This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracyi.e., the approximation error decays exponentially in the number of nonzero weights in the network-of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function-a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

We analyze approximation rates by deep ReLU networks of a class of multivariate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReLU architectures capable of efficiently approximating tensor products. Combining this with results concerning the approximation of well-behaved (i.e., fulfilling some smoothness properties) univariate functions, this provides insights into rates of deep ReLU approximation of multivariate functions with tensor structures. We apply this in particular to the model problem given by the price of a European maximum option on a basket of d assets within the Black–Scholes model for European maximum option pricing. We prove that the solution to the d-variate option pricing problem can be approximated up to an $$\varepsilon $$ ε -error by a deep ReLU network with depth $${\mathcal {O}}\big (\ln (d)\ln (\varepsilon ^{-1})+\ln (d)^2\big )$$ O ( ln ( d ) ln ( ε - 1 ) + ln ( d ) 2 ) and $${\mathcal {O}}\big (d^{2+\frac{1}{n}}\varepsilon ^{-\frac{1}{n}}\big )$$ O ( d 2 + 1 n ε - 1 n ) nonzero weights, where $$n\in {\mathbb {N}}$$ n ∈ N is arbitrary (with the constant implied in $${\mathcal {O}}(\cdot )$$ O ( · ) depending on n). The techniques developed in the constructive proof are of independent interest in the analysis of the expressive power of deep neural networks for solution manifolds of PDEs in high dimension.

Background: Due to the ongoing COVID-19 pandemic, demand for diagnostic testing has increased drastically, resulting in shortages of necessary materials to conduct the tests and overwhelming the capacity of testing laboratories. The supply scarcity and capacity limits affect test administration: priority must be given to hospitalized patients and symptomatic individuals, which can prevent the identification of asymptomatic and presymptomatic individuals and hence effective tracking and tracing policies. We describe optimized group testing strategies applicable to SARS-CoV-2 tests in scenarios tailored to the current COVID-19 pandemic and assess significant gains compared to individual testing.Methods: We account for biochemically realistic scenarios in the context of dilution effects on SARS-CoV-2 samples and consider evidence on specificity and sensitivity of PCR-based tests for the novel coronavirus. Because of the current uncertainty and the temporal and spatial changes in the prevalence regime, we provide analysis for several realistic scenarios and propose fast and reliable strategies for massive testing procedures.Key Findings: We find significant efficiency gaps between different group testing strategies in realistic scenarios for SARS-CoV-2 testing, highlighting the need for an informed decision of the pooling protocol depending on estimated prevalence, target specificity, and high- vs. low-risk population. For example, using one of the presented methods, all 1.47 million inhabitants of Munich, Germany, could be tested using only around 141 thousand tests if the infection rate is below 0.4% is assumed. Using 1 million tests, the 6.69 million inhabitants from the city of Rio de Janeiro, Brazil, could be tested as long as the infection rate does not exceed 1%. Moreover, we provide an interactive web application, available at www.grouptexting.com, for visualizing the different strategies and designing pooling schemes according to specific prevalence scenarios and test configurations.Interpretation: Altogether, this work may help provide a basis for an efficient upscaling of current testing procedures, which takes the population heterogeneity into account and is fine-grained towards the desired study populations, e.g., mild/asymptomatic individuals vs. symptomatic ones but also mixtures thereof.Funding: German Science Foundation (DFG), German Federal Ministry of Education and Research (BMBF), Chan Zuckerberg Initiative DAF, and Austrian Science Fund (FWF).

Background: Due to the ongoing COVID-19 pandemic, demand for diagnostic testing has increased drastically, resulting in shortages of necessary materials to conduct the tests and overwhelming the capacity of testing laboratories. The supply scarcity and capacity limits affect test administration: priority must be given to hospitalized patients and symptomatic individuals, which can prevent the identification of asymptomatic and presymptomatic individuals and hence effective tracking and tracing policies. We describe optimized group testing strategies applicable to SARS-CoV-2 tests in scenarios tailored to the current COVID-19 pandemic and assess significant gains compared to individual testing. Methods:We account for biochemically realistic scenarios in the context of dilution effects on SARS-CoV-2 samples and consider evidence on specificity and sensitivity of PCR-based tests for the novel coronavirus. Because of the current uncertainty and the temporal and spatial changes in the prevalence regime, we provide analysis for a number of realistic scenarios and propose fast and reliable strategies for massive testing procedures. Findings:We find significant efficiency gaps between different group testing strategies in realistic scenarios for SARS-CoV-2 testing, highlighting the need for an informed decision of the pooling protocol depending on estimated prevalence, target specificity, and high-vs. low-risk population. For example, using one of the presented methods, all 1·47 million inhabitants of Munich, Germany, could be tested using only around 141 thousand tests if the infection rate is below 0·4% is assumed. Using 1 million tests, the 6·69 million inhabitants from the city of Rio de Janeiro, Brazil, could be tested as long as the infection rate does not exceed 1%.Interpretation: Altogether this work may help provide a basis for efficient upscaling of current testing procedures, taking the population heterogeneity into account and fine grained towards the desired study populations, e.g. cross-sectional versus health-care workers and adapted mixtures thereof.

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