Comparison between cost functions to determine the "best" model?
tl;dr since you are referring to accuracy, I'm guessing you have a classification task.
There is no way to evaluate a model's actual performance, by its loss.
The goal of the loss function is to train the model and not to show how well this model classifies.
What you should do to see the best model is to evaluate it by a classification metric (e.g. accuracy). You should note that it is best to measure these on the validation set to avoid overfitting.

Some things to consider...

Different loss functions, don't measure the same thing. For example take into consideration MAE and MSE. MSE penalizes larger errors disproportionately, compared to MAE. Which is the best? Depends on the problem and what you want!

Classification depends on your model's predictions, which are by their nature continuous. However, when we are referring to the &quot;best model&quot; we want the one that classifies better. This can cause misleading effects, because a model can reduce its loss by becoming more &quot;confident&quot; in the samples it classifies as correct (usually due to overfitting).


You can look at the following as an example of this:
label | model1 | model2 
  0   | 0.995  |  0.5
  0   | 0.400  |  0.5
  1   | 0.600  |  0.5
  1   | 0.600  |  0.5

model1 is a better classifier ($3/4$ correct predictions), but model2 ($2/4$) would have the smallest loss*.
*technically this depends on the loss function, but this stands on MAE, MSE and cross entropy losses.