What are the impacts of choosing different loss functions in classification to approximate 0-1 loss We know that some objective functions are easier to optimize and some are hard. And there are many loss functions that we want to use but hard to use, for example 0-1 loss. So we find some proxy loss functions to do the work. For example, we use hinge loss or logistic loss to "approximate" 0-1 loss. 
Following plot is coming from Chris Bishop's PRML book. The Hinge Loss is plotted in blue, the Log Loss in red, the Square Loss in green and the 0/1 error in black.

I understand the reason we have such design (for hinge and logistic loss) is we want the objective function to be convex. 
By looking at hinge loss and logistic loss, it penalize more on strongly misclassified instances, and interestingly, it also penalize correctly classified instances if they are weakly classified. It is a really strange design. 
My question is what are the prices we need to pay by using different "proxy loss functions", such as hinge loss and logistic loss?
 A: Posting a late reply, since there is a very simple answer which has not been mentioned yet. 

what are the prices we need to pay by using different "proxy loss functions", such as hinge loss and logistic loss?

When you replace the non-convex 0-1 loss function by a convex
surrogate (e.g hinge-loss), you are actually now solving a different problem than the one you intended to solve (which is to minimize the number of classification mistakes). So you gain computational tractability (the problem becomes convex, meaning you can solve it efficiently using tools of convex optimization), but in the general case there is actually no way to relate the error of the classifier that minimizes a "proxy" loss and the error of the classifier that minimizes the 0-1 loss. If what you truly cared about was minimizing the number of misclassifications, I argue that this really is a big price to pay.
I should mention that this statement is worst-case, in the sense that it holds for any distribution $\mathcal D$. For some "nice" distributions, there are exceptions to this rule. The key example is of data distributions that have large margins w.r.t the decision boundary - see Theorem 15.4 in Shalev-Shwartz, Shai, and Shai Ben-David. Understanding machine learning: From theory to algorithms. Cambridge university press, 2014.
A: Ideally your loss function should reflect actual loss incurred by business. For instance, if you're classifying damaged goods, then the loss of misclassification could be like this:


*

*marking damaged goods that were not: lost profit on potential sale

*not marking damaged goods that were damaged: cost of return processing

A: Some of my thoughts, may not be correct though.

I understand the reason we have such design (for hinge and logistic loss) is we want the objective function to be convex.

Convexity is surely a nice property, but I think the most important reason is we want the objective function to have non-zero derivatives, so that we can make use of the derivatives to solve it. The objective function can be non-convex, in which case we often just stop at some local optima or saddle points.

and interestingly, it also penalize correctly classified instances if
  they are weakly classified. It is a really strange design.

I think such design sort of advises the model to not only make the right predictions, but also be confident about the predictions. If we don't want correctly classified instances to get punished, we can for example, move the hinge loss (blue) to the left by 1, so that they no longer get any loss. But I believe this often leads to worse result in practice.

what are the prices we need to pay by using different "proxy loss
  functions", such as hinge loss and logistic loss?

IMO by choosing different loss functions we are bringing different assumptions to the model. For example, the logistic regression loss (red) assumes a Bernoulli distribution, the MSE loss (green) assumes a Gaussian noise.

Following the least squares vs. logistic regression example in PRML, I added the hinge loss for comparison.

As shown in the figure, hinge loss and logistic regression / cross entropy / log-likelihood / softplus have very close results, because their objective functions are close (figure below), while MSE is generally more sensitive to outliers.  Hinge loss does not always have a unique solution because it's not strictly convex.

However one important property of hinge loss is, data points far away from the decision boundary contribute nothing to the loss, the solution will be the same with those points removed. 
The remaining points are called support vectors in the context of SVM. Whereas SVM uses a regularizer term to ensure the maximum margin property and a unique solution.
