The training and generalization dynamics of the Transformer's core mechanism, namely the Attention mechanism, remain under-explored. Besides, existing analyses primarily focus on single-head attention. Inspired by the demonstrated benefits of overparameterization when training fully-connected networks, we investigate the potential optimization and generalization advantages of using multiple attention heads. Towards this goal, we derive convergence and generalization guarantees for gradient-descent training of a single-layer multi-head self-attention model, under a suitable realizability condition on the data. We then establish primitive conditions on the initialization that ensure realizability holds. Finally, we demonstrate that these conditions are satisfied for a simple tokenized-mixture model. We expect the analysis can be extended to various data-model and architecture variations.

We propose in this paper, STANLEY, a STochastic gradient ANisotropic LangEvin dYnamics, for sampling high dimensional data. With the growing efficacy and potential of Energy-Based modeling, also known as non-normalized probabilistic modeling, for modeling a generative process of different natures of high dimensional data observations, we present an end-to-end learning algorithm for Energy-Based models (EBM) with the purpose of improving the quality of the resulting sampled data points. While the unknown normalizing constant of EBMs makes the training procedure intractable, resorting to Markov Chain Monte Carlo (MCMC) is in general a viable option. Realizing what MCMC entails for the EBM training, we propose in this paper, a novel high dimensional sampling method, based on an anisotropic stepsize and a gradient-informed covariance matrix, embedded into a discretized Langevin diffusion. We motivate the necessity for an anisotropic update of the negative samples in the Markov Chain by the nonlinearity of the backbone of the EBM, here a Convolutional Neural Network. Our resulting method, namely STANLEY, is an optimization algorithm for training Energy-Based models via our newly introduced MCMC method. We provide a theoretical understanding of our sampling scheme by proving that the sampler leads to a geometrically uniformly ergodic Markov Chain. Several image generation experiments are provided in our paper to show the effectiveness of our method.

The quality of explanations for the predictions of complex machine learning predictors is often measured using insertion and deletion metrics, which assess the faithfulness of the explanations, i.e., how correctly the explanations reflect the predictor's behavior. To improve the faithfulness, we propose insertion/deletion metric-aware explanation-based optimization (ID-ExpO), which optimizes differentiable predictors to improve both insertion and deletion scores of the explanations while keeping their predictive accuracy. Since the original insertion and deletion metrics are indifferentiable with respect to the explanations and directly unavailable for gradient-based optimization, we extend the metrics to be differentiable and use them to formalize insertion and deletion metric-based regularizers. The experimental results on image and tabular datasets show that the deep neural networks-based predictors fine-tuned using ID-ExpO enable popular post-hoc explainers to produce more faithful and easy-to-interpret explanations while keeping high predictive accuracy.

Stochastic processes defined on integer valued state spaces are popular within the physical A combination of factors such as non-linear dynamics, and biological sciences. These models are partial observation and a complicated latent structure necessary for capturing the dynamics of small makes the inference of parameter posteriors a systems where the individual nature of the challenging problem. The likelihood is generally intractable populations cannot be ignored and stochastic and although methods that can sample from effects are important. The inference of the the exact posteriors exist--many based on simulation parameters of such models, from time series of the model itself--these can become computationally data, is difficult due to intractability of the prohibitive in many situations (Doucet et al., 2015; likelihood; current methods, based on simulations Sherlock et al., 2015). Observations of a system with of the underlying model, can be so computationally low noise are particularly challenging for simulation expensive as to be prohibitive.

In the realm of deep learning, transformers have emerged as a dominant architecture, particularly in natural language processing tasks. However, with their widespread adoption, concerns regarding the security and privacy of the data processed by these models have arisen. In this paper, we address a pivotal question: Can the data fed into transformers be recovered using their attention weights and outputs? We introduce a theoretical framework to tackle this problem. Specifically, we present an algorithm that aims to recover the input data $X \in \mathbb{R}^{d \times n}$ from given attention weights $W = QK^\top \in \mathbb{R}^{d \times d}$ and output $B \in \mathbb{R}^{n \times n}$ by minimizing the loss function $L(X)$. This loss function captures the discrepancy between the expected output and the actual output of the transformer. Our findings have significant implications for the Localized Layer-wise Mechanism (LLM), suggesting potential vulnerabilities in the model's design from a security and privacy perspective. This work underscores the importance of understanding and safeguarding the internal workings of transformers to ensure the confidentiality of processed data.

Covariate shift occurs prevalently in practice, where the input distributions of the source and target data are substantially different. Despite its practical importance in various learning problems, most of the existing methods only focus on some specific learning tasks and are not well validated theoretically and numerically. To tackle this problem, we propose a unified analysis of general nonparametric methods in a reproducing kernel Hilbert space (RKHS) under covariate shift. Our theoretical results are established for a general loss belonging to a rich loss function family, which includes many commonly used methods as special cases, such as mean regression, quantile regression, likelihood-based classification, and margin-based classification. Two types of covariate shift problems are the focus of this paper and the sharp convergence rates are established for a general loss function to provide a unified theoretical analysis, which concurs with the optimal results in literature where the squared loss is used. Extensive numerical studies on synthetic and real examples confirm our theoretical findings and further illustrate the effectiveness of our proposed method.

Feature preprocessing continues to play a critical role when applying machine learning and statistical methods to tabular data. In this paper, we propose the use of the kernel density integral transformation as a feature preprocessing step. Our approach subsumes the two leading feature preprocessing methods as limiting cases: linear min-max scaling and quantile transformation. We demonstrate that, without hyperparameter tuning, the kernel density integral transformation can be used as a simple drop-in replacement for either method, offering protection from the weaknesses of each. Alternatively, with tuning of a single continuous hyperparameter, we frequently outperform both of these methods. Finally, we show that the kernel density transformation can be profitably applied to statistical data analysis, particularly in correlation analysis and univariate clustering.

Representation learning has significantly driven the field to develop pretrained models that can act as a valuable starting point when transferring to new datasets. With the rising demand for reliable machine learning and uncertainty quantification, there is a need for pretrained models that not only provide embeddings but also transferable uncertainty estimates. To guide the development of such models, we propose the Uncertainty-aware Representation Learning (URL) benchmark. Besides the transferability of the representations, it also measures the zero-shot transferability of the uncertainty estimate using a novel metric. We apply URL to evaluate eleven uncertainty quantifiers that are pretrained on ImageNet and transferred to eight downstream datasets. We find that approaches that focus on the uncertainty of the representation itself or estimate the prediction risk directly outperform those that are based on the probabilities of upstream classes. Yet, achieving transferable uncertainty quantification remains an open challenge. Our findings indicate that it is not necessarily in conflict with traditional representation learning goals. Code is provided under https://github.com/mkirchhof/url .

Attention-based neural networks such as transformers have demonstrated a remarkable ability to exhibit in-context learning (ICL): Given a short prompt sequence of tokens from an unseen task, they can formulate relevant per-token and next-token predictions without any parameter updates. By embedding a sequence of labeled training data and unlabeled test data as a prompt, this allows for transformers to behave like supervised learning algorithms. Indeed, recent work has shown that when training transformer architectures over random instances of linear regression problems, these models' predictions mimic those of ordinary least squares. Towards understanding the mechanisms underlying this phenomenon, we investigate the dynamics of ICL in transformers with a single linear self-attention layer trained by gradient flow on linear regression tasks. We show that despite non-convexity, gradient flow with a suitable random initialization finds a global minimum of the objective function. At this global minimum, when given a test prompt of labeled examples from a new prediction task, the transformer achieves prediction error competitive with the best linear predictor over the test prompt distribution. We additionally characterize the robustness of the trained transformer to a variety of distribution shifts and show that although a number of shifts are tolerated, shifts in the covariate distribution of the prompts are not. Motivated by this, we consider a generalized ICL setting where the covariate distributions can vary across prompts. We show that although gradient flow succeeds at finding a global minimum in this setting, the trained transformer is still brittle under mild covariate shifts. We complement this finding with experiments on large, nonlinear transformer architectures which we show are more robust under covariate shifts.

This paper introduces a new regularized version of the robust $\tau$-regression estimator for analyzing high-dimensional datasets subject to gross contamination in the response variables and covariates (explanatory variables). The resulting estimator, termed adaptive $\tau$-Lasso, is robust to outliers and high-leverage points. It also incorporates an adaptive $\ell_1$-norm penalty term, which enables the selection of relevant variables and reduces the bias associated with large true regression coefficients. More specifically, this adaptive $\ell_1$-norm penalty term assigns a weight to each regression coefficient. For a fixed number of predictors $p$, we show that the adaptive $\tau$-Lasso has the oracle property, ensuring both variable-selection consistency and asymptotic normality. Asymptotic normality applies only to the entries of the regression vector corresponding to the true support, assuming knowledge of the true regression vector support. We characterize its robustness via the finite-sample breakdown point and the influence function. We carry out extensive simulations and observe that the class of $\tau$-Lasso estimators exhibits robustness and reliable performance in both contaminated and uncontaminated data settings. We also validate our theoretical findings on robustness properties through simulation experiments. In the face of outliers and high-leverage points, the adaptive $\tau$-Lasso and $\tau$-Lasso estimators achieve the best performance or close-to-best performance in terms of prediction and variable selection accuracy compared to other competing regularized estimators for all scenarios considered in this study. Therefore, the adaptive $\tau$-Lasso and $\tau$-Lasso estimators can be effectively employed for a variety of sparse linear regression problems, particularly in high-dimensional settings and when the data is contaminated by outliers and high-leverage points.