Machine learning models with inputs in a Euclidean space Rd, when implemented on digital computers, generalize, and their generalization gap converges to 0 at a rate ofc/N1/2concerning the sample sizeN. However, the constantc> 0 obtained through classical methods can be large in terms of the ambient dimensiondand machine precision, posing a challenge whenNis small to realistically large. In this paper, we derive a family of generalization bounds {cm/N1/(2∨m)}∞m=1tailored for learning models on digital computers, which adapt to both the sample sizeNand the so-called geometric representation dimensionmof the discrete learning problem. Adjusting the parametermaccording toNresults in significantly tighter generalization bounds for practical sample sizesN, while settingmsmall maintains the optimal dimension-free worst-case rate ofO(1/N1/2). Notably,cm∈O(m1/2) for learning models on discretized Euclidean domains. Furthermore, our adaptive generalization bounds are formulated based on our new non-asymptotic result for concentration of measure in finite metric spaces, established via leveraging metric embedding arguments.

We study the learning problem associated with spiking neural networks. Specifically, we focus on spiking neural networks composed of simple spiking neurons having only positive synaptic weights, equipped with an affine encoder and decoder; we refer to these as affine spiking neural networks. These neural networks are shown to depend continuously on their parameters, which facilitates classical covering number-based generalization statements and supports stable gradient-based training. We demonstrate that the positivity of the weights enables a wide range of expressivity results, including rate-optimal approximation of smooth functions and dimension-independent approximation of Barron regular functions. In particular, we show in theory and simulations that affine spiking neural networks are capable of approximating shallow ReLU neural networks. Furthermore, we apply these affine spiking neural networks to standard machine learning benchmarks and reach competitive results (with respect to deep feedforward networks). Finally, we observe that from a generalization perspective, contrary to feedforward neural networks or previous results for general spiking neural networks, the depth has little to no adverse effect on theoretical guarantees for the generalization capabilities.