Deep Transfer $Q$-Learning for Offline Non-Stationary Reinforcement Learning

In dynamic decision-making scenarios across business and healthcare, leveraging sample trajectories from diverse populations can significantly enhance reinforcement learning (RL) performance for specific target populations, especially when sample sizes are limited. While existing transfer learning methods primarily focus on linear regression settings, they lack direct applicability to reinforcement learning algorithms. This paper pioneers the study of transfer learning for dynamic decision scenarios modeled by non-stationary finite-horizon Markov decision processes, utilizing neural networks as powerful function approximators and backward inductive learning. We demonstrate that naive sample pooling strategies, effective in regression settings, fail in Markov decision processes.To address this challenge, we introduce a novel ``re-weighted targeting procedure'' to construct ``transferable RL samples'' and propose ``transfer deep $Q^*$-learning'', enabling neural network approximation with theoretical guarantees. We assume that the reward functions are transferable and deal with both situations in which the transition densities are transferable or nontransferable. Our analytical techniques for transfer learning in neural network approximation and transition density transfers have broader implications, extending to supervised transfer learning with neural networks and domain shift scenarios. Empirical experiments on both synthetic and real datasets corroborate the advantages of our method, showcasing its potential for improving decision-making through strategically constructing transferable RL samples in non-stationary reinforcement learning contexts.

Transformers Simulate MLE for Sequence Generation in Bayesian Networks

Transformers have achieved significant success in various fields, notably excelling in tasks involving sequential data like natural language processing. Despite these achievements, the theoretical understanding of transformers' capabilities remains limited. In this paper, we investigate the theoretical capabilities of transformers to autoregressively generate sequences in Bayesian networks based on in-context maximum likelihood estimation (MLE). Specifically, we consider a setting where a context is formed by a set of independent sequences generated according to a Bayesian network. We demonstrate that there exists a simple transformer model that can (i) estimate the conditional probabilities of the Bayesian network according to the context, and (ii) autoregressively generate a new sample according to the Bayesian network with estimated conditional probabilities. We further demonstrate in extensive experiments that such a transformer does not only exist in theory, but can also be effectively obtained through training. Our analysis highlights the potential of transformers to learn complex probabilistic models and contributes to a better understanding of large language models as a powerful class of sequence generators.

Learning Spectral Methods by Transformers

Transformers demonstrate significant advantages as the building block of modern LLMs. In this work, we study the capacities of Transformers in performing unsupervised learning. We show that multi-layered Transformers, given a sufficiently large set of pre-training instances, are able to learn the algorithms themselves and perform statistical estimation tasks given new instances. This learning paradigm is distinct from the in-context learning setup and is similar to the learning procedure of human brains where skills are learned through past experience. Theoretically, we prove that pre-trained Transformers can learn the spectral methods and use the classification of bi-class Gaussian mixture model as an example. Our proof is constructive using algorithmic design techniques. Our results are built upon the similarities of multi-layered Transformer architecture with the iterative recovery algorithms used in practice. Empirically, we verify the strong capacity of the multi-layered (pre-trained) Transformer on unsupervised learning through the lens of both the PCA and the Clustering tasks performed on the synthetic and real-world datasets.

Localized exploration in contextual dynamic pricing achieves dimension-free regret

We study the problem of contextual dynamic pricing with a linear demand model. We propose a novel localized exploration-then-commit (LetC) algorithm which starts with a pure exploration stage, followed by a refinement stage that explores near the learned optimal pricing policy, and finally enters a pure exploitation stage. The algorithm is shown to achieve a minimax optimal, dimension-free regret bound when the time horizon exceeds a polynomial of the covariate dimension. Furthermore, we provide a general theoretical framework that encompasses the entire time spectrum, demonstrating how to balance exploration and exploitation when the horizon is limited. The analysis is powered by a novel critical inequality that depicts the exploration-exploitation trade-off in dynamic pricing, mirroring its existing counterpart for the bias-variance trade-off in regularized regression. Our theoretical results are validated by extensive experiments on synthetic and real-world data.

Benign Overfitting in Out-of-Distribution Generalization of Linear Models

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee [Tsigler and Bartlett, 2023], as well as under-parameterized OOD guarantee [Ge et al., 2024] when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $O(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $O(1/n)$, where $n$ is the number of samples.

One-Layer Transformer Provably Learns One-Nearest Neighbor In Context

Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning one of the most classical nonparametric estimators, the one-nearest neighbor prediction rule. Under a theoretical framework where the prompt contains a sequence of labeled training data and unlabeled test data, we show that, although the loss function is nonconvex when trained with gradient descent, a single softmax attention layer can successfully learn to behave like a one-nearest neighbor classifier. Our result gives a concrete example of how transformers can be trained to implement nonparametric machine learning algorithms, and sheds light on the role of softmax attention in transformer models.

Global Convergence in Training Large-Scale Transformers

Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and depth go to infinity, gradient flow converges to the Wasserstein gradient flow, which is represented by a partial differential equation. Then, we demonstrate that the gradient flow reaches a global minimum consistent with the PDE solution when the weight decay regularization parameter is sufficiently small. Our analysis is based on a series of novel mean-field techniques that adapt to Transformers. Compared with existing tools for deep networks (Lu et al., 2020) that demand homogeneity and global Lipschitz smoothness, we utilize a refined analysis assuming only $\textit{partial homogeneity}$ and $\textit{local Lipschitz smoothness}$. These new techniques may be of independent interest.

Analysis of the ICML 2023 Ranking Data: Can Authors' Opinions of Their Own Papers Assist Peer Review in Machine Learning?

We conducted an experiment during the review process of the 2023 International Conference on Machine Learning (ICML) that requested authors with multiple submissions to rank their own papers based on perceived quality. We received 1,342 rankings, each from a distinct author, pertaining to 2,592 submissions. In this paper, we present an empirical analysis of how author-provided rankings could be leveraged to improve peer review processes at machine learning conferences. We focus on the Isotonic Mechanism, which calibrates raw review scores using author-provided rankings. Our analysis demonstrates that the ranking-calibrated scores outperform raw scores in estimating the ground truth ``expected review scores'' in both squared and absolute error metrics. Moreover, we propose several cautious, low-risk approaches to using the Isotonic Mechanism and author-provided rankings in peer review processes, including assisting senior area chairs' oversight of area chairs' recommendations, supporting the selection of paper awards, and guiding the recruitment of emergency reviewers. We conclude the paper by addressing the study's limitations and proposing future research directions.

Factor Adjusted Spectral Clustering for Mixture Models

This paper studies a factor modeling-based approach for clustering high-dimensional data generated from a mixture of strongly correlated variables. Statistical modeling with correlated structures pervades modern applications in economics, finance, genomics, wireless sensing, etc., with factor modeling being one of the popular techniques for explaining the common dependence. Standard techniques for clustering high-dimensional data, e.g., naive spectral clustering, often fail to yield insightful results as their performances heavily depend on the mixture components having a weakly correlated structure. To address the clustering problem in the presence of a latent factor model, we propose the Factor Adjusted Spectral Clustering (FASC) algorithm, which uses an additional data denoising step via eliminating the factor component to cope with the data dependency. We prove this method achieves an exponentially low mislabeling rate, with respect to the signal to noise ratio under a general set of assumptions. Our assumption bridges many classical factor models in the literature, such as the pervasive factor model, the weak factor model, and the sparse factor model. The FASC algorithm is also computationally efficient, requiring only near-linear sample complexity with respect to the data dimension. We also show the applicability of the FASC algorithm with real data experiments and numerical studies, and establish that FASC provides significant results in many cases where traditional spectral clustering fails.

Covariate Assisted Entity Ranking with Sparse Intrinsic Scores

This paper addresses the item ranking problem with associate covariates, focusing on scenarios where the preference scores can not be fully explained by covariates, and the remaining intrinsic scores, are sparse. Specifically, we extend the pioneering Bradley-Terry-Luce (BTL) model by incorporating covariate information and considering sparse individual intrinsic scores. Our work introduces novel model identification conditions and examines the regularized penalized Maximum Likelihood Estimator (MLE) statistical rates. We then construct a debiased estimator for the penalized MLE and analyze its distributional properties. Additionally, we apply our method to the goodness-of-fit test for models with no latent intrinsic scores, namely, the covariates fully explaining the preference scores of individual items. We also offer confidence intervals for ranks. Our numerical studies lend further support to our theoretical findings, demonstrating validation for our proposed method