State-space models (SSMs) and transformers dominate the language modeling landscape. However, they are constrained to a lower computational complexity than classical recurrent neural networks (RNNs), limiting their expressivity. In contrast, RNNs lack parallelization during training, raising fundamental questions about the trade off between parallelization and expressivity. We propose implicit SSMs, which iterate a transformation until convergence to a fixed point. Theoretically, we show that implicit SSMs implement the non-linear state-transitions of RNNs. Empirically, we find that only approximate fixed-point convergence suffices, enabling the design of a scalable training curriculum that largely retains parallelization, with full convergence required only for a small subset of tokens. Our approach demonstrates superior state-tracking capabilities on regular languages, surpassing transformers and SSMs. We further scale implicit SSMs to natural language reasoning tasks and pretraining of large-scale language models up to 1.3B parameters on 207B tokens - representing, to our knowledge, the largest implicit model trained to date. Notably, our implicit models outperform their explicit counterparts on standard benchmarks. Our code is publicly available at github.com/microsoft/implicit_languagemodels

Holographic Storage is an old idea that has always promised high density and fast random access, but has never been commercially competitive with Hard Disk Drives (HDDs) and Solid State Devices (SSDs). In Project HSD at Microsoft Research we asked the question: “Does holographic storage finally make sense for cloud storage?” This article describes our journey toward answering this question. We achieved 1.8× higher density than the previous state-of-the-art, using commodity components available today and leveraging machine learning to compensate for the noise and distortions introduced by commodity components. This uncovered two new challenges which are the focus of this article: achieving high end-to-end energy efficiency without sacrificing capacity, and spatial multiplexing without mechanical movement. Improving end-to-end energy efficiency requires joint optimization across low-level media parameters and higher-level system parameters that govern background maintenance operations such as read refresh and garbage collection. We developed new physics models of the media; analytic and simulation models of the media access and background media maintenance; and workload-driven optimization to find optimal parameter combinations. These techniques resulted in a 14× improvement over the previous approach for typical workloads without sacrificing capacity. We also designed the first scalable and mechanical movement free spatial multiplexing system for holographic storage. Despite these advances, we conclude that currently, holographic storage is still far from the combination of density, capacity scaling, and energy efficiency needed to compete with the incumbent technologies. We need fundamental advances in the physical media that improve energy efficiency by another 1–2 orders of magnitude without reducing data density. Further advances in optics are also required to achieve spatial multiplexing that is simultaneously scalable, low-loss, and high-density.