The transformer architecture is ubiquitously used as the building block of most large-scale language models. However, it remains a painstaking guessing game of trial and error to set its myriad of architectural hyperparameters, e.g., number of layers, number of attention heads, and inner size of the feed forward network, and find architectures with the optimal trade-off between task performance like perplexity and compute constraints like memory and latency. This challenge is further exacerbated by the proliferation of various hardware. In this work, we leverage the somewhat surprising empirical observation that the number of non-embedding parameters in autoregressive transformers has a high rank correlation with task performance, irrespective of the architectural hyperparameters. Since architectural hyperparameters affect the latency and memory footprint in a hardware-dependent manner, the above observation organically induces a simple search algorithm that can be directly run on target devices. We rigorously show that the latency and perplexity pareto-frontier can be found without need for any model training, using non-embedding parameters as a proxy for perplexity. We evaluate our method, dubbed Lightweight Transformer Search (LTS), on diverse devices from ARM CPUs to Nvidia GPUs and show that the perplexity of Transformer-XL can be achieved with up to 2x lower latency. LTS extracts the pareto-frontier in less than 3 hours while running on a commodity laptop. We effectively remove the carbon footprint of training for hundreds of GPU hours, offering a strong simple baseline for future NAS methods in autoregressive language modeling.