High-dimensional Nonparametric Contextual Bandit Problem

We consider the kernelized contextual bandit problem with a large feature space. This problem involves $K$ arms, and the goal of the forecaster is to maximize the cumulative rewards through learning the relationship between the contexts and the rewards. It serves as a general framework for various decision-making scenarios, such as personalized online advertising and recommendation systems. Kernelized contextual bandits generalize the linear contextual bandit problem and offers a greater modeling flexibility. Existing methods, when applied to Gaussian kernels, yield a trivial bound of $O(T)$ when we consider $\Omega(\log T)$ feature dimensions. To address this, we introduce stochastic assumptions on the context distribution and show that no-regret learning is achievable even when the number of dimensions grows up to the number of samples. Furthermore, we analyze lenient regret, which allows a per-round regret of at most $\Delta > 0$. We derive the rate of lenient regret in terms of $\Delta$.

Minimax Rates of Estimation for Optimal Transport Map between Infinite-Dimensional Spaces

We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures, utilizing an optimal transport map as its key component. Estimating the optimal transport map from samples finds several applications, such as simulating dynamics between probability measures and functional data analysis. However, some transport maps on infinite-dimensional spaces require exponential-order data for estimation, which undermines their applicability. In this paper, we investigate the estimation of an optimal transport map between infinite-dimensional spaces, focusing on optimal transport maps characterized by the notion of $\gamma$-smoothness. Consequently, we show that the order of the minimax risk is polynomial rate in the sample size even in the infinite-dimensional setup. We also develop an estimator whose estimation error matches the minimax optimal rate. With these results, we obtain a class of reasonably estimable optimal transport maps on infinite-dimensional spaces and a method for their estimation. Our experiments validate the theory and practical utility of our approach with application to functional data analysis.

Precise gradient descent training dynamics for finite-width multi-layer neural networks

In this paper, we provide the first precise distributional characterization of gradient descent iterates for general multi-layer neural networks under the canonical single-index regression model, in the `finite-width proportional regime' where the sample size and feature dimension grow proportionally while the network width and depth remain bounded. Our non-asymptotic state evolution theory captures Gaussian fluctuations in first-layer weights and concentration in deeper-layer weights, and remains valid for non-Gaussian features. Our theory differs from existing neural tangent kernel (NTK), mean-field (MF) theories and tensor program (TP) in several key aspects. First, our theory operates in the finite-width regime whereas these existing theories are fundamentally infinite-width. Second, our theory allows weights to evolve from individual initializations beyond the lazy training regime, whereas NTK and MF are either frozen at or only weakly sensitive to initialization, and TP relies on special initialization schemes. Third, our theory characterizes both training and generalization errors for general multi-layer neural networks beyond the uniform convergence regime, whereas existing theories study generalization almost exclusively in two-layer settings. As a statistical application, we show that vanilla gradient descent can be augmented to yield consistent estimates of the generalization error at each iteration, which can be used to guide early stopping and hyperparameter tuning. As a further theoretical implication, we show that despite model misspecification, the model learned by gradient descent retains the structure of a single-index function with an effective signal determined by a linear combination of the true signal and the initialization.

Learning a Single Index Model from Anisotropic Data with vanilla Stochastic Gradient Descent

We investigate the problem of learning a Single Index Model (SIM)- a popular model for studying the ability of neural networks to learn features - from anisotropic Gaussian inputs by training a neuron using vanilla Stochastic Gradient Descent (SGD). While the isotropic case has been extensively studied, the anisotropic case has received less attention and the impact of the covariance matrix on the learning dynamics remains unclear. For instance, Mousavi-Hosseini et al. (2023b) proposed a spherical SGD that requires a separate estimation of the data covariance matrix, thereby oversimplifying the influence of covariance. In this study, we analyze the learning dynamics of vanilla SGD under the SIM with anisotropic input data, demonstrating that vanilla SGD automatically adapts to the data's covariance structure. Leveraging these results, we derive upper and lower bounds on the sample complexity using a notion of effective dimension that is determined by the structure of the covariance matrix instead of the input data dimension.

Approximation of Permutation Invariant Polynomials by Transformers: Efficient Construction in Column-Size

Transformers are a type of neural network that have demonstrated remarkable performance across various domains, particularly in natural language processing tasks. Motivated by this success, research on the theoretical understanding of transformers has garnered significant attention. A notable example is the mathematical analysis of their approximation power, which validates the empirical expressive capability of transformers. In this study, we investigate the ability of transformers to approximate column-symmetric polynomials, an extension of symmetric polynomials that take matrices as input. Consequently, we establish an explicit relationship between the size of the transformer network and its approximation capability, leveraging the parameter efficiency of transformers and their compatibility with symmetry by focusing on the algebraic properties of symmetric polynomials.

Federated Learning with Relative Fairness

This paper proposes a federated learning framework designed to achieve \textit{relative fairness} for clients. Traditional federated learning frameworks typically ensure absolute fairness by guaranteeing minimum performance across all client subgroups. However, this approach overlooks disparities in model performance between subgroups. The proposed framework uses a minimax problem approach to minimize relative unfairness, extending previous methods in distributionally robust optimization (DRO). A novel fairness index, based on the ratio between large and small losses among clients, is introduced, allowing the framework to assess and improve the relative fairness of trained models. Theoretical guarantees demonstrate that the framework consistently reduces unfairness. We also develop an algorithm, named \textsc{Scaff-PD-IA}, which balances communication and computational efficiency while maintaining minimax-optimal convergence rates. Empirical evaluations on real-world datasets confirm its effectiveness in maintaining model performance while reducing disparity.

Distillation of Discrete Diffusion through Dimensional Correlations

Diffusion models have demonstrated exceptional performances in various fields of generative modeling. While they often outperform competitors including VAEs and GANs in sample quality and diversity, they suffer from slow sampling speed due to their iterative nature. Recently, distillation techniques and consistency models are mitigating this issue in continuous domains, but discrete diffusion models have some specific challenges towards faster generation. Most notably, in the current literature, correlations between different dimensions (pixels, locations) are ignored, both by its modeling and loss functions, due to computational limitations. In this paper, we propose "mixture" models in discrete diffusion that are capable of treating dimensional correlations while remaining scalable, and we provide a set of loss functions for distilling the iterations of existing models. Two primary theoretical insights underpin our approach: first, that dimensionally independent models can well approximate the data distribution if they are allowed to conduct many sampling steps, and second, that our loss functions enables mixture models to distill such many-step conventional models into just a few steps by learning the dimensional correlations. We empirically demonstrate that our proposed method for discrete diffusions work in practice, by distilling a continuous-time discrete diffusion model pretrained on the CIFAR-10 dataset.

Effect of Random Learning Rate: Theoretical Analysis of SGD Dynamics in Non-Convex Optimization via Stationary Distribution

We consider a variant of the stochastic gradient descent (SGD) with a random learning rate and reveal its convergence properties. SGD is a widely used stochastic optimization algorithm in machine learning, especially deep learning. Numerous studies reveal the convergence properties of SGD and its simplified variants. Among these, the analysis of convergence using a stationary distribution of updated parameters provides generalizable results. However, to obtain a stationary distribution, the update direction of the parameters must not degenerate, which limits the applicable variants of SGD. In this study, we consider a novel SGD variant, Poisson SGD, which has degenerated parameter update directions and instead utilizes a random learning rate. Consequently, we demonstrate that a distribution of a parameter updated by Poisson SGD converges to a stationary distribution under weak assumptions on a loss function. Based on this, we further show that Poisson SGD finds global minima in non-convex optimization problems and also evaluate the generalization error using this method. As a proof technique, we approximate the distribution by Poisson SGD with that of the bouncy particle sampler (BPS) and derive its stationary distribution, using the theoretical advance of the piece-wise deterministic Markov process (PDMP).

Automatic Domain Adaptation by Transformers in In-Context Learning

Selecting or designing an appropriate domain adaptation algorithm for a given problem remains challenging. This paper presents a Transformer model that can provably approximate and opt for domain adaptation methods for a given dataset in the in-context learning framework, where a foundation model performs new tasks without updating its parameters at test time. Specifically, we prove that Transformers can approximate instance-based and feature-based unsupervised domain adaptation algorithms and automatically select an algorithm suited for a given dataset. Numerical results indicate that in-context learning demonstrates an adaptive domain adaptation surpassing existing methods.

Effect of Weight Quantization on Learning Models by Typical Case Analysis

This paper examines the quantization methods used in large-scale data analysis models and their hyperparameter choices. The recent surge in data analysis scale has significantly increased computational resource requirements. To address this, quantizing model weights has become a prevalent practice in data analysis applications such as deep learning. Quantization is particularly vital for deploying large models on devices with limited computational resources. However, the selection of quantization hyperparameters, like the number of bits and value range for weight quantization, remains an underexplored area. In this study, we employ the typical case analysis from statistical physics, specifically the replica method, to explore the impact of hyperparameters on the quantization of simple learning models. Our analysis yields three key findings: (i) an unstable hyperparameter phase, known as replica symmetry breaking, occurs with a small number of bits and a large quantization width; (ii) there is an optimal quantization width that minimizes error; and (iii) quantization delays the onset of overparameterization, helping to mitigate overfitting as indicated by the double descent phenomenon. We also discover that non-uniform quantization can enhance stability. Additionally, we develop an approximate message-passing algorithm to validate our theoretical results.