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Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias–variance trade-off, appears to be at odds with the observed behavior of methods used in modern machine-learning practice. The bias–variance trade-off implies that a model should balance underfitting and overfitting: Rich enough to express underlying structure in data and simple enough to avoid fitting spurious patterns. However, in modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered overfitted, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This “double-descent” curve subsumes the textbook U-shaped bias–variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine-learning models delineates the limits of classical analyses and has implications for both the theory and the practice of machine learning.

The large variation in seed mass among species inspired a vast array of theoretical and empirical research attempting to explain this variation. So far, seed mass variation was investigated by two classes of studies. One class focuses on species varying in seed mass within communities, while the second focuses on variation between communities, most often with respect to resource gradients. Here, we develop a model capable of simultaneously explaining variation in seed mass within and between communities. The model describes resource competition (for both soil and light resources) in annual communities and incorporates two fundamental aspects: light asymmetry (higher light acquisition per unit biomass for larger individuals) and growth allometry (negative dependency of relative growth rate on plant biomass). Results show that both factors are critical in determining patterns of seed mass variation. In general, growth allometry increases the reproductive success of small-seeded species while light asymmetry increases the reproductive success of large-seeded species. Increasing availability of soil resources increases light competition, thereby increasing the reproductive success of large-seeded species and ultimately the community (weighted) mean seed mass. An unexpected prediction of the model is that maximum variation in community seed mass (a measure of functional diversity) occurs under intermediate levels of soil resources. Extensions of the model incorporating size-dependent seed survival and disturbance also show patterns consistent with empirical observations. These overall results suggest that the mechanisms captured by the model are important in determining patterns of species and functional diversity.

Spoken language production involves selecting and assembling words and syntactic structures to convey one’s message. Here we probe this process by analyzing natural language productions of individuals with primary progressive aphasia (PPA) and healthy individuals. Based on prior neuropsychological observations, we hypothesize that patients who have difficulty producing complex syntax might choose semantically richer words to make their meaning clear, whereas patients with lexicosemantic deficits may choose more complex syntax. To evaluate this hypothesis, we first introduce a frequency-based method for characterizing the syntactic complexity of naturally produced utterances. We then show that lexical and syntactic complexity, as measured by their frequencies, are negatively correlated in a large (n = 79) PPA population. We then show that this syntax–lexicon trade-off is also present in the utterances of healthy speakers (n = 99) taking part in a picture description task, suggesting that it may be a general property of the process by which humans turn thoughts into speech.

Bacteria frequently encounter selection by both antibiotics and lytic bacteriophages. However, the evolutionary interactions between antibiotics and phages remain unclear, in particular, whether and when phages can drive evolutionary trade-offs with antibiotic resistance. Here, we describe Escherichia coli phage U136B, showing it relies on two host factors involved in different antibiotic resistance mechanisms: 1) the efflux pump protein TolC and 2) the structural barrier molecule lipopolysaccharide (LPS). Since TolC and LPS contribute to antibiotic resistance, phage U136B should select for their loss or modification, thereby driving a trade-off between phage resistance and either of the antibiotic resistance mechanisms. To test this hypothesis, we used fluctuation experiments and experimental evolution to obtain phage-resistant mutants. Using these mutants, we compared the accessibility of specific mutations (revealed in the fluctuation experiments) to their actual success during ecological competition and coevolution (revealed in the evolution experiments). Both tolC and LPS-related mutants arise readily during fluctuation assays, with tolC mutations becoming more common during the evolution experiments. In support of the trade-off hypothesis, phage resistance via tolC mutations occurs with a corresponding reduction in antibiotic resistance in many cases. However, contrary to the hypothesis, some phage resistance mutations pleiotropically confer increased antibiotic resistance. We discuss the molecular mechanisms underlying this surprising pleiotropic result, consideration for applied phage biology, and the importance of ecology in evolution of phage resistance. We envision that phages may be useful for the reversal of antibiotic resistance, but such applications will need to account for unexpected pleiotropy and evolutionary context.

A trade-off between growth and defence functions is commonly observed in plants. We propose that the association of plants with Epichloë fungal endophytes may eliminate this trade-off. This would be a consequence of the double role of these endophytes in host plants: the stimulation of plant growth hormones (e.g. gibberellins) and the fungal production of antiherbivore alkaloids. We put forward a model that integrates this dual effect of endophytes on plant growth and defence and test its predictions by means of meta-analysis of published literature. Our results support the notion that the enhanced plant resistance promoted by endophytes does not compromise plant growth. The limits and ecological benefits of this endophyte-mediated lack of plant growth–defence trade-off are discussed.

Positive (synergistic) and negative (trade-off) relationships among ecosystem services are influenced by drivers of change, such as policy interventions and environmental variability, and the mechanisms that link these drivers to ecosystem service outcomes. Failure to account for these drivers and mechanisms can result in poorly informed management decisions and reduced ecosystem service provision. Here, we review the literature to determine the extent to which drivers and mechanisms are considered in assessments of ecosystem service relationships. We show that only 19% of assessments explicitly identify the drivers and mechanisms that lead to ecosystem service relationships. While the proportion of assessments considering drivers has increased over time, most of these studies only implicitly consider the drivers of ecosystem service relationships. We recommend more assessments explicitly identify drivers of trade-offs and synergies, which can be achieved through a greater uptake of causal inference and process-based models, to ensure effective management of ecosystem services.

A longstanding idea in evolutionary physiology is that an enzyme cannot jointly optimize performance at both high and low temperatures due to a trade-off between stability and activity. Although a stability-activity trade-off has been observed for well-characterized examples, such a trade-off is not imposed by any physical chemical constraint. To better understand the pervasiveness of this trade-off, I investigated the stability-activity relationship for comparative biochemical studies of purified orthologous enzymes identified by a literature search. The nature of this relationship varied greatly among studies. Notably, studies of enzymes with low mean synonymous nucleotide sequence divergence were less likely to exhibit the predicted negative correlation between stability and activity. Similarly, a survey of directed evolution investigations of the stability-activity relationship indicated that these traits are often uncoupled among nearly identical yet phenotypically divergent enzymes. This suggests that the presumptive trade-off often reported for investigations of enzymes with high mean sequence divergence may in some cases instead be a consequence of the degeneration over time of enzyme function in unselected environments, rather than a direct effect of thermal adaptation. The results caution against the general assertion of a stability-activity trade-off during enzyme adaptation.

Sexual selection theory predicts a trade-off between premating (ornaments and armaments) and postmating (testes and ejaculates) sexual traits, assuming that growing and maintaining these traits is costly and that total reproductive investments are limited. The number of males in competition, the reproductive gains from investing in premating sexual traits, and the level of sperm competition are all predicted to influence how males allocate their finite resources to these traits. Yet, empirical examination of these predictions is currently scarce. Here, we studied relative expenditure on pre- and postmating sexual traits among frog species varying in their population density, operational sex ratio, and the number of competing males for each clutch of eggs. We found that the intensifying struggle to monopolize fertilizations as more and more males clasp the same female to fertilize her eggs shifts male reproductive investment toward sperm production and away from male weaponry. This shift, which is mediated by population density and the associated level of male–male competition, likely also explains the trade-off between pre- and postmating sexual traits in our much broader sample of anuran species. Our results highlight the power of such a multilevel approach in resolving the evolution of traits and allocation trade-offs.

1. A fundamental trade-off among vascular plants between traits inferring rapid resource acquisition and those leading to conservation of resources has now been accepted broadly, but is based on empirical data with a strong bias towards leaf traits. Here, we test whether interspecific variation in traits of different plant organs obeys this same trade-off and whether within-plant trade-offs are consistent between organs. 2. Thereto, we measured suites of the same chemical and structural traits from the main vegetative organs for a species set representing aquatic, riparian and terrestrial environments including the main vascular higher taxa and growth forms of a subarctic flora. The traits were chosen to have consistent relevance for plant defence and growth across organs and environments: carbon, nitrogen, phosphorus, lignin, dry matter content, pH. 3. Our analysis shows several new trait correlations across leaves, stems and roots and a striking pattern of whole-plant integrative resource economy, leading to tight correspondence between the local leaf economics spectrum and the root (r = 0.64), stem (r = 0.78) and whole-plant (r = 0.93) economics spectra. 4. Synthesis. Our findings strongly suggest that plant resource economics is consistent across species' organs in a subarctic flora. We provide thus the first evidence for a 'plant economics spectrum' closely related to the local subarctic 'leaf economics spectrum'. Extending that concept to other biomes is, however, necessary before any generalization might be made. In a world facing rapid vegetation change, these results nevertheless bear considerable prospects of predicting below-ground plant functions from the above-ground components alone.

The biotic mechanisms underlying ecosystem functioning and stability have been extensively—but separately—explored in the literature, making it difficult to understand the relationship between functioning and stability. In this study, we used community models to examine how complementarity and selection, the two major biodiversity mechanisms known to enhance ecosystem biomass production, affect ecosystem stability. Our analytic and simulation results show that although complementarity promotes stability, selection impairs it. The negative effects of selection on stability operate through weakening portfolio effects and selecting species that have high productivity but low tolerance to perturbations (“risk-prone” species). In contrast, complementarity enhances stability by increasing portfolio effects and reducing the relative abundance of risk-prone species. Consequently, ecosystem functioning and stability exhibit either a synergy, if complementarity effects prevail, or trade-off, if selection effects prevail. Across species richness levels, ecosystem functioning and stability tend to be positively related, but negative relationships can occur when selection co-varies with richness. Our findings provide novel insights for understanding the functioning-stability relationship, with potential implications for both ecological research and ecosystem management.