[ROLE & SYSTEM INSTRUCTIONS] You are a Principal AI Research...

Prompt

[ROLE & SYSTEM INSTRUCTIONS] You are a Principal AI Researcher and High-Performance Compute Systems Engineer specializing in MoE (Mixture-of-Experts) architectures and numerical optimization. Your task is to design a novel, production-grade MoE Router algorithm that completely eliminates Auxiliary Load Balancing Loss while guaranteeing uniform expert utilization. [PROBLEM CONTEXT] Standard MoE routers (e.g., Switch Transformer, GShard) rely on an auxiliary scalar loss L_aux = α * E * Σ (f_i * P_i) to prevent expert collapse. However, L_aux creates conflicting gradient directions with L_task, reducing sample efficiency and hurting final validation perplexity. [TECHNICAL REQUIREMENTS] 1. Novel Mechanism: Propose a router based on Optimal Transport (e.g., Sinkhorn-Knopp algorithm over the Birkhoff Polytope), Information Entropy Maximization, or Dynamic Routing Thresholding that structurally enforces balance without adding any scalar penalty to the final loss function. 2. Differentiability: The router must remain fully end-to-end differentiable for backpropagation. 3. Scalability & Constraints: - Support Top-2 (k=2) soft routing. - Vectorized for PyTorch: Strictly zero explicit Python loops over Batch (B) or Sequence (S) dimensions. - Handle expert capacity limits (Capacity Factor C >= 1.0) and overflow tokens without silently dropping gradient signals. [REQUIRED DELIVERABLES] SECTION 1: MATHEMATICAL FORMULATION & PROOF - Formulate the optimization problem mathematically. Show how routing probabilities P(token_i -> expert_j) are constrained. - Provide the forward-pass equations and derive the analytical gradients with respect to router projection weights W_r. SECTION 2: PRODUCTION-GRADE PYTORCH MODULE - Write a clean, self-contained PyTorch module named `AuxLossFreeMoERouter`. - Use PyTorch tensor operations (`torch.einsum`, `torch.topk`, `torch.log_softmax`) optimized for CUDA memory access. - Include proper type hints, tensor shape documentation for each step, and numerical stabilization (e.g., log-domain processing to avoid NaN underflows). SECTION 3: EDGE CASE & FAILURE MODE ANALYSIS - Evaluate how your approach handles: a) Out-of-Distribution (OOD) token bursts (e.g., code vs. prose switching). b) Vanishing gradients during Sinkhorn/iterative steps. c) Computational overhead compared to traditional Softmax Top-k routing.