  In a physical neural system, learning rules must be local both in space and
time. In order for learning to occur, non-local information must be
communicated to the deep synapses through a communication channel, the deep
learning channel. We identify several possible architectures for this learning
channel (Bidirectional, Conjoined, Twin, Distinct) and six symmetry challenges:
1) symmetry of architectures; 2) symmetry of weights; 3) symmetry of neurons;
4) symmetry of derivatives; 5) symmetry of processing; and 6) symmetry of
learning rules. Random backpropagation (RBP) addresses the second and third
symmetry, and some of its variations, such as skipped RBP (SRBP) address the
first and the fourth symmetry. Here we address the last two desirable
symmetries showing through simulations that they can be achieved and that the
learning channel is particularly robust to symmetry variations. Specifically,
random backpropagation and its variations can be performed with the same
non-linear neurons used in the main input-output forward channel, and the
connections in the learning channel can be adapted using the same algorithm
used in the forward channel, removing the need for any specialized hardware in
the learning channel. Finally, we provide mathematical results in simple cases
showing that the learning equations in the forward and backward channels
converge to fixed points, for almost any initial conditions. In symmetric
architectures, if the weights in both channels are small at initialization,
adaptation in both channels leads to weights that are essentially symmetric
during and after learning. Biological connections are discussed.
