The $k$-nearest-neighbor language model ($k$NN-LM), one of the retrieval-augmented language models, improves the perplexity for given text by directly accessing a large datastore built from any text data during inference. A widely held hypothesis for the success of $k$NN-LM is that its explicit memory, i.e., the datastore, enhances predictions for long-tail phenomena. However, prior works have primarily shown its ability to retrieve long-tail contexts, leaving the model's performance remain underexplored in estimating the probabilities of long-tail target tokens during inference. In this paper, we investigate the behavior of $k$NN-LM on low-frequency tokens, examining prediction probability, retrieval accuracy, token distribution in the datastore, and approximation error of the product quantization. Our experimental results reveal that $k$NN-LM does not improve prediction performance for low-frequency tokens but mainly benefits high-frequency tokens regardless of long-tail contexts in the datastore.

Gboard Decoder produces suggestions by looking for paths that best match input touch points on the context aware search space, which is backed by the language Finite State Transducers (FST). The language FST is currently an N-gram language model (LM). However, N-gram LMs, limited in context length, are known to have sparsity problem under device model size constraint. In this paper, we propose \textbf{Neural Search Space} which substitutes the N-gram LM with a Neural Network LM (NN-LM) and dynamically constructs the search space during decoding. Specifically, we integrate the long range context awareness of NN-LM into the search space by converting its outputs given context, into the language FST at runtime. This involves language FST structure redesign, pruning strategy tuning, and data structure optimizations. Online experiments demonstrate improved quality results, reducing Words Modified Ratio by [0.26\%, 1.19\%] on various locales with acceptable latency increases. This work opens new avenues for further improving keyboard decoding quality by enhancing neural LM more directly.