Is it possible to validate a deep learning model by training small data subset?
Short answer: No, because Deep Learning works well on huge amount of data.
Long answer: No. The problem is that learning only one face could overfit your model on that specific face, without learning features not present in your examples. Because for example, the model has learn to detect your face thanks to a specific, very simple, pattern in that face (that's called overfitting). 
Making a stupid simple example, your model has learn to detect that face because there is a mole on your right cheek, and it has learn to identify it
To make your model perform well on the general case, you need an huge amount of data, making your model capable to learn different kind of patterns
Suggestion:
Because the training of a deep neural network is a time consuming task, usually one does not train one single neural network at time, but many neural network are trained in parallel, with different hyperparameters (layers, nodes, activation functions, learning rate, etc).
Edit because of the discussion below:
If your dataset is small is quite impossible to have a good performance on the general case, because the neural network will learn the easiest pattern, which is usually not the general/better one. 
Adding data you force the neural network to extract good patterns, that work on the general case. 
It's a tradeoff, but usually a training on a small dataset would not lead to a good classifier on the general case
edit2: refrasing everything to make it more clear. A good performance on a small dataset don't tell you if your model when trained on all the dataset is a good model. That's why you train to 
the majority of your dataset and test/validate on a smaller dataset
I am looking to train a large model (resnet or vgg) for face identification.
Is it valid strategy to train on few faces (1..3) to validate a model?
In other words - if a model learns one face well - is it evidence that the model is good for the task?
point here is that I don't want to spend a week of GPU expensive time only to find out that my model is no good or data has errors or my TF coding has a bug