We show theoretically and empirically that the linear Transformer, when applied to graph data, can implement algorithms that solve canonical problems such as electric flow and eigenvector decomposition. The input to the Transformer is simply the graph incidence matrix; no other explicit positional encoding information is provided. We present explicit weight configurations for implementing each such graph algorithm, and we bound the errors of the constructed Transformers by the errors of the underlying algorithms. Our theoretical findings are corroborated by experiments on synthetic data. Additionally, on a real-world molecular regression task, we observe that the linear Transformer is capable of learning a more effective positional encoding than the default one based on Laplacian eigenvectors. Our work is an initial step towards elucidating the inner-workings of the Transformer for graph data.

Recent research has sought to understand Transformers through the lens of in-context learning with functional data. We extend that line of work with the goal of moving closer to language models, considering categorical outcomes, nonlinear underlying models, and nonlinear attention. The contextual data are of the form $\textsf{C}=(x_1,c_1,\dots,x_N,c_{N})$ where each $c_i\in\{0,\dots,C-1\}$ is drawn from a categorical distribution that depends on covariates $x_i\in\mathbb{R}^d$. Contextual outcomes in the $m$th set of contextual data, $\textsf{C}_m$, are modeled in terms of latent function $f_m(x)\in\textsf{F}$, where $\textsf{F}$ is a functional class with $(C-1)$-dimensional vector output. The probability of observing class $c\in\{0,\dots,C-1\}$ is modeled in terms of the output components of $f_m(x)$ via the softmax. The Transformer parameters may be trained with $M$ contextual examples, $\{\textsf{C}_m\}_{m=1,M}$, and the trained model is then applied to new contextual data $\textsf{C}_{M+1}$ for new $f_{M+1}(x)\in\textsf{F}$. The goal is for the Transformer to constitute the probability of each category $c\in\{0,\dots,C-1\}$ for a new query $x_{N_{M+1}+1}$. We assume each component of $f_m(x)$ resides in a reproducing kernel Hilbert space (RKHS), specifying $\textsf{F}$. Analysis and an extensive set of experiments suggest that on its forward pass the Transformer (with attention defined by the RKHS kernel) implements a form of gradient descent of the underlying function, connected to the latent vector function associated with the softmax. We present what is believed to be the first real-world demonstration of this few-shot-learning methodology, using the ImageNet dataset.

Zero-shot learning (ZSL) is a promising approach to generalizing a model to categories unseen during training by leveraging class attributes, but challenges remain. Recently, methods using generative models to combat bias towards classes seen during training have pushed state of the art, but these generative models can be slow or computationally expensive to train. Also, these generative models assume that the attribute vector of each unseen class is available a priori at training, which is not always practical. Additionally, while many previous ZSL methods assume a one-time adaptation to unseen classes, in reality, the world is always changing, necessitating a constant adjustment of deployed models. Models unprepared to handle a sequential stream of data are likely to experience catastrophic forgetting. We propose a Meta-learned Attribute self-Interaction Network (MAIN) for continual ZSL. By pairing attribute self-interaction trained using meta-learning with inverse regularization of the attribute encoder, we are able to outperform state-of-the-art results without leveraging the unseen class attributes while also being able to train our models substantially faster (>100x) than expensive generative-based approaches. We demonstrate this with experiments on five standard ZSL datasets (CUB, aPY, AWA1, AWA2, and SUN) in the generalized zero-shot learning and continual (fixed/dynamic) zero-shot learning settings. Extensive ablations and analyses demonstrate the efficacy of various components proposed.

Open world classification is a task in natural language processing with key practical relevance and impact. Since the open or {\em unknown} category data only manifests in the inference phase, finding a model with a suitable decision boundary accommodating for the identification of known classes and discrimination of the open category is challenging. The performance of existing models is limited by the lack of effective open category data during the training stage or the lack of a good mechanism to learn appropriate decision boundaries. We propose an approach based on \underline{a}daptive \underline{n}egative \underline{s}amples (ANS) designed to generate effective synthetic open category samples in the training stage and without requiring any prior knowledge or external datasets. Empirically, we find a significant advantage in using auxiliary one-versus-rest binary classifiers, which effectively utilize the generated negative samples and avoid the complex threshold-seeking stage in previous works. Extensive experiments on three benchmark datasets show that ANS achieves significant improvements over state-of-the-art methods.

The recent introduction of thermodynamic integration techniques has provided a new framework for understanding and improving variational inference (VI). In this work, we present a careful analysis of the thermodynamic variational objective (TVO), bridging the gap between existing variational objectives and shedding new insights to advance the field. In particular, we elucidate how the TVO naturally connects the three key variational schemes, namely the importance-weighted VI, Renyi-VI, and MCMC-VI, which subsumes most VI objectives employed in practice. To explain the performance gap between theory and practice, we reveal how the pathological geometry of thermodynamic curves negatively affects TVO. By generalizing the integration path from the geometric mean to the weighted Holder mean, we extend the theory of TVO and identify new opportunities for improving VI. This motivates our new VI objectives, named the Holder bounds, which flatten the thermodynamic curves and promise to achieve a one-step approximation of the exact marginal log-likelihood. A comprehensive discussion on the choices of numerical estimators is provided. We present strong empirical evidence on both synthetic and real-world datasets to support our claims.

We examine interval estimation of the effect of a treatment T on an outcome Y given the existence of an unobserved confounder U. Using H\"older's inequality, we derive a set of bounds on the confounding bias |E[Y|T=t]-E[Y|do(T=t)]| based on the degree of unmeasured confounding (i.e., the strength of the connection U->T, and the strength of U->Y). These bounds are tight either when U is independent of T or when U is independent of Y given T (when there is no unobserved confounding). We focus on a special case of this bound depending on the total variation distance between the distributions p(U) and p(U|T=t), as well as the maximum (over all possible values of U) deviation of the conditional expected outcome E[Y|U=u,T=t] from the average expected outcome E[Y|T=t]. We discuss possible calibration strategies for this bound to get interval estimates for treatment effects, and experimentally validate the bound using synthetic and semi-synthetic datasets.

Although recent works have developed methods that can generate estimations (or imputations) of the missing entries in a dataset to facilitate downstream analysis, most depend on assumptions that may not align with real-world applications and could suffer from poor performance in subsequent tasks. This is particularly true if the data have large missingness rates or a small population. More importantly, the imputation error could be propagated into the prediction step that follows, causing the gradients used to train the prediction models to be biased. Consequently, in this work, we introduce the importance guided stochastic gradient descent (IGSGD) method to train multilayer perceptrons (MLPs) and long short-term memories (LSTMs) to directly perform inference from inputs containing missing values without imputation. Specifically, we employ reinforcement learning (RL) to adjust the gradients used to train the models via back-propagation. This not only reduces bias but allows the model to exploit the underlying information behind missingness patterns. We test the proposed approach on real-world time-series (i.e., MIMIC-III), tabular data obtained from an eye clinic, and a standard dataset (i.e., MNIST), where our imputation-free predictions outperform the traditional two-step imputation-based predictions using state-of-the-art imputation methods.

InfoNCE-based contrastive representation learners, such as SimCLR, have been tremendously successful in recent years. However, these contrastive schemes are notoriously resource demanding, as their effectiveness breaks down with small-batch training (i.e., the log-K curse, whereas K is the batch-size). In this work, we reveal mathematically why contrastive learners fail in the small-batch-size regime, and present a novel simple, non-trivial contrastive objective named FlatNCE, which fixes this issue. Unlike InfoNCE, our FlatNCE no longer explicitly appeals to a discriminative classification goal for contrastive learning. Theoretically, we show FlatNCE is the mathematical dual formulation of InfoNCE, thus bridging the classical literature on energy modeling; and empirically, we demonstrate that, with minimal modification of code, FlatNCE enables immediate performance boost independent of the subject-matter engineering efforts. The significance of this work is furthered by the powerful generalization of contrastive learning techniques, and the introduction of new tools to monitor and diagnose contrastive training. We substantiate our claims with empirical evidence on CIFAR10, ImageNet, and other datasets, where FlatNCE consistently outperforms InfoNCE.

Successful applications of InfoNCE and its variants have popularized the use of contrastive variational mutual information (MI) estimators in machine learning. While featuring superior stability, these estimators crucially depend on costly large-batch training, and they sacrifice bound tightness for variance reduction. To overcome these limitations, we revisit the mathematics of popular variational MI bounds from the lens of unnormalized statistical modeling and convex optimization. Our investigation not only yields a new unified theoretical framework encompassing popular variational MI bounds but also leads to a novel, simple, and powerful contrastive MI estimator named as FLO. Theoretically, we show that the FLO estimator is tight, and it provably converges under stochastic gradient descent. Empirically, our FLO estimator overcomes the limitations of its predecessors and learns more efficiently. The utility of FLO is verified using an extensive set of benchmarks, which also reveals the trade-offs in practical MI estimation.

Federated learning has emerged as an important distributed learning paradigm, where a server aggregates a global model from many client-trained models while having no access to the client data. Although it is recognized that statistical heterogeneity of the client local data yields slower global model convergence, it is less commonly recognized that it also yields a biased federated global model with a high variance of accuracy across clients. In this work, we aim to provide federated learning schemes with improved fairness. To tackle this challenge, we propose a novel federated learning system that employs zero-shot data augmentation on under-represented data to mitigate statistical heterogeneity and encourage more uniform accuracy performance across clients in federated networks. We study two variants of this scheme, Fed-ZDAC (federated learning with zero-shot data augmentation at the clients) and Fed-ZDAS (federated learning with zero-shot data augmentation at the server). Empirical results on a suite of datasets demonstrate the effectiveness of our methods on simultaneously improving the test accuracy and fairness.