The Nature of Mathematical Modeling and Probabilistic Optimization Engineering in Generative AI

TL;DR

This paper analyzes mathematical and probabilistic optimization techniques in generative AI, proposing novel solutions for sub-word encoding, hyperparameter tuning, positional encoding, attention mechanisms, and quantization to enhance model performance.

Contribution

It introduces new optimization methods and model enhancements for key components in Transformer-based generative AI models, advancing algorithmic and probabilistic approaches.

Findings

Optimal sub-word encoding solution similar to BPE and WordPiece.

Cross entropy method effectively optimizes hyperparameters for word2vec.

Proposed probabilistic FlashAttention improves attention efficiency.

Abstract

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.