Deterministic classifier and input features I call the function estimation based classifiers as deterministic, the ones which estimates the $f(x) = a'x+b$ directly, rather than estimating the conditional or joint probabilities directly.  For example, SVM outputs a score which corresponds to $f(x) = a'x+b$ and the classification rule is $sign(f(x))$. Now, my question is, since the original formulation is based on estimating a function $f(x)$, does this implicitly means that the input is non-ambiguous ?.  To be clear,  support I got two input features, $x_i$ and $x_j$, and corresponding labels $y_i$ and $y_j$.  Suppose $x_i = x_j$ but $y_i \neq y_j$.  In case of density estimation based approaches like logistic regression, it makes sense to use this data, as output is a probability function.  But in case of function estimator based approaches, how can we use this data ?.   I know there are ways to convert svm scores to probability values, but the original formulation is based on function estimation.
 A: SVM works through the minimization of a loss function, and in this case, the loss function would just try to minimize whatever data you have (even if "logically impossible") - So I don't really see the problem? 
I don't think SVM assumes that there is no noise present (which is how I understand what you are saying), but rather it just doesn't explicitly assume a model on the noise (like you would in the probabilistic case). 
Loss functions and probability models on the noise are very similar in some sense, some are even equivalent in some sense (squared loss and Gaussian-noise models often coincide).
A: With a discriminative approach $f(x) = a'x+b$ your data should be linear separable. If they are not, you can make nonlinear mapping $\phi$ to transform input data and make them linear separable. In such case you will use $f(x) = a'\phi(x)+b$. 
In case that you have $x_i = x_j$ but $y_i \neq y_j$, your data is not linear separable, and you cannot define any mapping $\phi$ which will transform it to linear separable. So, in theory you should not use linear classifier $f$. However, many discriminative classifiers can be used with such data (like soft-margin SVM) as they not require that all data must be correctly classified during the training process.
A: No, there is no assumption of the inputs being deterministic. We always assume that the data are random variables, so are "noisy".
Regression models aim at making predictions as close as possible to the labels, so in the regression case, the prediction would lie somewhere in between the two identical points with different labels, where the meaning of "in-between" would depend on the loss function you used. In the classification case, the model returns a score and minimizes the loss between the score and the labels. The score could be probability than it is easier to interpret, but also can be something else like in your example. As you noticed, there are ways to turn those scores into values between zero and one that can be interpreted as probabilities.
When doing hard classifications, you would also have a decision rule (the sign function in your example) that turns the score into discrete labels. While with soft classifications the model could accommodate contradictory labels by predicting the "undecided" score when doing hard classifications you might need to make an arbitrary decision and pick one of the labels.

I know there are ways to convert svm scores to probability values, but the original formulation is based on function estimation.

Yes, so what? You need to distinguish between a mathematical model and its implementation. The mathematical model assumes that we are dealing with random variables, etc. The implementation of the algorithm deals with numerical data and doesn't know, nor care, if it is deterministic or "random". You can use least squares to fit a regression line to a set of deterministic points, or you can use it as a statistical model. It would work the same in both cases, but the interpretation of the results would be different.
