Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Auxiliary Balancing Loss › Visualizing the Balancing Effect

Michael BrenndoerferNovember 17, 20252 min read

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Part 33 of 67Mixture of Experts

Chapter 244Auxiliary Balancing Loss › Visualizing the Balancing Effect

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Let's visualize how the auxiliary loss affects routing over training iterations:

In[23]:

Code

import numpy as np
import torch
import torch.nn.functional as F

# Simulate training with auxiliary loss
torch.manual_seed(42)
np.random.seed(42)

hidden_size, num_experts, expert_size = 32, 4, 64
model = MoELayer(hidden_size, num_experts, expert_size)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)

aux_losses = []
imbalance_ratios = []

for step in range(100):
    x = torch.randn(4, 16, hidden_size)
    y = torch.randn(4, 16, hidden_size)

    optimizer.zero_grad()
    output = model(x)

    task_loss = F.mse_loss(output, y)
    total_loss = (
        task_loss + 0.1 * model.aux_loss
    )  # Higher coef for faster visualization

    total_loss.backward()
    optimizer.step()

    # Track metrics
    aux_losses.append(model.aux_loss.item())

    # Compute utilization
    with torch.no_grad():
        router_probs = F.softmax(model.router(x), dim=-1)
        expert_indices = router_probs.argmax(dim=-1)
        util = compute_expert_utilization(
            router_probs, expert_indices, num_experts
        )
        imbalance_ratios.append(util["imbalance_ratio"])

Out[24]:

Visualization

Line plot of auxiliary loss vs training step. — Auxiliary loss evolution during training. The loss decreases from its initial random state toward the target value of 1.0, showing the router is learning to balance probabilities and routing.

Line plot of max/min expert ratio vs training step. — Expert imbalance ratio evolution. The ratio of maximum to minimum expert usage improves concurrently with the auxiliary loss, reducing from high imbalance to a more stable distribution.

The auxiliary loss pushes the model toward more balanced routing. As training progresses, the auxiliary loss approaches 1.0 (perfect balance), and the imbalance ratio decreases.