Can we say that RNN for time series is an example of semi-supervised learning? I am learning neural nets, esp. focusing on RNN for my research problem. This question has nothing exactly to do with my research. 
With my understanding of RNN, I can think of it as an example of semi-supervised learning rather than supervised learning. 
It is because we do not have an exact data set (unsupervised, since no actual labels), but we use the shifted value of the input as the data set (makeshift labels). Hence this makes RNN a semi-supervised learning algorithm (at least for time series).
Am I correct in this understanding?
 A: An RNN (or any neural network for that matter) is basically just a big function of the inputs and parameters. There are supervised models which use RNNs, unsupervised models which use RNNs, and semi-supervised models which use RNNs. How something is supervised is generally independent of the architecture used.
The most "classic" use of RNNs is in language modeling, where we model $p(x) = \prod_i p(x_{i} | x_{j<i})$, and each conditional factor is computed by the RNN. This is an unsupervised model.
A: Semi-supervised refers to using a combination of a labeled dataset (usually quite small) and a (usually much larger) unlabeled dataset.
When the labels are automatically derived from the data itself, this is these days called self-supervised learning or self-supervision. Examples in time series include predicting the next time step, or filling in a gap in the sequence.
This website by Lillian Weng is a good reference for some of the many possible strategies. https://lilianweng.github.io/lil-log/2019/11/10/self-supervised-learning.html
A: 
It is because we do not have an exact data set (unsupervised, since
no actual labels), but we use the shifted value of the input as the
data set (makeshift labels).

If by this you mean that you predict something like
$$
y_t = f(y_{t-1}, y_{t-2}, \dots)
$$
then it is supervised because you have the values that are predicted. If you observed the thing that the model predicts, it is a supervised learning problem. Unsupervised learning would be when the predicted values would not be observed, like in clustering where there are no given cluster labels that are predicted.
Semi-supervised learning would be something different:

Semi-supervised learning is a class of supervised learning tasks and
techniques that also make use of unlabeled data for training –
typically a small amount of labeled data with a large amount of
unlabeled data.

So most of the time-series problems would be supervised learning problems, though a little bit different from others because the same data (but shifted) would be used both as inputs and outputs. Of course, there are unsupervised time-series models, e.g. for clustering, detecting changepoints, etc.
