Do autoregressive Transformer language models require explicit positional encodings (PEs)? The answer is ‘no’ provided they have more than one layer—they can distinguish sequences with permuted tokens without the need for explicit PEs. This follows from the fact that a cascade of (permutation invariant) set processors can collectively exhibit sequence-sensitive behavior in the autoregressive setting. This property has been known since early efforts (contemporary with GPT-2) adopting the Transformer for language modeling. However, this result does not appear to have been well disseminated, leading to recent rediscoveries. This may be partially due to a sudden growth of the language modeling community after the advent of GPT-2/3, but perhaps also due to the lack of a clear explanation in prior work, despite being commonly understood by practitioners in the past. Here we review the long-forgotten explanation why explicit PEs are nonessential for multi-layer autoregressive Transformers (in contrast, one-layer models require PEs to discern order information of their inputs), as well as the origin of this result, and hope to re-establish it as a common knowledge.

Recent studies of the computational power of recurrent neural networks (RNNs) reveal a hierarchy of RNN architectures, given real-time and finite-precision assumptions. Here we study auto-regressive Transformers with linearised attention, a.k.a. linear Transformers (LTs) or Fast Weight Programmers (FWPs). LTs are special in the sense that they are equivalent to RNN-like sequence processors with a fixed-size state, while they can also be expressed as the now-popular self-attention networks. We show that many well-known results for the standard Transformer directly transfer to LTs/FWPs. Our formal language recognition experiments demonstrate how recently proposed FWP extensions such as recurrent FWPs and self-referential weight matrices successfully overcome certain limitations of the LT, e.g., allowing for generalisation on the parity problem. Our code is public.

How to reduce compute and memory requirements of neural networks (NNs) without sacrificing performance? Many recent works use sparse Mixtures of Experts (MoEs) to build resource-efficient large language models (LMs). Here we introduce several novel perspectives on MoEs, presenting a general framework that *unifies* various methods to *approximate two-layer NNs* (e.g., feedforward blocks of Transformers), including product-key memories (PKMs). Leveraging insights from this framework, we propose methods to improve both MoEs and PKMs. Unlike prior work that compares MoEs with dense baselines under the *compute-equal* condition, our evaluation condition is *parameter-equal*, which is crucial to properly evaluate LMs. We show that our MoEs are competitive with the *dense* Transformer-XL on both the WikiText-103 and enwiki8 datasets at two different scales, while being much more resource efficient. This demonstrates that MoEs are relevant not only to extremely large LMs but also to any-scale resource-efficient LMs. Our code is public.

Well-designed diagnostic tasks have played a key role in studying the failure of neural nets (NNs) to generalize systematically. Famous examples include SCAN and Compositional Table Lookup (CTL). Here we introduce CTL++, a new diagnostic dataset based on compositions of unary symbolic functions. While the original CTL is used to test length generalization or productivity, CTL++ is designed to test systematicity of NNs, that is, their capability to generalize to unseen compositions of known functions. CTL++ splits functions into groups and tests performance on group elements composed in a way not seen during training. We show that recent CTL-solving Transformer variants fail on CTL++. The simplicity of the task design allows for fine-grained control of task difficulty, as well as many insightful analyses. For example, we measure how much overlap between groups is needed by tested NNs for learning to compose. We also visualize how learned symbol representations in outputs of functions from different groups are compatible in case of success but not in case of failure. These results provide insights into failure cases reported on more complex compositions in the natural language domain. Our code is public.

Recently, many datasets have been proposed to test the systematic generalization ability of neural networks. The companion baseline Transformers, typically trained with default hyper-parameters from standard tasks, are shown to fail dramatically. Here we demonstrate that by revisiting model configurations as basic as scaling of embeddings, early stopping, relative positional embedding, and Universal Transformer variants, we can drastically improve the performance of Transformers on systematic generalization. We report improvements on five popular datasets: SCAN, CFQ, PCFG, COGS, and Mathematics dataset. Our models improve accuracy from 50% to 85% on the PCFG productivity split, and from 35% to 81% on COGS. On SCAN, relative positional embedding largely mitigates the EOS decision problem (Newman et al., 2020), yielding 100% accuracy on the length split with a cutoff at 26. Importantly, performance differences between these models are typically invisible on the IID data split. This calls for proper generalization validation sets for developing neural networks that generalize systematically. We publicly release the code to reproduce our results.