Distributed optimization of deeply nested systems (1212.5921v1)

Abstract: In science and engineering, intelligent processing of complex signals such as images, sound or language is often performed by a parameterized hierarchy of nonlinear processing layers, sometimes biologically inspired. Hierarchical systems (or, more generally, nested systems) offer a way to generate complex mappings using simple stages. Each layer performs a different operation and achieves an ever more sophisticated representation of the input, as, for example, in an deep artificial neural network, an object recognition cascade in computer vision or a speech front-end processing. Joint estimation of the parameters of all the layers and selection of an optimal architecture is widely considered to be a difficult numerical nonconvex optimization problem, difficult to parallelize for execution in a distributed computation environment, and requiring significant human expert effort, which leads to suboptimal systems in practice. We describe a general mathematical strategy to learn the parameters and, to some extent, the architecture of nested systems, called the method of auxiliary coordinates (MAC). This replaces the original problem involving a deeply nested function with a constrained problem involving a different function in an augmented space without nesting. The constrained problem may be solved with penalty-based methods using alternating optimization over the parameters and the auxiliary coordinates. MAC has provable convergence, is easy to implement reusing existing algorithms for single layers, can be parallelized trivially and massively, applies even when parameter derivatives are not available or not desirable, and is competitive with state-of-the-art nonlinear optimizers even in the serial computation setting, often providing reasonable models within a few iterations.

• The paper introduces the Method of Auxiliary Coordinates (MAC) to transform deeply nested optimization challenges into tractable constrained problems.
• MAC effectively decouples system layers by employing alternating penalty-based optimization, significantly simplifying deep neural network training.
• Empirical results demonstrate rapid convergence and competitive performance, offering a practical solution for distributed and parallel computing environments.

Distributed Optimization of Deeply Nested Systems

The paper "Distributed Optimization of Deeply Nested Systems," authored by Miguel A. Carreira-Perpiñán and Weiran Wang, addresses a complex and pressing issue in the field of machine learning: optimizing hierarchical systems. These systems, often represented as deep neural networks, pose significant challenges due to their nonlinear and deeply nested nature. The computational complexity involved in jointly estimating the parameters of all layers and selecting an optimal architecture is daunting. The paper proposes a method called the Method of Auxiliary Coordinates (MAC) as a novel approach to tackle these challenges.

Summary of the Method

MAC is a strategy designed to simplify the optimization of nested systems by transforming the original, deeply nested problem into a constrained optimization problem in an augmented space that does not involve nesting. This is achieved by introducing auxiliary coordinates, which serve as stand-ins for intermediate states within the nested functions, effectively decoupling the system and making the optimization problem more tractable.

The optimization of MAC leverages alternating optimization methods via penalty-based strategies. MAC's strengths lie in its proven convergence, ease of implementation, and ability to be trivially parallelized. The approach is compatible even with functions where derivatives are unavailable or undesirable, thus broadening its application scope significantly.

Numerical Results and Claims

The authors demonstrate the efficacy of MAC through empirical results, showing that it is competitive with state-of-the-art nonlinear optimizers, even when computations are performed serially. In experimental settings, MAC consistently provides reasonable models within a few iterations, making it an efficient solution for real-world applications.

Implications and Future Directions

Practically, MAC can be employed in various domains where distributed and cloud computing environments are leveraged due to its ability to massively parallelize computations. Theoretically, this approach opens new avenues for optimization in complex hierarchical models, presenting an alternative to traditional backpropagation-based methods.

In terms of future developments in AI, MAC's capability to unlock efficient optimization for deep networks suggests a potential shift in strategies for training models with numerous parameters and layers. Continued exploration might include fine-tuning the introduction of auxiliary coordinates for specific architectures or extending MAC's application to recurrent or heterogeneous networks.

In conclusion, while the paper refrains from sensationalizing its contributions, the approach outlined is a significant step towards addressing the computational challenges posed by deeply nested systems. By enabling a more efficient way to handle these problems, MAC contributes robustly to the advancement of machine learning frameworks and methodologies, suggesting promising directions for future research and applications in AI.