Model training requires significantly more memory, compared with inference. Parameter efficient fine-tuning (PEFT) methods provide a means of adapting large models to downstream tasks using less memory. However, existing methods such as adapters, prompt tuning or low-rank adaptation (LoRA) either introduce latency overhead at inference time or achieve subpar downstream performance compared with full fine-tuning. In this work we propose Random Subspace Adaptation (ROSA), a method that outperforms previous PEFT methods by a significant margin, while maintaining a zero latency overhead during inference time. In contrast to previous methods, ROSA is able to adapt subspaces of arbitrarily large dimension, better approximating full-finetuning. We demonstrate both theoretically and experimentally that this makes ROSA strictly more expressive than LoRA, without consuming additional memory during runtime. As PEFT methods are especially useful in the natural language processing domain, where models operate on scales that make full fine-tuning very expensive, we evaluate ROSA in two common NLP scenarios: natural language generation (NLG) and natural language understanding (NLU) with GPT-2 and RoBERTa, respectively. We show that on almost every GLUE task ROSA outperforms LoRA by a significant margin, while also outperforming LoRA on NLG tasks. Our code is available at https://github.com/rosa-paper/rosa

Second-order Recurrent Neural Networks (2RNNs) extend RNNs by leveraging second-order interactions for sequence modelling. These models are provably more expressive than their first-order counterparts and have connections to well-studied models from formal language theory. However, their large parameter tensor makes computations intractable. To circumvent this issue, one approach known as MIRNN consists in limiting the type of interactions used by the model. Another is to leverage tensor decomposition to diminish the parameter count. In this work, we study the model resulting from parameterizing 2RNNs using the CP decomposition, which we call CPRNN. Intuitively, the rank of the decomposition should reduce expressivity. We analyze how rank and hidden size affect model capacity and show the relationships between RNNs, 2RNNs, MIRNNs, and CPRNNs based on these parameters. We support these results empirically with experiments on the Penn Treebank dataset which demonstrate that, with a fixed parameter budget, CPRNNs outperforms RNNs, 2RNNs, and MIRNNs with the right choice of rank and hidden size.