# Multiple solutions with same minima in MLP with same weights

I came across an excercise on deep learning from here. It goes as follows: Consider a simple MLP with a single hidden layer of $$d$$ dimensions in the hidden layer and a single output. Show that for any local minimum there are at least $$d!$$ equivalent solutions that behave identically.

As the network is a MLP with one hidden layer, the equation would be: $$O = W^{(2)}(W^{(1)}x + b_1) + b_2$$

Assuming I am correct, where do I need to go from here to get to the solution?

If one permutes the connections of the hidden layer ($$d!$$ ways to do that), and move and rename connections appropriately, then one effectively has the same MLP with the exact same minima, yet the configuration has changed (in a trivial sense). Thus there are (at least) $$d!$$ configurations only trivialy different with the exact same minima.

To see it in your notation, effectively the hidden layer output is the following sum:

$$O_{\text{hidden}} = \sum_{i=1}^d w_{\pi_i} \cdot x + b_{\pi_i}$$

Where $$\pi_i$$ is some order of the connections. For example $$\pi_i = i$$

But for $$d$$ items there are $$d!$$ permutations thus $$d!$$ order functions $$\pi(i) = \pi_i$$. Yet the difference is only in re-ordering the configuration and is trivial. The rest follow from that.

This is one reason why Neural Networks are non-convex models.

• Excellent explanation May 6, 2021 at 4:23