What do eps and tol do in LassoCV (scikit-learn) Using scikit-learn:
http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LassoCV.html
Specifically, I am interested in:
1) If eps grows, does the accuracy(precision) increase or decrease?
2) If tol grows, does the accuracy(precision) increase or decrease?
 A: Here is an example of LassoCV's affect on MSE with varying eps and tol (using the diabetes dataset), for various $\alpha$'s. Note that this is the average MSE (each CV run will have a different MSE): 


It appears that eps has a significant impact for some penalty parameters, but with a large enough penalty it doesn't matter. tol doesn't seem to play a large role (at least as far as scikit has implement LassoCV).
See below for code.
import matplotlib.pyplot as plt
from matplotlib.pyplot import cm
%matplotlib inline
import numpy as np
from sklearn import datasets
from sklearn.linear_model import LassoCV

# load data
diabetes = datasets.load_diabetes()
X = diabetes.data
y = diabetes.target

# Plot of epsilons
epss = [0.0001, 0.001, 0.01, 0.1]

plt.figure(figsize=(10,6))
color = cm.rainbow(np.linspace(0,1,len(epss)))

for i,c in zip(epss,color):
    model = LassoCV(eps=i).fit(X, y)

    ymin, ymax = 2300, 3800
    plt.plot(m_log_alphas, model.mse_path_.mean(axis=-1), color=c,
             label='eps = {}'.format(i), linewidth=2)
    plt.legend()

    plt.xlabel('-log(alpha)')
    plt.ylabel('Mean square error')
    plt.axis('tight')
    plt.ylim(ymin, ymax)


# Plot of tols
plt.figure(figsize=(10,6))
tols = [0.0001, 0.001, 0.01, 0.1, 1]

color = cm.rainbow(np.linspace(0,1,len(tols)))

for i,c in zip(tols,color):
    model = LassoCV(tol=i).fit(X, y)

    ymin, ymax = 2300, 3800
    plt.plot(m_log_alphas, model.mse_path_.mean(axis=-1), color=c,
             label='tol = {}'.format(i), linewidth=2)
    plt.legend()

    plt.xlabel('-log(alpha)')
    plt.ylabel('Mean square error')
    plt.axis('tight')
    plt.ylim(ymin, ymax)

