In L_aux = α·N·Σ f_i·p_i, which term receives gradients from the balancing loss?

The correct answer is p_i only, the mean softmax routing probability for expert i across the batch. In L_aux = α·N·Σ f_i·p_i, p_i is a smooth function of router weights (softmax output averaged over the batch). Gradients flow through p_i into the router's linear projection weights. f_i is the hard fraction of tokens actually dispatched to expert i after top-k selection. The discrete argmax/top-k blocks gradients, f_i is treated as a constant during the aux loss backward pass.

Only p_i receives gradients from the balancing loss. The soft term p_i is the batch-averaged softmax routing probability, differentiable w.r.t. router weights. f_i is the hard dispatch fraction from top-k counts. The discrete routing decision blocks backprop. Switch Transformer routes aux-loss gradients through p_i only, not f_i. Option A wrongly claims f_i backprops via straight-through. Option C claims full top-k differentiability, false. Option D claims aux loss is logging-only, it is added to L_total and trained jointly.