A supervised learning algorithm searches over a set of functions A → B parametrised by a space P to find the best approximation to some ideal function f : A → B. It does this by taking examples (a, f (a)) ∈ A × B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent-with respect to a fixed step size and an error function satisfying a certain property-defines a monoidal functor from a category of parametrised functions to this category of update rules. A key contribution is the notion of request function. This provides a structural perspective on backpropagation, giving a broad generalisation of neural networks and linking it with structures from bidirectional programming and open games.

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Metabolism plays a central role in evolution, as resource conservation is a selective pressure for fitness and survival. Resource-driven adaptations offer a good model to study evolutionary innovation more broadly. It remains unknown how resource-driven optimization of genome function integrates chromatin architecture with transcriptional phase transitions. Here we show that tuning of genome architecture and heterotypic transcriptional condensates mediate resilience to nutrient limitation. Network genomic integration of phenotypic, structural, and functional relationships reveals that fat tissue promotes organismal adaptations through metabolic acceleration chromatin domains and heterotypic PGC1A condensates. We find evolutionary adaptations in several dimensions; low conservation of amino acid residues within protein disorder regions, nonrandom chromatin location of metabolic acceleration domains, condensate-chromatin stability through cis-regulatory anchoring and encoding of genome plasticity in radial chromatin organization. We show that environmental tuning of these adaptations leads to fasting endurance, through efficient nuclear compartmentalization of lipid metabolic regions, and, locally, human-specific burst kinetics of lipid cycling genes. This process reduces oxidative stress, and fatty-acid mediated cellular acidification, enabling endurance of condensate chromatin conformations. Comparative genomics of genetic and diet perturbations reveal mammalian convergence of phenotype and structural relationships, along with loss of transcriptional control by diet-induced obesity. Further, we find that radial transcriptional organization is encoded in functional divergence of metabolic disease variant-hubs, heterotypic condensate composition, and protein residues sensing metabolic variation. During fuel restriction, these features license the formation of large heterotypic condensates that buffer proton excess, and shift viscoelasticity for condensate endurance. This mechanism maintains physiological pH, reduces pH-resilient inflammatory gene programs, and enables genome plasticity through transcriptionally driven cell-specific chromatin contacts. In vivo manipulation of this circuit promotes fasting-like adaptations with heterotypic nuclear compartments, metabolic and cell-specific homeostasis. In sum, we uncover here a general principle by which transcription uses environmental fluctuations for genome function, and demonstrate how resource conservation optimizes transcriptional self-organization through robust feedback integrators, highlighting obesity as an inhibitor of genome plasticity relevant for many diseases.

Metabolism plays a central role in evolution, as resource conservation is a selective pressure for fitness and survival. Resource-driven adaptations offer a good model to study evolutionary innovation more broadly. It remains unknown how resource-driven optimization of genome function integrates chromatin architecture with transcriptional phase transitions. Here we show that tuning of genome architecture and heterotypic transcriptional condensates mediate resilience to nutrient limitation. Network genomic integration of phenotypic, structural, and functional relationships reveals that fat tissue promotes organismal adaptations through metabolic acceleration chromatin domains and heterotypic PGC1A condensates. We find evolutionary innovation in several dimensions; low conservation of amino acid residues within protein disorder regions, nonrandom chromatin location of metabolic acceleration domains, condensate-chromatin stability through cis-regulatory archoring and encoding of genome plasticity in radial chromatin organization. We show that environmental tuning of these adaptations leads to fasting endurance, through efficient nuclear compartmentalization of lipid metabolic regions, and, locally, human-specific burst kinetics of lipid cycling genes. This process reduces oxidative stress, and fatty-acid mediated cellular acidification, enabling endurance of condensate chromatin conformations. Comparative genomics of genetic and diet perturbations reveal mammalian convergence of phenotype and structural relationships, along with loss of transcriptional control by diet-induced obesity. Further, we find that radial transcriptional organization is encoded in functional divergence of metabolic disease variant-hubs, heterotypic condensate composition, and evolutionary tuned protein residues sensing metabolic variation. During fuel restriction, these features license the formation of large heterotypic condensates that buffer proton excess, and shift viscoelasticity for condensate endurance. This mechanism maintains physiological pH, reduces pH-resilient inflammatory gene programs, and enables genome plasticity through transcriptionally driven cell-specific chromatin contacts. In vivo manipulation of this circuit promotes fasting-like adaptations with heterotypic nuclear compartments, metabolic and cell-specific homeostasis. In sum, we uncover here a general principle by which transcription uses environmental fluctuations for genome function, and demonstrate how resource conservation optimizes transcriptional self-organization through robust feedback integrators, highlighting obesity as an inhibitor of genome plasticity relevant for many diseases.