FA_Ex

Last updated Feb 27, 2024 Edit Source

• Hebbian
• Phd
• Differential

#hebbian #phd
2024-02-10 16:57

Honestly just seems like it’s a fancy way to say, “Multiply the hebbian rule by your error signal”. It was bringing backprop and Feedback alignment into it, idk. The mathematical notation is weak in this paper. It’s super unclear whether the output is used in the weights update or if it’s the derivative of the output. It may actually, if the latter case is true, be more like differential hebbian learning with a global training/reward signal tacked on.

Here’s a diagram of what they essentially are doing: FA_Ex-J5ETYZL4

the caption: Schematic illustration of three learning rules:

1. BP (backpropagation): Requires the information in $W_2$ to backpropagate.
2. FA (feedback alignment): Requires heterogeneity in the tentative impact of the middle layer neurons on the output.
3. FA_Ex-100% (feedback alignment with 100% excitatory neurons in middle layer): Most biologically plausible, as it computes using locally available information at a synaptic triad. Its performance is fairly good and comparable to BP.

With the notations:

• $y_i^{(1)} := f\left( \sum_j W_{ij}^{(1)} x_j \right)$
• $y^{(2)} := f\left( \sum_i W_i^{(2)} y_i^{(1)} \right)$
• $J := \frac{e^2}{2} = \frac{(y^{(2)} - y)^2}{2}$

The gradient for BP is given by: $$ \frac{\partial J}{\partial W_{ij}^{(1)}} = e W_i^{(2)} f’\left( \sum_j W_{ij}^{(1)} x_j \right) \cdot x_j $$

In FA, $W_i^{(2)}$ is replaced by a random number $B_i$, and in FA_Ex-100%, it’s replaced by 1. Therefore, for FA_Ex-100%, the synaptic weights in the middle layer are updated by the following rule: $$ (\Delta W){ij} = -0.01 \frac{\partial J}{\partial W{ij}^{(1)}} = -0.01 x_j \theta(I_i) e $$

Here, $\theta$ is a step function and $I_i$ is the current input to neuron $i$. This can be interpreted as: $$ (\Delta W)_{ij} \propto \text{pre}_i \times \text{post}_j \times \text{dopamine} $$

Interestingly, simplified and biologically plausible learning rules like FA_Ex-100% work robustly as long as the error signal, possibly conveyed by dopaminergic neurons, is accurate.

The original FA_Ex-100% rule is given by: $$ (1W){ij} = -0.01 \frac{\partial J}{\partial W{ij}^1} = -0.01 x_j \theta(I_i) e $$ where:

• $(1W)_{ij}$: Change in weight for the synapse from neuron $j$ to neuron $i$.
• $-0.01$: Learning rate.
• $x_j$: Pre-synaptic activity (input to the synapse).
• $\theta(I_i)$: Step function of the input to neuron $i$, representing post-synaptic activity.
• $e$: Error signal, related to the difference between actual and desired output.

The revised FA_Ex-100% rule using standard error signal notation is: $$ \Delta W_{ij} = \eta \cdot x_j \cdot \theta(I_i) \cdot \frac{\partial L}{\partial y_i} $$ where:

• $\Delta W_{ij}$: Change in weight for the synapse from neuron $j$ to neuron $i$.
• $\eta$: Learning rate.
• $x_j$: Pre-synaptic activity.
• $\theta(I_i)$: Step function of the input to neuron $i$, representing post-synaptic activity.
• $\frac{\partial L}{\partial y_i}$: Error signal, represented as the derivative of the loss function with respect to the output of neuron $i$.