Apollo: An Adaptive Parameter-wise Diagonal Quasi-Newton Method for Nonconvex Stochastic Optimization (2009.13586v6)

Abstract: In this paper, we introduce Apollo, a quasi-Newton method for nonconvex stochastic optimization, which dynamically incorporates the curvature of the loss function by approximating the Hessian via a diagonal matrix. Importantly, the update and storage of the diagonal approximation of Hessian is as efficient as adaptive first-order optimization methods with linear complexity for both time and memory. To handle nonconvexity, we replace the Hessian with its rectified absolute value, which is guaranteed to be positive-definite. Experiments on three tasks of vision and language show that Apollo achieves significant improvements over other stochastic optimization methods, including SGD and variants of Adam, in term of both convergence speed and generalization performance. The implementation of the algorithm is available at https://github.com/XuezheMax/apollo.

• The paper introduces a diagonal quasi-Newton approach that adjusts learning rates per parameter for enhanced efficiency.
• The paper addresses nonconvex challenges by rectifying the Hessian to maintain positive definiteness and guarantee stable convergence.
• The paper demonstrates Apollo's superiority over SGD and Adam variants in convergence speed and generalization through extensive experiments.

Overview of the Apollo Optimization Method

The paper introduces Apollo, a quasi-Newton method specifically designed to enhance the efficiency of nonconvex stochastic optimization. Apollo innovatively addresses several challenges associated with training deep neural networks through a diagonally approximated Hessian matrix, incorporating curvature dynamically in a manner reminiscent of adaptive first-order optimization methods. This paper systematically examines Apollo's efficacy compared to standard methods such as stochastic gradient descent (SGD) and various Adam variants, highlighting both theoretical advancements and empirical results.

Technical Contributions

1. Diagonal Quasi-Newton Approach: Apollo differentiates itself from traditional quasi-Newton methods by utilizing a parameter-wise diagonal approximation of the Hessian matrix. This strategy maintains the computational simplicity seen in adaptive first-order methods while affording Apollo the ability to dynamically adjust its learning rates across different parameters.
2. Handling Nonconvexity: Nonconvex optimization is inherently challenging due to the potential negative curvature of the Hessian. Apollo addresses this by employing the rectified absolute value of the Hessian, ensuring the matrix remains positive definite. This crucial step guarantees stable convergence across nonconvex landscapes without resorting to costly line search methods.
3. Efficiency and Scalability: The method achieves linear complexity for both time and memory, making it highly applicable to large-scale machine learning tasks. Apollo's storage requirements and computational costs are significantly reduced compared to traditional second-order methods, aligning them with those of first-order methods.
4. Convergence and Experimentation: The authors provide rigorous convergence analysis, demonstrating Apollo's effectiveness not only in convex optimization scenarios but also in more complex nonconvex tasks. Empirical results across computer vision and natural language processing tasks exhibit Apollo's superior convergence speed and generalization performance compared to existing methodologies, including AdaHessian and adaptive variants of SGD.

Empirical Results

Apollo's performance was evaluated on three diverse tasks comprised of computer vision and language processing applications. Noteworthy improvements over SGD and Adam variants were observed in both convergence speed and the ability to generalize. This underscores Apollo's potential for real-world adoption where nonconvex landscapes are common.

Implications and Future Directions

The research into Apollo suggests substantial implications for accelerating the training of deep neural networks, particularly in resource-intensive applications such as image classification and LLMing. While the paper focuses on the diagonal approximation of the Hessian, future work might investigate full-matrix approximations or additional techniques to further reduce bifurcation issues in large-scale networks. Furthermore, extending Apollo's framework to accommodate distributed learning environments could also be an advantageous pursuit.

In conclusion, Apollo represents a significant step forward in the quest to optimize complex nonconvex functions efficiently. By marrying quasi-Newton updates with parameter-wise adaptability, Apollo stands as a promising alternative to existing optimization methods, unlocking potential enhancements in speed and accuracy for large-scale machine learning models.