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Autoregressive language models are the currently dominant paradigm for text generation, but they have some fundamental limitations that cannot be remedied by scale—for example inherently sequential and unidirectional generation. While alternate classes of models have been explored, we have limited mathematical understanding of their fundamental power and limitations. In this paper we focus on Generative Masked Language Models (GMLMs), a non-autoregressive paradigm in which we train a model to fit conditional probabilities of the data distribution via masking, which are subsequently used as inputs to a Markov Chain to draw samples from the model. These models empirically strike a promising speed-quality tradeoff as each step can be typically parallelized by decoding the entire sequence in parallel. We develop a mathematical framework for analyzing and improving such models which sheds light on questions of sample complexity and inference speed and quality. Empirically, we adapt the T5 model for iteratively-refined parallel decoding, achieving 2-3x speedup in machine translation with minimal sacrifice in quality compared with autoregressive models. We run careful ablation experiments to give recommendations on key design choices, and make fine-grained observations on the common error modes in connection with our theory. Our mathematical analyses and empirical observations characterize both potentials and limitations of this approach, and can be applied to future works on improving understanding and performance of GMLMs. 

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling. However, a theoretical understanding of the exact role of augmentation remains limited. Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression. Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining. Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation, and prove two generalization bounds that are free of model complexity. Our first bound works for an arbitrary encoder, where the prediction error is decomposed as the sum of an estimation error incurred by fitting a linear probe with RKHS regression, and an approximation error entailed by RKHS approximation. Our second bound specifically addresses the case where the encoder is near-optimal, that is it approximates the top-d eigenspace of the RKHS induced by the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance. 

The premise of identifiable and causal representation learning is to improve the current representation learning paradigm in terms of generalizability or robustness. Despite recent progress in questions of identifiability, more theoretical results demonstrating concrete advantages of these methods for downstream tasks are needed. In this paper, we consider the task of intervention extrapolation: predicting how interventions affect an outcome, even when those interventions are not observed at training time, and show that identifiable representations can provide an effective solution to this task even if the interventions affect the outcome non-linearly. Our setup includes an outcome variable Y , observed features X, which are generated as a non-linear transformation of latent features Z, and exogenous action variables A, which influence Z. The objective of intervention extrapolation is then to predict how interventions on A that lie outside the training support of A affect Y . Here, extrapolation becomes possible if the effect of A on Z is linear and the residual when regressing Z on A has full support. As Z is latent, we combine the task of intervention extrapolation with identifiable representation learning, which we call Rep4Ex: we aim to map the observed features X into a subspace that allows for non-linear extrapolation in A. We show that the hidden representation is identifiable up to an affine transformation in Z-space, which, we prove, is sufficient for intervention extrapolation. The identifiability is characterized by a novel constraint describing the linearity assumption of A on Z. Based on this insight, we propose a flexible method that enforces the linear invariance constraint and can be combined with any type of autoencoder. We validate our theoretical findings through a series of synthetic experiments and show that our approach can indeed succeed in predicting the effects of unseen interventions. 

Unlabeled data is a key component of modern machine learning. In general, the role of unlabeled data is to impose a form of smoothness, usually from the similarity information encoded in a base kernel, such as the ε-neighbor kernel or the adjacency matrix of a graph. This work revisits the classical idea of spectrally transformed kernel regression (STKR), and provides a new class of general and scalable STKR estimators able to leverage unlabeled data. Intuitively, via spectral transformation, STKR exploits the data distribution for which unlabeled data can provide additional information. First, we show that STKR is a principled and general approach, by characterizing a universal type of “target smoothness”, and proving that any sufficiently smooth function can be learned by STKR. Second, we provide scalable STKR implementations for the inductive setting and a general transformation function, while prior work is mostly limited to the transductive setting. Third, we derive statistical guarantees for two scenarios: STKR with a known polynomial transformation, and STKR with kernel PCA when the transformation is unknown. Overall, we believe that this work helps deepen our understanding of how to work with unlabeled data, and its generality makes it easier to inspire new methods. 

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling. However, a theoretical understanding of the exact role of augmentation remains limited. Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression. Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining. Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation, and prove two generalization bounds that are free of model complexity. Our first bound works for an arbitrary encoder, where the prediction error is decomposed as the sum of an estimation error incurred by fitting a linear probe with RKHS regression, and an approximation error entailed by RKHS approximation. Our second bound specifically addresses the case where the encoder is near-optimal, that is it approximates the top-d eigenspace of the RKHS induced by the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance. 

We study the problem of learning causal representations from unknown, latent interventions in a general setting, where the latent distribution is Gaussian but the mixing function is completely general. We prove strong identifiability results given unknown single-node interventions, i.e., without having access to the intervention targets. This generalizes prior works which have focused on weaker classes, such as linear maps or paired counterfactual data. This is also the first instance of causal identifiability from non-paired interventions for deep neural network embeddings. Our proof relies on carefully uncovering the high-dimensional geometric structure present in the data distribution after a non-linear density transformation, which we capture by analyzing quadratic forms of precision matrices of the latent distributions. Finally, we propose a contrastive algorithm to identify the latent variables in practice and evaluate its performance on various tasks. 

Structural causal models (SCMs) are widely used in various disciplines to repre- sent causal relationships among variables in complex systems. Unfortunately, the underlying causal structure is often unknown, and estimating it from data remains a challenging task. In many situations, however, the end goal is to localize the changes (shifts) in the causal mechanisms between related datasets instead of learn- ing the full causal structure of the individual datasets. Some applications include root cause analysis, analyzing gene regulatory network structure changes between healthy and cancerous individuals, or explaining distribution shifts. This paper focuses on identifying the causal mechanism shifts in two or more related datasets over the same set of variables—without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key technical contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for the identification of shifts under general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/iSCAN. 

We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational auto-encoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i.e. unsupervised) models with universal approximation capabilities, the side information u is not necessary: We prove identifiability of the entire generative model where we do not observe u and only observe the data x. The models we consider match auto-encoder architectures used in practice that leverage mixture priors in the latent space and ReLU/leaky-ReLU activations in the encoder, such as VaDE and MFC-VAE. Our main result is an identifiability hierarchy that significantly generalizes previous work and exposes how different assumptions lead to different “strengths” of identifiability, and includes certain “vanilla” VAEs with isotropic Gaussian priors as a special case. For example, our weakest result establishes (unsupervised) identifiability up to an affine transformation, and thus partially resolves an open problem regarding model identifiability raised in prior work. These theoretical results are augmented with experiments on both simulated and real data. 

We describe a novel neuro-symbolic model architecture we term "neuro-causal models," that uses a synthesis of deep generative models and causal graphical models to automatically infer higher level symbolic information from lower level "raw features", while also allowing for rich relationships among the symbolic variables. 

Concept-based interpretations of black-box models are often more intuitive for humans to understand. The most widely adopted approach for concept-based interpretation is Concept Activation Vector (CAV). CAV relies on learning a linear relation between some latent representation of a given model and concepts. The linear separability is usually implicitly assumed but does not hold true in general. In this work, we started from the original intent of concept-based interpretation and proposed Concept Gradient (CG), extending concept-based interpretation beyond linear concept functions. We showed that for a general (potentially non-linear) concept, we can mathematically evaluate how a small change of concept affecting the model’s prediction, which leads to an extension of gradient-based interpretation to the concept space. We demonstrated empirically that CG outperforms CAV in both toy examples and real world datasets.