gradient descent, $v$ does not decrease monotonically under RRR evolution. An easy way to show that the RRR flow field $F$ cannot be the gradient of a function is to construct toy examples (particular sets $A$ and $B$) in two dimensions where the flow has limit-cycle behavior. Limit cycles, if abundant, pose a possible trapping mechanism for RRR, analogous to the local minima faced by SGD. Empirically we have seen very little evidence of limit-cycle trapping, even in challenging small models where gradient methods consistently fail because of the local minimum problem.

There are roughly three kinds of constraints that define the sets $A$ and $B$ in our application of RRR to neural network training. These are constraints associated with (i) neuron inputs, (ii) neuron outputs, and (iii) “consensus" for replicated variables. The input constraints, one for each neuron, have the form $$\label{inputcon}
x\cdot w=y,$$ where $w$ and $x$ are the vectors of weights and outputs of other neurons that combine to give the neuron’s pre-activation value $y$. We consign this constraint to set $B$, so that $P_B$ applied to an arbitrary $(w,x,y)$ finds the distance minimizing change $(w,x,y)\to (w_B,x_B,y_B)$ that satisfies . We see that weights $w$ and neuron 