Citadel·Machine Learning Engineer·Technical Phone Screen·Senior

Senior

Jun 2026

Summary

Citadel ML Engineer interview hit me with a deep-dive on MoE routing gradients. One question, very technical, felt more like a research discussion than a standard interview.

Questions Asked (1)

You have a Mixture-of-Experts router using hard routing decisions like argmax, which aren't differentiable. How would you modify it to make the routing learnable via gradient descent? Walk through at least two approaches, compare them on training stability, output quality, compute cost, and implementation complexity, and explain how you'd handle things like load balancing and expert collapse.

System DesignTechnical Trade-offsAlgorithms & Data Structures

Author's notes

This one took me a second to organize because there's genuinely a lot to say.

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AI HintsAI Generated

Suggested Approach

Start by explaining why argmax is non-differentiable and the need for a differentiable relaxation. Then present two concrete approaches: (1) softmax with temperature annealing and (2) Gumbel-Softmax (or Concrete) relaxation. Compare them across the requested dimensions and discuss load balancing and expert collapse mitigation.

Pro tip: Emphasize that the choice depends on whether you need hard routing at inference; Gumbel-Softmax allows straight-through estimation for hard decisions while keeping gradients, but temperature annealing is critical for stability.

1. Explain the problem

Argmax is non-differentiable, so gradients cannot flow to the router. We need a differentiable approximation that still yields near-hard routing.

2. Approach 1: Softmax with temperature

Replace argmax with softmax over router logits, optionally with a temperature parameter that is annealed to sharpen the distribution. This makes routing fully differentiable.

3. Approach 2: Gumbel-Softmax (Concrete) relaxation

Use Gumbel-Softmax to sample a differentiable approximation of a one-hot vector. With straight-through, you can use hard routing in the forward pass and soft gradients in the backward pass.

4. Compare approaches

Discuss training stability, output quality, compute cost, and implementation complexity for each. Softmax is simpler but may produce soft mixtures; Gumbel-Softmax is more complex but enables hard routing.

5. Handle load balancing and expert collapse

Introduce auxiliary losses: load balancing loss to encourage equal expert usage, and importance loss to prevent collapse. Also consider noise injection or capacity constraints.

Key Points to Mention

Softmax with temperature annealing: simple, but routing remains soft unless annealed; can lead to expert collapse if not balanced.
Gumbel-Softmax with straight-through estimator: allows hard routing at inference, but adds noise and requires tuning temperature.
Load balancing loss: auxiliary loss to penalize uneven expert utilization, often based on coefficient of variation or entropy.
Expert collapse: when a few experts dominate; mitigated by load balancing, noise, and capacity factors.
Compute cost: softmax is cheaper; Gumbel-Softmax adds sampling overhead but can be efficient with vectorized ops.
Training stability: Gumbel-Softmax can be unstable due to noise; temperature annealing and gradient clipping help.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.