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One of the demanding frontiers in ultracold quantum science is identifying laser cooling schemes for complex atoms and molecules out of their vast spectra of internal states. Motivated by the prospect of expanding the set of available ultracold molecules for applications in fundamental physics, chemistry, astrochemistry, and quantum simulation, we propose and demonstrate an automated graph-based search approach for viable laser cooling schemes. The method is time efficient, reproduces the results of previous manual searches, and reveals a plethora of new potential laser cooling schemes. We discover laser cooling schemes for YO, ${\mathrm{C}}_2, {\mathrm{OH}}^{+}$, CN, and ${\mathrm{CO}}_2$, including surprising schemes that start from highly excited states or do not rely on a strong main transition. A central insight of this work is that the reinterpretation of quantum states and transitions between them as a graph can dramatically enhance the ability to identify new quantum control schemes for complex quantum systems. As such, this approach will also apply to complex atoms and, in fact, any complex many-body quantum system with a discrete spectrum of internal states. 

The second-order properties of the training loss have a massive impact on the optimization dynamics of deep learning models. Fort & Scherlis (2019) discovered that a large excess of positive curvature and local convexity of the loss Hessian is associated with highly trainable initial points located in a region coined the "Goldilocks zone". Only a handful of subsequent studies touched upon this relationship, so it remains largely unexplained. In this paper, we present a rigorous and comprehensive analysis of the Goldilocks zone for homogeneous neural networks. In particular, we derive the fundamental condition resulting in excess of positive curvature of the loss, explaining and refining its conventionally accepted connection to the initialization norm. Further, we relate the excess of positive curvature to model confidence, low initial loss, and a previously unknown type of vanishing cross-entropy loss gradient. To understand the importance of excessive positive curvature for trainability of deep networks, we optimize fully-connected and convolutional architectures outside the Goldilocks zone and analyze the emergent behaviors. We find that strong model performance is not perfectly aligned with the Goldilocks zone, calling for further research into this relationship.