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Tasks across diverse application domains can be posed as large-scale optimization problems, these include graphics, vision, machine learning, imaging, health, scheduling, planning, and energy system forecasting. Independently of the application domain, proximal algorithms have emerged as a formal optimization method that successfully solves a wide array of existing problems, often exploiting problem-specific structures in the optimization. Although model-based formal optimization provides a principled approach to problem modeling with convergence guarantees, at first glance, this seems to be at odds with black-box deep learning methods. A recent line of work shows that, when combined with learning-based ingredients, model-based optimization methods are effective, interpretable, and allow for generalization to a wide spectrum of applications with little or no extra training data. However, experimenting with such hybrid approaches for different tasks by hand requires domain expertise in both proximal optimization and deep learning, which is often error-prone and time-consuming. Moreover, naively unrolling these iterative methods produces lengthy compute graphs, which when differentiated via autograd techniques results in exploding memory consumption, making batch-based training challenging. In this work, we introduce ∇-Prox, a domain-specific modeling language and compiler for large-scale optimization problems using differentiable proximal algorithms. ∇-Prox allows users to specify optimization objective functions of unknowns concisely at a high level, and intelligently compiles the problem into compute and memory-efficient differentiable solvers. One of the core features of ∇-Prox is its full differentiability, which supports hybrid model- and learning-based solvers integrating proximal optimization with neural network pipelines. Example applications of this methodology include learning-based priors and/or sample-dependent inner-loop optimization schedulers, learned with deep equilibrium learning or deep reinforcement learning. With a few lines of code, we show ∇-Prox can generate performant solvers for a range of image optimization problems, including end-to-end computational optics, image deraining, and compressive magnetic resonance imaging. We also demonstrate ∇-Prox can be used in a completely orthogonal application domain of energy system planning, an essential task in the energy crisis and the clean energy transition, where it outperforms state-of-the-art CVXPY and commercial Gurobi solvers.more » « lessFree, publicly-accessible full text available August 1, 2024
Free, publicly-accessible full text available June 1, 2024
Early evolution of mutualism is characterized by big and predictable adaptive changes, including the specialization of interacting partners, such as through deleterious mutations in genes not required for metabolic cross-feeding. We sought to investigate whether these early mutations improve cooperativity by manifesting in synergistic epistasis between genomes of the mutually interacting species. Specifically, we have characterized evolutionary trajectories of syntrophic interactions of Desulfovibrio vulgaris (Dv) with Methanococcus maripaludis (Mm) by longitudinally monitoring mutations accumulated over 1000 generations of nine independently evolved communities with analysis of the genotypic structure of one community down to the single-cell level. We discovered extensive parallelism across communities despite considerable variance in their evolutionary trajectories and the perseverance within many evolution lines of a rare lineage of Dv that retained sulfate-respiration (SR+) capability, which is not required for metabolic cross-feeding. An in-depth investigation revealed that synergistic epistasis across pairings of Dv and Mm genotypes had enhanced cooperativity within SR− and SR+ assemblages, enabling their coexistence within the same community. Thus, our findings demonstrate that cooperativity of a mutualism can improve through synergistic epistasis between genomes of the interacting species, enabling the coexistence of mutualistic assemblages of generalists and their specialized variants.