Mathematical Foundations2 articles

Mathematical Foundations

Articles

  • Speculative Decoding: Mathematical Foundations, Exact Distribution Preservation via Rejection Sampling, and Tree-Structured Verification

    Autoregressive decoding in large language models exhibits a severe computational asymmetry. Generating text token by token requires loading every parameter of a multi-billion-parameter network into high-bandwidth memory (HBM) for every forward pass. In low-batch or single-stream inference regimes, the compute units spend the majority of execution cycles waiting for memory transfers, yielding an arithmetic intensity of order O(1) FLOP per byte. Speculative decoding resolves this bottleneck by de

    1 min
  • The Score Function Estimator: Mathematical Foundations of REINFORCE, Log-Derivative Tricks, and Baseline Variance Reduction

    In modern artificial intelligence, standard backpropagation relies on continuous differentiability: every operation between model parameters and the final loss must provide well-behaved analytical Jacobian matrices. However, many of the most critical optimization challenges in machine learning break this continuity. Autoregressive token generation in large language models, discrete tool invocation, programmatic compiler execution, and black-box reward environments are fundamentally non-different

    1 min