Neural nets with implicit layers

A unifying framework for various networks, including neural ODEs, where our layers are not simple forward operations but who exacluation is represented as some optimisation problem.

For some info see the NeurIPS 2020 tutorial, Deep Implicit Layers - Neural ODEs, Deep Equilibirum Models, and Beyond, by Zico Kolter, David Duvenaud, and Matt Johnson.

NB: This is different to the implicit representation method. Since implicit layers and implicit representation layers also occur in the same problems (such as ML PDES this terminological confusion will haunt us.

Differentiable Convex Optimization Layers introduces cvxpylayers:

Optimization layers add domain-specific knowledge or learnable hard constraints to machine learning models. Many of these layers solve convex and constrained optimization problems of the form

\[ \begin{array}{rl} x^{\star}(\theta)=\operatorname{argmin}_{x} & f(x ; \theta) \\ \text { subject to } g(x ; \theta) & \leq 0 \\ h(x ; \theta) & =0 \end{array} \]

with parameters θ, objective f, and constraint functions g,h and do end-to-end learning through them with respect to θ.

In this tutorial we introduce our new library cvxpylayers for easily creating differentiable new convex optimization layers. This lets you express your layer with the CVXPY domain specific language as usual and then export the CVXPY object to an efficient batched and differentiable layer with a single line of code. This project turns every convex optimization problem expressed in CVXPY into a differentiable layer.

A different, although AFAICT equivalent, terminology is used Stephen Gould in Gould, Hartley, and Campbell (2019), under the banner of Deep Declarative Networks. Fun applications he highlight: robust losses in pooling layers, projection onto shapes, convex programming and warping, matching problems, (relaxed) graph alignment, noisy point-cloud surface reconstruction… (I am sitting in his seminar as I write this.) They have example code (pytorch). He relates some minimax-like optimisations to “Stackelberg games” which are an optimisation problem embedded in game theory.

That provokes certain other ideas: Learning basis decomposition, hyperparameter optimisation… That last one Stephen relates to this one by discussing both problems as “bi-level optimisation problems”.

Related: Deep equilibrium networks (Bai, Kolter, and Koltun 2019; Bai, Koltun, and Kolter 2020).