title: The Nature of Mathematical Modeling and Probabilistic Optimization
    Engineering in Generative AI
summary: In this paper, we give an in-depth analysis on the mathematical problem
  formulations and the probabilistic optimization explorations for some of the
  key components in Transformer model [33] in the field of generative AI. We
  explore and discuss some potential further enhancement for current state of the
  art methods for some key underlying technologies of generative AI models from
  algorithmic and probabilistic optimization perspective. In particular, we
  present an optimal solution for sub-word encoding (SWE) based on similar
  initial settings as that of byte-pair encoding (BPE) algorithm in [9] with
  similar objectives as that of WordPiece approach in [28, 31] to maximize the
  likelihood of the training data. We also present cross entropy optimization
  method to optimize hyperparameters for word2vec model [17]. In addition, we
  propose a factored combination of rotary positional encoding (RoPE) [32] and
  attention with linear biases (ALiBi) [23] with a harmonic series. We also
  present a probabilistic FlashAttention [6, 7] (PrFlashAttention) method with a
  probability distribution over block distances in the matrix to decide which
  block is likely to participate in a given round of attention computation while
  maintaining the lower triangle shape of the tensor for autoregressive language
  models by re-shaping the tensors. Finally, we present staircase adaptive
  quantization (SAQ) of key-value (KV) cache for multi-query attention (MQA)
  based on the framework presented in [16] to have gradual quantization
  degradation while achieving reasonable model quality and cost savings.
paper_authors._id: 671af9f682e53a391ca9cfed
paper_authors.name: Fulu Li
paper_authors.status: claimed_verified
paper_authors.statusLastChangedAt: 2024-10-25T09:30:12.356Z
paper_authors._id: 670d586a5cde0d9af5008ad0
paper_authors.avatarUrl: /avatars/be510dee91e6101bb1d79978e76e65ee.svg
paper_authors.fullname: Fulu Li
paper_authors.type: user
paper_authors.user: Fulu2024
paper_title: The Nature of Mathematical Modeling and Probabilistic Optimization
    Engineering in Generative AI
paper_summary: In this paper, we give an in-depth analysis on the mathematical problem
  formulations and the probabilistic optimization explorations for some of the
  key components in Transformer model [33] in the field of generative AI. We
  explore and discuss some potential further enhancement for current state of the
  art methods for some key underlying technologies of generative AI models from
  algorithmic and probabilistic optimization perspective. In particular, we
  present an optimal solution for sub-word encoding (SWE) based on similar
  initial settings as that of byte-pair encoding (BPE) algorithm in [9] with
  similar objectives as that of WordPiece approach in [28, 31] to maximize the
  likelihood of the training data. We also present cross entropy optimization
  method to optimize hyperparameters for word2vec model [17]. In addition, we
  propose a factored combination of rotary positional encoding (RoPE) [32] and
  attention with linear biases (ALiBi) [23] with a harmonic series. We also
  present a probabilistic FlashAttention [6, 7] (PrFlashAttention) method with a
  probability distribution over block distances in the matrix to decide which
  block is likely to participate in a given round of attention computation while
  maintaining the lower triangle shape of the tensor for autoregressive language
  models by re-shaping the tensors. Finally, we present staircase adaptive
  quantization (SAQ) of key-value (KV) cache for multi-query attention (MQA)
  based on the framework presented in [16] to have gradual quantization
  degradation while achieving reasonable model quality and cost savings.
paper_ai_summary: The paper explores mathematical formulations and probabilistic optimization techniques for various components of Transformer models in generative AI, proposing enhancements such as sub-word encoding, cross entropy optimization, factored rotary positional encoding, probabilistic FlashAttention, and staircase adaptive quantization.
paper_ai_keywords: sub-word encoding byte-pair encoding WordPiece cross entropy optimization hyperparameters word2vec rotary positional encoding attention with linear biases FlashAttention probabilistic FlashAttention staircase adaptive quantization key-value cache multi-query attention