Recent advances in artificial intelligence have created a paradigm shift in which models are trained on large amounts of data and can be adapted to different tasks, dubbed foundation models (Bommasani et al., 2021). These models, which often employ self-supervision, can extract valuable knowledge from various types of data, including natural language (Devlin et al., 2018; Brown et al., 2020), images (Radford et al., 2021), and biological sequencing counts (Theodoris et al., 2023). This acquired knowledge allows the model to generalize when asked to perform tasks in novel scenarios. With vast amounts of data becoming increasingly available from diverse sources, such models are of interest to leverage information that can be learned in order to build more intelligent systems (Bubeck et al., 2023). A critical aspect of intelligent systems is the ability to reason about cause-and-effect relationships (Zhang et al., 2023), which is vital to making informed decisions across various domains, including healthcare, economics, and statistics (Kube et al., 2019; Geffner et al., 2022; Zhang et al., 2022). Relying solely on correlation-based models (Harrison and March, 1984) can lead to misleading conclusions, as they do not account for the underlying causal mechanisms. This limitation is also observed in the realm of foundation models (Bubeck et al., 2023; Mahowald et al., 2023; Wolfram, 2023).

It has been observed in recent years that transformers have problems with length generalization for certain types of reasoning and arithmetic tasks. In particular, the performance of a transformer model trained on tasks (say addition) up to a certain length (e.g., 5 digit numbers) drops sharply when applied to longer instances of the same problem. This work proposes an approach based on task hinting towards addressing length generalization. Our key idea is that while training the model on task-specific data, it is helpful to simultaneously train the model to solve a simpler but related auxiliary task as well. We study the classical sorting problem as a canonical example to evaluate our approach. We design a multitask training framework and show that task hinting significantly improve length generalization. For sorting we show that it is possible to train models on data consisting of sequences having length at most $20$, and improve the test accuracy on sequences of length $100$ from less than 1% (for standard training) to more than 92% (via task hinting). Our study uncovers several interesting aspects of length generalization. We observe that while several auxiliary tasks may seem natural a priori, their effectiveness in improving length generalization differs dramatically. We further use probing and visualization-based techniques to understand the internal mechanisms via which the model performs the task, and propose a theoretical construction consistent with the observed learning behaviors of the model. Based on our construction, we show that introducing a small number of length dependent parameters into the training procedure can further boost the performance on unseen lengths. Finally, we also show the efficacy of our task hinting based approach beyond sorting, giving hope that these techniques will be applicable in broader contexts.

Batch reinforcement learning (RL) defines the task of learning from a fixed batch of data lacking exhaustive exploration. Worst-case optimality algorithms, which calibrate a value-function model class from logged experience and perform some type of pessimistic evaluation under the learned model, have emerged as a promising paradigm for batch RL. However, contemporary works on this stream have commonly overlooked the hierarchical decision-making structure hidden in the optimization landscape. In this paper, we adopt a game-theoretical viewpoint and model the policy learning diagram as a two-player general-sum game with a leader-follower structure. We propose a novel stochastic gradient-based learning algorithm: StackelbergLearner, in which the leader player updates according to the total derivative of its objective instead of the usual individual gradient, and the follower player makes individual updates and ensures transition-consistent pessimistic reasoning. The derived learning dynamic naturally lends StackelbergLearner to a game-theoretic interpretation and provides a convergence guarantee to differentiable Stackelberg equilibria. From a theoretical standpoint, we provide instance-dependent regret bounds with general function approximation, which shows that our algorithm can learn a best-effort policy that is able to compete against any comparator policy that is covered by batch data. Notably, our theoretical regret guarantees only require realizability without any data coverage and strong function approximation conditions, e.g., Bellman closedness, which is in contrast to prior works lacking such guarantees. Through comprehensive experiments, we find that our algorithm consistently performs as well or better as compared to state-of-the-art methods in batch RL benchmark and real-world datasets.

We study high-confidence off-policy evaluation in the context of infinite-horizon Markov decision processes, where the objective is to establish a confidence interval (CI) for the target policy value using only offline data pre-collected from unknown behavior policies. This task faces two primary challenges: providing a comprehensive and rigorous error quantification in CI estimation, and addressing the distributional shift that results from discrepancies between the distribution induced by the target policy and the offline data-generating process. Motivated by an innovative unified error analysis, we jointly quantify the two sources of estimation errors: the misspecification error on modeling marginalized importance weights and the statistical uncertainty due to sampling, within a single interval. This unified framework reveals a previously hidden tradeoff between the errors, which undermines the tightness of the CI. Relying on a carefully designed discriminator function, the proposed estimator achieves a dual purpose: breaking the curse of the tradeoff to attain the tightest possible CI, and adapting the CI to ensure robustness against distributional shifts. Our method is applicable to time-dependent data without assuming any weak dependence conditions via leveraging a local supermartingale/martingale structure. Theoretically, we show that our algorithm is sample-efficient, error-robust, and provably convergent even in non-linear function approximation settings. The numerical performance of the proposed method is examined in synthetic datasets and an OhioT1DM mobile health study.

Generalization properties are a central aspect of the design and analysis of learning algorithms. One notion that has been considered in many previous works as leading to good generalization is flat minima, which informally describes a loss surface that is insensitive to noise perturbations. However, the design of efficient algorithms (that are easy to analyze) to find them is relatively under-explored. In this paper, we propose a new algorithm to address this issue, which minimizes a stochastic optimization objective that averages noise perturbations injected into the weights of a function. This algorithm is shown to enjoy both theoretical and empirical advantages compared to existing algorithms involving worst-case perturbations. Theoretically, we show tight convergence rates of our algorithm to find first-order stationary points of the stochastic objective. Empirically, the algorithm induces a penalty on the trace of the Hessian, leading to iterates that are flatter than SGD and other alternatives, with tighter generalization gaps. Altogether, this work contributes a provable and practical algorithm to find flat minima by optimizing the noise stability properties of a function.

It is widely recognized that the generalization ability of neural networks can be greatly enhanced through carefully tuning the training procedure. The current state-of-the-art training approach involves utilizing stochastic gradient descent (SGD) or Adam optimization algorithms along with a combination of additional regularization techniques such as weight decay, dropout, or noise injection. Optimal generalization can only be achieved by tuning a multitude of hyper-parameters extensively, which can be time-consuming and necessitates the additional validation dataset. To address this issue, we present a nearly tuning-free PAC-Bayes training framework that requires no extra regularization. This framework achieves test performance comparable to that of SGD/Adam, even when the latter are optimized through a complete grid search and supplemented with additional regularization terms. To understand the underlying benefits of these strategies, numerous studies have focused on studying individual strategies. For instance, it has been shown that larger learning rates (Cohen et al., 2021; Barrett & Dherin, 2020), momentum (Ghosh et al., 2022), smaller batch sizes (Lee & Jang, 2022) and batch normalization (Luo et al., 2018) individually induce higher degrees of implicit regularization on the sharpness of the loss function, yielding better generalization. Additionally, the intensity of explicit regularization techniques such as weight decay (Loshchilov & Hutter, 2017), dropout (Wei et al., 2020), parameter noise injection (Neelakantan et al., 2015; Orvieto et al., 2022), label noise (Damian et al., 2021) can significantly affect generalization. Despite these observations and explanations, it's unclear why seeking optimal combinations of these regularizations is still crucial in practice. Adjusting the intensity of each regularization based on different scenarios can be a tedious job, especially when previous research has indicated that some techniques can conflict with each other (Li et al., 2019). We summarize this challenge for conventional training in (Q1). Alternatively, PAC-Bayes generalization bounds provide foundational insights into generalization in the absence of validation and testing data (Shawe-Taylor & Williamson, 1997). Jiang et al. (2019) further suggests that PAC-Bayes bounds are among the best for evaluating generalization capabilities. Although PAC-Bayes bounds were traditionally used only in the post-training stage for quality control (Vapnik, 1998; McAllester, 1999), the recent work (Dziugaite & Roy, 2017b) has opened the door to using these bounds during the training phase. They showed that one can directly train a network via optimizing the PAC-Bayes bound, a strategy we refer to as PAC-Bayes training, and obtain reasonable performances.

We consider an important problem in scientific discovery, namely identifying sparse governing equations for nonlinear dynamical systems. This involves solving sparse ridge regression problems to provable optimality in order to determine which terms drive the underlying dynamics. We propose a fast algorithm, OKRidge, for sparse ridge regression, using a novel lower bound calculation involving, first, a saddle point formulation, and from there, either solving (i) a linear system or (ii) using an ADMM-based approach, where the proximal operators can be efficiently evaluated by solving another linear system and an isotonic regression problem. We also propose a method to warm-start our solver, which leverages a beam search. Experimentally, our methods attain provable optimality with run times that are orders of magnitude faster than those of the existing MIP formulations solved by the commercial solver Gurobi.

Joint multimodal functional data acquisition, where functional data from multiple modes are measured simultaneously from the same subject, has emerged as an exciting modern approach enabled by recent engineering breakthroughs in the neurological and biological sciences. One prominent motivation to acquire such data is to enable new discoveries of the underlying connectivity by combining multimodal signals. Despite the scientific interest, there remains a gap in principled statistical methods for estimating the graph underlying multimodal functional data. To this end, we propose a new integrative framework that models the data generation process and identifies operators mapping from the observation space to the latent space. We then develop an estimator that simultaneously estimates the transformation operators and the latent graph. This estimator is based on the partial correlation operator, which we rigorously extend from the multivariate to the functional setting. Our procedure is provably efficient, with the estimator converging to a stationary point with quantifiable statistical error. Furthermore, we show recovery of the latent graph under mild conditions. Our work is applied to analyze simultaneously acquired multimodal brain imaging data where the graph indicates functional connectivity of the brain. We present simulation and empirical results that support the benefits of joint estimation.

This paper studies the problem of conducting self-supervised learning for node representation learning on graphs. Most existing self-supervised learning methods assume the graph is homophilous, where linked nodes often belong to the same class or have similar features. However, such assumptions of homophily do not always hold in real-world graphs. We address this problem by developing a decoupled self-supervised learning (DSSL) framework for graph neural networks. DSSL imitates a generative process of nodes and links from latent variable modeling of the semantic structure, which decouples different underlying semantics between different neighborhoods into the self-supervised learning process. Our DSSL framework is agnostic to the encoders and does not need prefabricated augmentations, thus is flexible to different graphs. To effectively optimize the framework, we derive the evidence lower bound of the self-supervised objective and develop a scalable training algorithm with variational inference. We provide a theoretical analysis to justify that DSSL enjoys the better downstream performance. Extensive experiments on various types of graph benchmarks demonstrate that our proposed framework can achieve better performance compared with competitive baselines.

Active learning provides a framework to adaptively query the most informative experiments towards learning an unknown black-box function. Various approaches of active learning have been proposed in the literature, however, they either focus on exploration or exploitation in the design space. Methods that do consider exploration-exploitation simultaneously employ fixed or ad-hoc measures to control the trade-off that may not be optimal. In this paper, we develop a Bayesian hierarchical approach, referred as BHEEM, to dynamically balance the exploration-exploitation trade-off as more data points are queried. To sample from the posterior distribution of the trade-off parameter, We subsequently formulate an approximate Bayesian computation approach based on the linear dependence of queried data in the feature space. Simulated and real-world examples show the proposed approach achieves at least 21% and 11% average improvement when compared to pure exploration and exploitation strategies respectively. More importantly, we note that by optimally balancing the trade-off between exploration and exploitation, BHEEM performs better or at least as well as either pure exploration or pure exploitation.