Error Update operation

The error update calculation follows the equations below.

Hn=Hn−1+λ∗f−1(xn∗en)H_n = H_{n-1} + \lambda*f^{-1}(x_n * e_n)

bn=bn−1+λ∗f−1(en)b_n = b_{n-1} + \lambda*f^{-1}(e_n)

The operation uses the same structure as the feedforward operation with a different input configuration. For this case, the data uses the normal port, the error uses the tap port and the tap use the bias port. This allows the same usage of the circuit with some extra glue logic.

The block diagram of the operation is shown below.

graph LR Add(+) Mult(*) subgraph MACN Mult-->Add Data0-->Mult Tap-->Mult end subgraph MAC0 Mult1(*)-->Add1(+) DataN-->Mult1 Tap1-->Mult1 end subgraph DelayLine Add-->Rn Add1-->R0 end Data-->Data0 Data-->DataN R0 -->BiasAdd(+) Bias-->BiasAdd BiasAdd-->Nonlinearity Nonlinearity-->Output

The operation shown above is very similar to the feedforward operation with the following differences

  1. The tap input is the error (This is made more efficient by storing the tap and error in the same memory)
  2. The output of the block is parallel and is written to the tap memory
  3. The bias update is a scalar addition not shown in the diagram which uses the error directly

Data Flow

The data flow for this operation is as follows :

  1. The error data is fed to the block through the tap input and stored
  2. The tap data is fed through the tap port like normal operation
  3. The input data (scala) is multiplied by error (vector)
  4. The bias input is fed with the tap inputs
  5. The output of the accumulator is fed back to the memory

The ordering of the operations is shown below

Type 0 1 K K+1 N
Tap (Vector) E0 E0 E0 E1 E1
Data D0 D1 DK D0 DK
Bias B0 B1 BK B0 BK