varE: 0.022 lambda: 113.796
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Iter: 106 time/iter: 0.001 varE: 0.022 lambda: 120.017
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Iter: 107 time/iter: 0 varE: 0.022 lambda: 42.666
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Iter: 108 time/iter: 0 varE: 0.022 lambda: 208.264
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Iter: 109 time/iter: 0.001 varE: 0.022 lambda: 602.515
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Iter: 110 time/iter: 0 varE: 0.022 lambda: 1921.265
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Iter: 111 time/iter: 0 varE: 0.022 lambda: 2103.287
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Iter: 112 time/iter: 0 varE: 0.022 lambda: 1170.763
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Iter: 113 time/iter: 0 varE: 0.022 lambda: 4685.368
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Iter: 114 time/iter: 0 varE: 0.022 lambda: 5399.16
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Iter: 115 time/iter: 0 varE: 0.022 lambda: 4447.316
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Iter: 116 time/iter: 0.001 varE: 0.022 lambda: 6423.038
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Iter: 117 time/iter: 0 varE: 0.022 lambda: 9076.635
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Iter: 118 time/iter: 0 varE: 0.022 lambda: 253.022
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Iter: 119 time/iter: 0 varE: 0.022 lambda: 322.896
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Iter: 120 time/iter: 0 varE: 0.022 lambda: 1027.708
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Iter: 121 time/iter: 0 varE: 0.022 lambda: 1647.459
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Iter: 122 time/iter: 0 varE: 0.022 lambda: 1687.051
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Iter: 123 time/iter: 0.001 varE: 0.022 lambda: 736.892
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Iter: 124 time/iter: 0 varE: 0.022 lambda: 854.023
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Iter: 125 time/iter: 0 varE: 0.022 lambda: 555.158
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Iter: 126 time/iter: 0.001 varE: 0.022 lambda: 404.102
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Iter: 127 time/iter: 0 varE: 0.022 lambda: 279.216
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Iter: 128 time/iter: 0 varE: 0.022 lambda: 175.487
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Iter: 129 time/iter: 0 varE: 0.022 lambda: 252.52
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Iter: 130 time/iter: 0 varE: 0.022 lambda: 173.788
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Iter: 131 time/iter: 0 varE: 0.022 lambda: 250.226
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Iter: 132 time/iter: 0 varE: 0.022 lambda: 244.941
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Iter: 133 time/iter: 0.001 varE: 0.022 lambda: 376.349
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Iter: 134 time/iter: 0 varE: 0.022 lambda: 181.139
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Iter: 135 time/iter: 0 varE: 0.022 lambda: 385.314
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Iter: 136 time/iter: 0 varE: 0.022 lambda: 750.134
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Iter: 137 time/iter: 0 varE: 0.022 lambda: 915.788
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Iter: 138 time/iter: 0 varE: 0.022 lambda: 765.129
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Iter: 139 time/iter: 0.001 varE: 0.022 lambda: 311.379
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Iter: 140 time/iter: 0 varE: 0.022 lambda: 463.942
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Iter: 141 time/iter: 0 varE: 0.022 lambda: 373.727
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Iter: 142 time/iter: 0 varE: 0.022 lambda: 673.945
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Iter: 143 time/iter: 0.001 varE: 0.022 lambda: 122.193
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Iter: 144 time/iter: 0 varE: 0.022 lambda: 36.414
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Iter: 145 time/iter: 0 varE: 0.022 lambda: 57.65
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Iter: 146 time/iter: 0 varE: 0.022 lambda: 68.418
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Iter: 147 time/iter: 0 varE: 0.022 lambda: 85.69
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Iter: 148 time/iter: 0 varE: 0.022 lambda: 39.633
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Iter: 149 time/iter: 0 varE: 0.022 lambda: 27.73
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Iter: 150 time/iter: 0.001 varE: 0.022 lambda: 13.804
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Iter: 151 time/iter: 0.003 varE: 0.022 lambda: 12.436
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Iter: 152 time/iter: 0 varE: 0.022 lambda: 16.072
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Iter: 153 time/iter: 0 varE: 0.022 lambda: 15.016
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Iter: 154 time/iter: 0 varE: 0.022 lambda: 17.67
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Iter: 155 time/iter: 0 varE: 0.022 lambda: 19.091
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Iter: 156 time/iter: 0.001 varE: 0.022 lambda: 23.258
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Iter: 157 time/iter: 0 varE: 0.022 lambda: 27.779
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Iter: 158 time/iter: 0 varE: 0.022 lambda: 47.661
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Iter: 159 time/iter: 0 varE: 0.022 lambda: 24.498
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Iter: 160 time/iter: 0 varE: 0.022 lambda: 55.796
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Iter: 161 time/iter: 0.001 varE: 0.022 lambda: 26.39
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Iter: 162 time/iter: 0 varE: 0.022 lambda: 31.065
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Iter: 163 time/iter: 0 varE: 0.022 lambda: 13.695
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Iter: 164 time/iter: 0 varE: 0.022 lambda: 25.334
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Iter: 165 time/iter: 0 varE: 0.022 lambda: 25.808
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Iter: 166 time/iter: 0.001 varE: 0.022 lambda: 20.976
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Iter: 167 time/iter: 0 varE: 0.022 lambda: 9.649
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Iter: 168 time/iter: 0 varE: 0.022 lambda: 12.589
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Iter: 169 time/iter: 0 varE: 0.022 lambda: 9.742
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Iter: 170 time/iter: 0 varE: 0.022 lambda: 13.752
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Iter: 171 time/iter: 0 varE: 0.022 lambda: 36.077
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Iter: 172 time/iter: 0.001 varE: 0.022 lambda: 19.563
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Iter: 173 time/iter: 0 varE: 0.022 lambda: 35.96
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Iter: 174 time/iter: 0 varE: 0.022 lambda: 40.309
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Iter: 175 time/iter: 0.001 varE: 0.022 lambda: 49.65
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Iter: 176 time/iter: 0 varE: 0.022 lambda: 15.747
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Iter: 177 time/iter: 0 varE: 0.022 lambda: 15.153
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Iter: 178 time/iter: 0 varE: 0.022 lambda: 48.756
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Iter: 179 time/iter: 0 varE: 0.022 lambda: 69.916
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Iter: 180 time/iter: 0.001 varE: 0.022 lambda: 103.127
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Iter: 181 time/iter: 0 varE: 0.022 lambda: 98.12
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Iter: 182 time/iter: 0 varE: 0.022 lambda: 39.687
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Iter: 183 time/iter: 0 varE: 0.022 lambda: 36.219
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Iter: 184 time/iter: 0 varE: 0.022 lambda: 19.397
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Iter: 185 time/iter: 0 varE: 0.022 lambda: 107.989
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Iter: 186 time/iter: 0.001 varE: 0.022 lambda: 38.974
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Iter: 187 time/iter: 0 varE: 0.022 lambda: 58.731
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Iter: 188 time/iter: 0 varE: 0.022 lambda: 72.948
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Iter: 189 time/iter: 0 varE: 0.022 lambda: 196.729
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Iter: 190 time/iter: 0.001 varE: 0.022 lambda: 167.251
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Iter: 191 time/iter: 0 varE: 0.022 lambda: 465.696
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Iter: 192 time/iter: 0 varE: 0.022 lambda: 166.978
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Iter: 193 time/iter: 0 varE: 0.022 lambda: 290.227
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Iter: 194 time/iter: 0 varE: 0.022 lambda: 362.825
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Iter: 195 time/iter: 0 varE: 0.022 lambda: 283.874
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Iter: 196 time/iter: 0.001 varE: 0.022 lambda: 362.644
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Iter: 197 time/iter: 0 varE: 0.022 lambda: 323.437
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Iter: 198 time/iter: 0 varE: 0.022 lambda: 179.06
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Iter: 199 time/iter: 0 varE: 0.022 lambda: 139.532
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Iter: 200 time/iter: 0 varE: 0.022 lambda: 42.176
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Iter: 201 time/iter: 0 varE: 0.022 lambda: 6.525
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Iter: 202 time/iter: 0.001 varE: 0.022 lambda: 3.213
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Iter: 203 time/iter: 0 varE: 0.022 lambda: 24.349
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Iter: 204 time/iter: 0 varE: 0.022 lambda: 12.1
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Iter: 205 time/iter: 0 varE: 0.022 lambda: 25.685
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Iter: 206 time/iter: 0.001 varE: 0.022 lambda: 20.607
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Iter: 207 time/iter: 0 varE: 0.022 lambda: 52.916
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Iter: 208 time/iter: 0 varE: 0.022 lambda: 30.877
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Iter: 209 time/iter: 0 varE: 0.022 lambda: 18.797
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Iter: 210 time/iter: 0.001 varE: 0.022 lambda: 19.363
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Iter: 211 time/iter: 0 varE: 0.022 lambda: 11.668
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Iter: 212 time/iter: 0 varE: 0.022 lambda: 21.394
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Iter: 213 time/iter: 0 varE: 0.022 lambda: 17.363
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Iter: 214 time/iter: 0 varE: 0.022 lambda: 6.272
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Iter: 215 time/iter: 0 varE: 0.022 lambda: 4.39
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Iter: 216 time/iter: 0.001 varE: 0.022 lambda: 14.455
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Iter: 217 time/iter: 0 varE: 0.022 lambda: 10.684
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Iter: 218 time/iter: 0 varE: 0.022 lambda: 8.715
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Iter: 219 time/iter: 0 varE: 0.022 lambda: 2.193
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Iter: 220 time/iter: 0.001 varE: 0.022 lambda: 3.275
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Iter: 221 time/iter: 0 varE: 0.022 lambda: 4.551
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Iter: 222 time/iter: 0 varE: 0.022 lambda: 0.567
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Iter: 223 time/iter: 0 varE: 0.022 lambda: 1.601
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Iter: 224 time/iter: 0 varE: 0.022 lambda: 0.72
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Iter: 225 time/iter: 0.001 varE: 0.022 lambda: 0.952
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Iter: 226 time/iter: 0 varE: 0.022 lambda: 7.968
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Iter: 227 time/iter: 0 varE: 0.022 lambda: 1.002
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Iter: 228 time/iter: 0 varE: 0.022 lambda: 0.749
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Iter: 229 time/iter: 0 varE: 0.022 lambda: 1.088
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Iter: 230 time/iter: 0.001 varE: 0.022 lambda: 0.571
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Iter: 231 time/iter: 0 varE: 0.022 lambda: 0.99
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Iter: 232 time/iter: 0 varE: 0.022 lambda: 1.019
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Iter: 233 time/iter: 0.001 varE: 0.022 lambda: 0.167
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Iter: 234 time/iter: 0 varE: 0.022 lambda: 0.415
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Iter: 235 time/iter: 0 varE: 0.022 lambda: 0.308
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Iter: 236 time/iter: 0 varE: 0.022 lambda: 0.677
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Iter: 237 time/iter: 0 varE: 0.022 lambda: 0.681
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Iter: 238 time/iter: 0.001 varE: 0.022 lambda: 1.198
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Iter: 239 time/iter: 0 varE: 0.022 lambda: 1.036
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Iter: 240 time/iter: 0 varE: 0.022 lambda: 0.653
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Iter: 241 time/iter: 0 varE: 0.022 lambda: 3.515
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Iter: 242 time/iter: 0 varE: 0.022 lambda: 8.574
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Iter: 243 time/iter: 0.001 varE: 0.022 lambda: 19.037
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Iter: 244 time/iter: 0 varE: 0.022 lambda: 2.939
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Iter: 245 time/iter: 0 varE: 0.022 lambda: 3.787
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Iter: 246 time/iter: 0 varE: 0.022 lambda: 7.823
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Iter: 247 time/iter: 0.001 varE: 0.022 lambda: 4.912
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Iter: 248 time/iter: 0 varE: 0.022 lambda: 2.163
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Iter: 249 time/iter: 0 varE: 0.022 lambda: 5.942
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Iter: 250 time/iter: 0 varE: 0.022 lambda: 4.598
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Iter: 251 time/iter: 0 varE: 0.022 lambda: 16.835
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Iter: 252 time/iter: 0.001 varE: 0.022 lambda: 13.799
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Iter: 253 time/iter: 0 varE: 0.022 lambda: 71.999
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Iter: 254 time/iter: 0 varE: 0.022 lambda: 83.471
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Iter: 255 time/iter: 0.001 varE: 0.022 lambda: 63.122
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Iter: 256 time/iter: 0 varE: 0.022 lambda: 61.058
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Iter: 258 time/iter: 0 varE: 0.022 lambda: 19.113
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Iter: 259 time/iter: 0 varE: 0.022 lambda: 1.226
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Iter: 260 time/iter: 0.001 varE: 0.022 lambda: 1.72
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Iter: 261 time/iter: 0 varE: 0.022 lambda: 4.106
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Iter: 262 time/iter: 0 varE: 0.022 lambda: 7.041
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Iter: 263 time/iter: 0.001 varE: 0.022 lambda: 42.213
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Iter: 264 time/iter: 0 varE: 0.022 lambda: 20.917
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Iter: 265 time/iter: 0 varE: 0.022 lambda: 47.152
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Iter: 266 time/iter: 0 varE: 0.022 lambda: 55.378
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Iter: 267 time/iter: 0.001 varE: 0.022 lambda: 61.182
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Iter: 268 time/iter: 0 varE: 0.022 lambda: 36.742
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Iter: 269 time/iter: 0 varE: 0.022 lambda: 31.673
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Iter: 270 time/iter: 0.001 varE: 0.022 lambda: 166.322
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Iter: 271 time/iter: 0 varE: 0.022 lambda: 97.387
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Iter: 272 time/iter: 0 varE: 0.022 lambda: 86.979
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Iter: 273 time/iter: 0 varE: 0.022 lambda: 68.911
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Iter: 274 time/iter: 0 varE: 0.022 lambda: 27.488
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Iter: 275 time/iter: 0 varE: 0.022 lambda: 20.686
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Iter: 276 time/iter: 0.001 varE: 0.022 lambda: 352.923
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Iter: 277 time/iter: 0 varE: 0.022 lambda: 272.384
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Iter: 278 time/iter: 0 varE: 0.022 lambda: 818.335
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Iter: 279 time/iter: 0.001 varE: 0.022 lambda: 667.588
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Iter: 280 time/iter: 0 varE: 0.022 lambda: 364.746
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Iter: 281 time/iter: 0 varE: 0.022 lambda: 298.504
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Iter: 282 time/iter: 0 varE: 0.022 lambda: 245.648
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Iter: 283 time/iter: 0.001 varE: 0.022 lambda: 215.112
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Iter: 284 time/iter: 0 varE: 0.022 lambda: 225.496
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Iter: 285 time/iter: 0 varE: 0.022 lambda: 305.076
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Iter: 286 time/iter: 0 varE: 0.022 lambda: 302.927
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Iter: 287 time/iter: 0 varE: 0.022 lambda: 125.62
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Iter: 288 time/iter: 0.001 varE: 0.022 lambda: 62.59
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Iter: 289 time/iter: 0 varE: 0.022 lambda: 95.348
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Iter: 290 time/iter: 0 varE: 0.022 lambda: 159.534
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Iter: 291 time/iter: 0.001 varE: 0.022 lambda: 119.051
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Iter: 292 time/iter: 0 varE: 0.022 lambda: 121.006
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Iter: 293 time/iter: 0 varE: 0.022 lambda: 86.737
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Iter: 294 time/iter: 0 varE: 0.022 lambda: 93.341
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Iter: 1094 time/iter: 0 varE: 0.022 lambda: 10644511.253
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Iter: 1099 time/iter: 0 varE: 0.022 lambda: 34767696.449
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Iter: 1100 time/iter: 0 varE: 0.022 lambda: 44552018.27
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