# How do I interpret the following parameters as a network

I have a background in OR and I am new to AI. I know the basics and I currently try to understand the article "Learning Combinatorial Optimization Algorithms over Graphs".

So far I understand the overall procedure, however, part $$x$$ of EQ (3), is hard to grasp (I reformulated the formula in a more general way that does not require knowledge about the article)

$$\mu = relu(\theta_1 a + \theta_2 b + \theta_3 \underbrace{\sum_{i \in I} relu(\theta_4 c_i)}_{x})$$

with $$\theta_1 \in \mathbb{R}^p, \theta_2,\theta_3 \in \mathbb{R}^{p\times p}, \theta_4 \in \mathbb{R}^p$$ and $$I$$ being a set of varying cardinality for each input to be evaluated and $$a,b,c_i$$ being given inputs.

The authors claim to train the network end to end. If we replace $$x$$ by some input value (without $$\theta_4$$) I would be able to follow the idea.

However as far as I understand $$x$$, $$relu(\theta_4 c_i)$$ is a little NN on its own with input $$c_i$$ and output of length $$p$$. Sure, we can compute the values and sum them up and use them as input, however, I do not understand how the end to end training will work for such a setup where basically two networks exist.

I would be more than thankful for help on this topic.