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In the contemporary big data realm, Deep Neural Networks (DNNs) are evolving towards more complex architectures to achieve higher inference accuracy. Model compression techniques can be leveraged to efficiently deploy such compute-intensive architectures on resource-limited mobile devices. Such methods comprise various hyper-parameters that require per-layer customization to ensure high accuracy. Choosing such hyper-parameters is cumbersome as the pertinent search space grows exponentially with model layers. This paper introduces GeneCAI, a novel optimization method that automatically learns how to tune per-layer compression hyper-parameters. We devise a bijective translation scheme that encodes compressed DNNs to the genotype space. The optimality of each genotype is measured using a multi-objective score based on accuracy and number of floating point operations. We develop customized genetic operations to iteratively evolve the non-dominated solutions towards the optimal Pareto front, thus, capturing the optimal trade-off between model accuracy and complexity. GeneCAI optimization method is highly scalable and can achieve a near-linear performance boost on distributed multi-GPU platforms. Our extensive evaluations demonstrate that GeneCAI outperforms existing rule-based and reinforcement learning methods in DNN compression by finding models that lie on a better accuracy-complexity Pareto curve.