Dung Le

Fanqi Yan, Huy Nguyen, Dung Le et al.

We conduct the convergence analysis of parameter estimation in the contaminated mixture of experts. This model is motivated from the prompt learning problem where ones utilize prompts, which can be formulated as experts, to fine-tune a large-scale pre-trained model for learning downstream tasks. There are two fundamental challenges emerging from the analysis: (i) the proportion in the mixture of the pre-trained model and the prompt may converge to zero during the training, leading to the prompt vanishing issue; (ii) the algebraic interaction among parameters of the pre-trained model and the prompt can occur via some partial differential equations and decelerate the prompt learning. In response, we introduce a distinguishability condition to control the previous parameter interaction. Additionally, we also investigate various types of expert structure to understand their effects on the convergence behavior of parameter estimation. In each scenario, we provide comprehensive convergence rates of parameter estimation along with the corresponding minimax lower bounds. Finally, we run several numerical experiments to empirically justify our theoretical findings.

Fanqi Yan, Huy Nguyen, Dung Le et al.

The softmax-contaminated mixture of experts (MoE) model is deployed when a large-scale pre-trained model, which plays the role of a fixed expert, is fine-tuned for learning downstream tasks by including a new contamination part, or prompt, functioning as a new, trainable expert. Despite its popularity and relevance, the theoretical properties of the softmax-contaminated MoE have remained unexplored in the literature. In the paper, we study the convergence rates of the maximum likelihood estimator of gating and prompt parameters in order to gain insights into the statistical properties and potential challenges of fine-tuning with a new prompt. We find that the estimability of these parameters is compromised when the prompt acquires overlapping knowledge with the pre-trained model, in the sense that we make precise by formulating a novel analytic notion of distinguishability. Under distinguishability of the pre-trained and prompt models, we derive minimax optimal estimation rates for all the gating and prompt parameters. By contrast, when the distinguishability condition is violated, these estimation rates become significantly slower due to their dependence on the prompt convergence rate to the pre-trained model. Finally, we empirically corroborate our theoretical findings through several numerical experiments.