Maximum Principle Based Algorithms for Deep Learning (1710.09513v4)

Abstract: The continuous dynamical system approach to deep learning is explored in order to devise alternative frameworks for training algorithms. Training is recast as a control problem and this allows us to formulate necessary optimality conditions in continuous time using the Pontryagin's maximum principle (PMP). A modification of the method of successive approximations is then used to solve the PMP, giving rise to an alternative training algorithm for deep learning. This approach has the advantage that rigorous error estimates and convergence results can be established. We also show that it may avoid some pitfalls of gradient-based methods, such as slow convergence on flat landscapes near saddle points. Furthermore, we demonstrate that it obtains favorable initial convergence rate per-iteration, provided Hamiltonian maximization can be efficiently carried out - a step which is still in need of improvement. Overall, the approach opens up new avenues to attack problems associated with deep learning, such as trapping in slow manifolds and inapplicability of gradient-based methods for discrete trainable variables.

• The paper introduces a continuous control formulation for neural network training using Pontryagin’s Maximum Principle.
• It develops an extended method of successive approximations (E-MSA) to overcome limitations of traditional gradient methods.
• Empirical tests on MNIST benchmarks show that E-MSA delivers improved per-iteration performance compared to SGD and Adam.

Maximum Principle Based Algorithms for Deep Learning

The paper "Maximum Principle Based Algorithms for Deep Learning," explores an innovative perspective on training deep neural networks by employing techniques from optimal control theory. The authors propose to recast the problem of training as a control problem, leveraging the continuous dynamical systems approach. Under this formulation, the authors apply Pontryagin's Maximum Principle (PMP) to derive necessary optimality conditions for this setup. This fresh approach aims not only to provide rigorous error estimates and convergence guarantees but also to address specific limitations of traditional gradient-based optimization methods\null.

The central thesis involves utilizing the PMP, a well-established method in control theory, to derive an alternative training algorithm for deep learning. This proposition is rooted in the continuous-time perspective, where the dynamics of neural networks are explored as ordinary differential equations (ODEs), a viewpoint that parallels the structure of Residual Networks. The paper presents a detailed theoretical framework and algorithmic adaptations necessary to embed PMP within standard deep learning workflows. Additionally, a method inspired by the classical "method of successive approximations" (MSA) is adapted and advanced (termed E-MSA) to provide improved convergence properties.

Key Contributions

1. Continuous Dynamical Systems Approach: The authors introduce a theoretical framework where neural network training is modeled using continuous dynamical systems, allowing for the embedding of optimal control techniques.
2. Pontryagin's Maximum Principle: By adopting PMP, the training problem is framed as an optimization of the Hamiltonian, effectively separating forward and backward propagations, akin to gradient computations in traditional neural networks.
3. Modified Method of Successive Approximations (E-MSA): A novel algorithm, E-MSA, is designed to overcome the divergence issues of basic MSA. The extended version introduces an augmented Hamiltonian, which includes penalty terms to maintain feasibility and convergence.
4. Error Estimates and Convergence: The authors provide rigorous analysis of error estimates, highlighting that E-MSA not only offers fast initial descent but also shows resilience against slow convergence typical of gradient methods near saddle points or flat landscapes.
5. Empirical Validation: Through numerical experiments, E-MSA demonstrates promising results on both synthetic problems and established benchmarks such as MNIST and Fashion MNIST, indicating better per-iteration performance compared to models optimized using traditional methods like SGD and Adam.

Implications and Future Directions

The proposed framework and resulting algorithms offer significant implications for the field of deep learning. The separation of optimization from propagation—facilitated by PMP—suggests potential for parallelization and reduced dependency on precise gradient information, addressing some inherent challenges of current gradient-based approaches. Further, the ability to work with discrete parameter spaces hints at applicability to networks with discrete weights, relevant for memory-constrained environments such as edge devices.

Looking forward, the paper opens several avenues for research:

• Efficiency of Hamiltonian Maximization: Developing specialized algorithms for more efficient Hamiltonian maximization is crucial for real-world scalability.
• Application to More Complex Models: Testing E-MSA on deeper, more complex architectures and datasets such as ImageNet to validate practical advantages.
• Extension to Discrete and Hybrid Neural Architectures: Adapting PMP-based training to networks with mixed or discrete weights, broadening the approach's utility.

In conclusion, the integration of optimal control principles within deep learning training, as articulated in this paper, offers a promising avenue towards more efficient and potentially powerful training algorithms. By departing from the traditional reliance on gradient information, the approach could signal a substantive shift in how neural networks are optimized and deployed.