Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a wellstudied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist, and if it exists the marginal model does not always respect the latent projection; they do not always satisfy a Markov property; and their graphs are not always consistent with their causal semantics. We prove that for SCMs in general each of these properties does hold under certain solvability conditions. Our work generalizes results for SCMs with cycles that were only known for certain special cases so far. We introduce the class of simple SCMs that extends the class of acyclic SCMs to the cyclic setting, while preserving many of the convenient properties of acyclic SCMs. With this paper, we aim to provide the foundations for a general theory of statistical causal modeling with SCMs.

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With the introduction of SNIP [40], it has been demonstrated that modern neural networks can effectively be pruned before training. Yet, its sensitivity criterion has since been criticized for not propagating training signal properly or even disconnecting layers. As a remedy, GraSP [70] was introduced, compromising on simplicity. However, in this work we show that by applying the sensitivity criterion iteratively in smaller steps -still before training -we can improve its performance without difficult implementation. As such, we introduce 'SNIP-it'.We then demonstrate how it can be applied for both structured and unstructured pruning, before and/or during training, therewith achieving state-of-the-art sparsityperformance trade-offs. That is, while already providing the computational benefits of pruning in the training process from the start. Furthermore, we evaluate our methods on robustness to overfitting, disconnection and adversarial attacks as well.Preprint. Under review.

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