Physarum Powered Differentiable Linear Programming Layers and Applications

Consider a learning algorithm, which involves an internal call to an optimization routine such as a generalized eigenvalue problem, a cone programming problem or even sorting. Integrating such a method as layers within a trainable deep network in a numerically stable way is not simple -- for instance, only recently, strategies have emerged for eigendecomposition and differentiable sorting. We propose an efficient and differentiable solver for general linear programming problems which can be used in a plug and play manner within deep neural networks as a layer. Our development is inspired by a fascinating but not widely used link between dynamics of slime mold (physarum) and mathematical optimization schemes such as steepest descent. We describe our development and demonstrate the use of our solver in a video object segmentation task and meta-learning for few-shot learning. We review the relevant known results and provide a technical analysis describing its applicability for our use cases. Our solver performs comparably with a customized projected gradient descent method on the first task and outperforms the very recently proposed differentiable CVXPY solver on the second task. Experiments show that our solver converges quickly without the need for a feasible initial point. Interestingly, our scheme is easy to implement and can easily serve as layers whenever a learning procedure needs a fast approximate solution to a LP, within a larger network

On the Convergence Time of a Natural Dynamics for Linear Programming

We consider a system of nonlinear ordinary differential equations for the solution of linear programming (LP) problems that was first proposed in the mathematical biology literature as a model for the foraging behavior of acellular slime mold Physarum polycephalum, and more recently considered as a method to solve LP instances. We study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and derive a new time bound to approximate optimality that depends on the relative entropy between projected versions of the optimal point and of the initial point. The bound scales logarithmically with the LP cost coefficients and linearly with the inverse of the relative accuracy, establishing the efficiency of the dynamics for arbitrary LP instances with positive costs

Two Results on Slime Mold Computations

We present two results on slime mold computations. In wet-lab experiments (Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems. Biologists proposed a mathematical model, a system of differential equations, for the slime's adaption process (J. Theoretical Biology'07). It was shown that the process convergences to the shortest path (J. Theoretical Biology'12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector. Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can $\varepsilon$-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on $\varepsilon$, and a number of steps with quartic dependence on $\mathrm{opt}/(\varepsilon\Phi)$, where $\Phi$ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost ($\mathrm{opt}$). We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of $\varepsilon$, and the number of steps depends logarithmically on $1/\varepsilon$ and quadratically on $\mathrm{opt}/\Phi$

Physarum Inspired Dynamics to Solve Semi-Definite Programs

Physarum Polycephalum is a Slime mold that can solve the shortest path problem. A mathematical model based on the Physarum's behavior, known as the Physarum Directed Dynamics, can solve positive linear programs. In this paper, we will propose a Physarum based dynamic based on the previous work and introduce a new way to solve positive Semi-Definite Programming (SDP) problems, which are more general than positive linear programs. Empirical results suggest that this extension of the dynamic can solve the positive SDP showing that the nature-inspired algorithm can solve one of the hardest problems in the polynomial domain. In this work, we will formulate an accurate algorithm to solve positive and some non-negative SDPs and formally prove some key characteristics of this solver thus inspiring future work to try and refine this method