[Paper Review] Uncertainty Quantification in Deep Learning through Stochastic Maximum Principle.

Richard Archibald, Feng Bao+2 more|arXiv (Cornell University)|Nov 28, 2020

Stochastic Gradient Optimization Techniques2 references4 citations

TL;DR

This paper proposes a stochastic neural network framework grounded in stochastic optimal control theory, leveraging the stochastic maximum principle to derive an efficient stochastic gradient descent algorithm. The method enables uncertainty quantification in deep learning with theoretical convergence guarantees and empirical validation across applications.

ABSTRACT

We develop a probabilistic machine learning method, which formulates a class of stochastic neural networks by a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced under the stochastic maximum principle framework. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

Motivation & Objective

• To address the challenge of uncertainty quantification in deep learning models.
• To formulate deep learning as a stochastic optimal control problem for principled uncertainty estimation.
• To develop an efficient optimization algorithm based on the stochastic maximum principle.
• To establish theoretical convergence for the proposed stochastic gradient descent method.
• To validate the method through numerical experiments on real-world applications.

Proposed method

• The method formulates deep neural networks as stochastic optimal control problems to embed uncertainty in model parameters.
• It applies the stochastic maximum principle to derive necessary conditions for optimality, enabling gradient-based training.
• An efficient stochastic gradient descent algorithm is designed using the derived optimality conditions.
• The framework allows for end-to-end training with uncertainty-aware parameter updates.
• Theoretical convergence analysis is conducted under standard assumptions for stochastic control.
• Numerical experiments are performed to evaluate performance and uncertainty calibration.

Experimental results

Research questions

• RQ1 How can deep learning models be systematically formulated as stochastic optimal control problems to quantify uncertainty?
• RQ2 What efficient optimization algorithm can be derived from the stochastic maximum principle for training such models?
• RQ3 Does the proposed stochastic gradient descent method converge under standard assumptions?
• RQ4 How well does the method perform in practice compared to baseline uncertainty quantification techniques?
• RQ5 Can the framework be effectively applied to real-world deep learning tasks with reliable uncertainty estimates?

Key findings

• The proposed stochastic neural network framework successfully integrates uncertainty quantification into deep learning through stochastic optimal control.
• The stochastic gradient descent algorithm derived from the stochastic maximum principle demonstrates theoretical convergence.
• Numerical experiments confirm the method's effectiveness in practical applications with reliable uncertainty estimates.
• The approach provides a principled alternative to existing uncertainty quantification methods in deep learning.
• The framework enables end-to-end training with uncertainty-aware optimization, improving model robustness.
• Empirical results show competitive performance in uncertainty calibration and predictive accuracy.