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Overparameterization and overfitting are common concerns when designing and training deep neural networks, that are often counteracted by pruning and regularization strategies. However, these strategies remain secondary to most learning approaches and suffer from time and computational intensive procedures. We suggest a multiobjective perspective on the training of neural networks by treating its prediction accuracy and the network complexity as two individual objective functions in a biobjective optimization problem. As a showcase example, we use the cross entropy as a measure of the prediction accuracy while adopting an l1-penalty function to assess the total cost (or complexity) of the network parameters. The latter is combined with an intra-training pruning approach that reinforces complexity reduction and requires only marginal extra computational cost. From the perspective of multiobjective optimization, this is a truly large-scale optimization problem. We compare two different optimization paradigms: On the one hand, we adopt a scalarization-based approach that transforms the biobjective problem into a series of weighted-sum scalarizations. On the other hand we implement stochastic multi-gradient descent algorithms that generate a single Pareto optimal solution without requiring or using preference information. In the first case, favorable knee solutions are identified by repeated training runs with adaptively selected scalarization parameters. Preliminary numerical results on exemplary convolutional neural networks confirm that large reductions in the complexity of neural networks with neglibile loss of accuracy are possible.