Autoregressive models have driven remarkable progress in language modeling. Their foundational reliance on discrete tokens, unidirectional context, and singlepass decoding, while central to their success, also inspires the exploration of a design space that could offer new axes of modeling flexibility. In this work, we explore an alternative paradigm, shifting language modeling from a discrete token space to a continuous latent space. We propose a novel framework TarFlowLM, that employs transformer-based autoregressive normalizing flows [73] to model these continuous representations. This approach unlocks substantial flexibility, enabling the construction of models that can capture global bi-directional context through stacked, alternating-direction autoregressive transformations, support block-wise generation with flexible token patch sizes, and facilitate a hierarchical multi-pass generation process. We further propose new mixture-based coupling transformations designed to capture complex dependencies within the latent space shaped by discrete data, and demonstrate theoretical connections to conventional discrete autoregressive models. Extensive experiments on language modeling benchmarks demonstrate strong likelihood performance and highlight the flexible modeling capabilities inherent in our framework.

For example, transactions infinancial systems, serverlogs, and user activity traces can allnaturally be represented as discrete events in continuous time.

We will understand how our assumptions on the data enable us to create meaningful optimization problems. In fact, we will derive commonly used criteria such as cross-entropy in classification and mean square error in regression. Finally, I am trying to answer an interview question that I encountered: What would happen if we use MSE on binary classification? To begin, let's start with a fundamental question: what is the difference between likelihood and probability? The data xxx are connected to the possible models θ\thetaθ by means of a probability P(x,θ)P(x,\theta)P(x,θ) or a probability density function (pdf) p(x,θ)p(x,\theta)p(x,θ).