Ridge regression -- why does the model only care to control large outliers? One of the purposes of ridge regression is to curb the effects of outliers which may cause the regression coefficients to be so large and hence cause a highly biased model.
That's why the constraint $\Sigma\beta_j^2<s$ is imposed, forcing the coefficients to not exceed a certain value. 
Here is my issue. An outlier could be a value that is either too large or too small. I think this should mean than outliers could cause the $\beta_j$'s to be too small or too large. The formulation of this constraint inequality seems to only care about controlling those outliers which might make the $\beta_j$'s too large. 
I believe Ordinary Least Squares Regression equally suffers from the effects of very small outliers. 
Can someone please explain if or how Ridge regression controls the "small" outliers as well?
 A: The ridge estimator is very susceptible to outliers, much like the OLS estimator. The reason for that is that we still depend on the least squares minimization technique and this does not allow large residuals. Hence the regression line, plane or hyperplane will be drawn towards the outliers.
There has been considerable work in recent years in order to "robustify" ridge regression. In case you want to read further, here is one paper that advocates an approach based on M-estimators.

Maronna, Ricardo A. "Robust ridge regression for high-dimensional data." Technometrics 53.1 (2011): 44-53.

A: Ridge regression is a modification of linear regression. Linear regression is the best linear unbiased estimator (BLUE). They key word is "unbiased." Linear regression should give the smallest mean square error (MSE) of any unbiased linear estimator. Ridge regression adds bias in exchange to reduce variance, thus creating a biased estimator with potentially lower MSE. It does this with a regularization or "smoothing" parameter to shrink the coefficients as you describe. This has the effect of decorrellating variables for differering values along the parameter.
Techniques that are based on the squared loss function are particularly sensitive to outliers. To alleviate this, least absolute square error or even least median squares can be used as the loss function. However, ridge regression by itself is not meant to be used to reduce the effect of outliers because it is just a slightly modified version of linear regression estimated with an identical loss function, but penalized to adjust some of the assumptions in the linear regression model.  There is nothing here that explicitly deals with outliers.
