Differentiable Convex Optimization Layers (1910.12430v1)

Abstract: Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver's solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program. We implement our methodology in version 1.1 of CVXPY, a popular Python-embedded DSL for convex optimization, and additionally implement differentiable layers for disciplined convex programs in PyTorch and TensorFlow 2.0. Our implementation significantly lowers the barrier to using convex optimization problems in differentiable programs. We present applications in linear machine learning models and in stochastic control, and we show that our layer is competitive (in execution time) compared to specialized differentiable solvers from past work.

• The paper introduces Disciplined Parametrized Programming (DPP) and the Affine-Solver-Affine form to differentiate convex programs efficiently.
• It implements these methods in CVXPY 1.1 with support from PyTorch and TensorFlow, significantly lowering usability barriers.
• The approach achieves competitive performance in applications like linear machine learning models and stochastic control while enhancing model expressiveness.

Differentiable Convex Optimization Layers

The paper entitled "Differentiable Convex Optimization Layers" by Agrawal et al. presents innovative advancements in the integration of differentiable convex optimization into deep learning architectures. This work expands upon recent research in which optimization layers are embedded into neural networks, allowing gradients to propagate through optimization problems and thus enhancing model expressiveness and performance.

Overview and Contributions

The research delineates a methodology to efficiently differentiate through disciplined convex programs, a particularly useful subclass of optimization problems easily handled by domain-specific languages (DSLs) for convex optimization. The approach addresses the limitations of existing differentiation software that are typically rigid and hard to adapt to novel scenarios.

The authors introduce a new grammar, Disciplined Parametrized Programming (DPP), which extends Disciplined Convex Programming (DCP) to parameterize optimization problems in a way that maintains differentiability. Central to the approach is the development of the Affine-Solver-Affine (ASA) form that standardizes the way convex programs are represented, ensuring that differentiability is preserved across problem instances.

The significant contributions of this work include:

1. Development of DPP and ASA Form: The paper details a novel approach to parameterized convex optimization problems, enabling efficient differentiation. This allows the creation of differentiable layers without direct backpropagation through the canonicalization process.
2. Implementation and Integration: The methodology is implemented in CVXPY 1.1, a prominent Python DSL for convex optimization, with additional support through PyTorch and TensorFlow 2.0. This implementation greatly reduces usability barriers, facilitating the seamless incorporation of convex optimization in differentiable programs.
3. Applications and Performance: Robust applications in linear machine learning models and stochastic control are showcased. Importantly, the research finds that their differentiable layers are competitive with specialized solvers, such as OptNet, in terms of execution time, reflecting the efficiency of the proposed method.

Practical and Theoretical Implications

The integration of differentiable convex optimization layers into machine learning models provides significant theoretical and practical implications:

• Inductive Bias: Incorporating these layers introduces an inductive bias that can be highly beneficial for specific learning tasks. This bias can guide the learning process, resulting in models that generalize better to unseen data.
• Complex Problem Handling: Differentiable convex optimization layers extend the capacity to handle complex constraints and objectives within neural networks, which are typically challenging to address through conventional means.
• Broader Applications: The work opens up new possibilities for applications such as adversarial attacks in machine learning, sensitivity analysis, and approximate dynamic programming, showcasing the versatility of this approach.

Future Directions

Looking forward, the research points to several exciting avenues for further exploration and development:

• Extension to Other Optimization Classes: Expanding the framework to support other optimization problem classes, such as nonlinear programs, could enhance its applicability across even more domains.
• Solver Interfacing: Efforts to interface the framework with specialized solvers (e.g., for quadratic programs) could optimize performance further, enabling faster solutions while maintaining accuracy.
• Nonconvex Problem Differentiation: Investigating similar methods for differentiating through nonconvex optimization problems presents an intriguing challenge, potentially broadening the applicability of differentiable optimization layers.

This paper provides a solid foundation for advancing the integration of convex optimization into deep learning, offering both concrete software tools and a conceptual framework that could inspire further innovations in the intersection of optimization and machine learning.