Mixture of experts (MoE) has recently emerged as an effective framework to advance the efficiency and scalability of machine learning models by softly dividing complex tasks among multiple specialized sub-models termed experts. Central to the success of MoE is an adaptive softmax gating mechanism which takes responsibility for determining the relevance of each expert to a given input and then dynamically assigning experts their respective weights. Despite its widespread use in practice, a comprehensive study on the effects of the softmax gating on the MoE has been lacking in the literature. To bridge this gap in this paper, we perform a convergence analysis of parameter estimation and expert estimation under the MoE equipped with the standard softmax gating or its variants, including a dense-to-sparse gating and a hierarchical softmax gating, respectively. Furthermore, our theories also provide useful insights into the design of sample-efficient expert structures. In particular, we demonstrate that it requires polynomially many data points to estimate experts satisfying our proposed \emph{strong identifiability} condition, namely a commonly used two-layer feed-forward network. In stark contrast, estimating linear experts, which violate the strong identifiability condition, necessitates exponentially many data points as a result of intrinsic parameter interactions expressed in the language of partial differential equations. All the theoretical results are substantiated with a rigorous guarantee.

Markov chains are fundamental to statistical machine learning, underpinning key methodologies such as Markov Chain Monte Carlo (MCMC) sampling and temporal difference (TD) learning in reinforcement learning (RL). Given their widespread use, it is crucial to establish rigorous probabilistic guarantees on their convergence, uncertainty, and stability. In this work, we develop novel, high-dimensional concentration inequalities and Berry-Esseen bounds for vector- and matrix-valued functions of Markov chains, addressing key limitations in existing theoretical tools for handling dependent data. We leverage these results to analyze the TD learning algorithm, a widely used method for policy evaluation in RL. Our analysis yields a sharp high-probability consistency guarantee that matches the asymptotic variance up to logarithmic factors. Furthermore, we establish a $O(T^{-\frac{1}{4}}\log T)$ distributional convergence rate for the Gaussian approximation of the TD estimator, measured in convex distance. These findings provide new insights into statistical inference for RL algorithms, bridging the gaps between classical stochastic approximation theory and modern reinforcement learning applications.

Transformer models [54] have been known as the state-of-the-art architecture for a wide range of machine learning and deep learning applications, including language modeling [16, 3, 47, 51], computer vision [17, 4, 46, 35], and reinforcement learning [5, 31, 25], etc. One of the central components that contribute to the success of the Transformer models is the self-attention mechanism, which enables sequence-to-sequence models to concentrate on relevant parts of the input data. In particular, for each token in an input sequence, the self-attention mechanism computes a context vector formulated as a weighted sum of the tokens, where more relevant tokens to the context are assigned larger weights than others (see Section 2.1 for a formal definition). Therefore, self-attention is able to capture long-range dependencies and complex relationships within the data. However, since the weights in the context vector are normalized by the softmax function, there might be an undesirable competition among the tokens, that is, an increase in the weight of a token leads to a decrease in the weights of others. As a consequence, the traditional softmax self-attention mechanism might focus only on a few aspects of the data and possibly ignore other informative features [48]. Additionally, [22] also discovered that the tokens' inner dependence on the attention scores owing to the softmax normalization partly causes the attention sink phenomenon occurring

Diffusion models have emerged as a powerful tool for image generation and denoising. Typically, generative models learn a trajectory between the starting noise distribution and the target data distribution. Recently Liu et al. (2023b) designed a novel alternative generative model Rectified Flow (RF), which aims to learn straight flow trajectories from noise to data using a sequence of convex optimization problems with close ties to optimal transport. If the trajectory is curved, one must use many Euler discretization steps or novel strategies, such as exponential integrators, to achieve a satisfactory generation quality. In contrast, RF has been shown to theoretically straighten the trajectory through successive rectifications, reducing the number of function evaluations (NFEs) while sampling. It has also been shown empirically that RF may improve the straightness in two rectifications if one can solve the underlying optimization problem within a sufficiently small error. In this paper, we make two key theoretical contributions: 1) we provide the first theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution. Our error rate is characterized by the number of discretization steps and a \textit{new formulation of straightness} stronger than that in the original work. 2) under a mild regularity assumption, we show that for a rectified flow from a Gaussian to any general target distribution with finite first moment (e.g. mixture of Gaussians), two rectifications are sufficient to achieve a straight flow, which is in line with the previous empirical findings. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings.

Statistical inference tasks, such as constructing confidence regions or simultaneous confidence intervals, are often addressed by deriving distributional theory such as central limit theorems (CLTs) for the estimator in use. Due to the high dimensionality of modern science and engineering applications, there has been a surge of interests in deriving convergence results that are valid in a finite-sample manner. In Reinforcement Learning (RL), a discipline that underpins many recent machine learning breakthroughs (Agarwal et al. (2019); Sutton and Barto (2018)) one central question is to evaluate with confidence the quality of a given policy, measured by its value function. As RL is often modeled as decision making in Markov decision processes (MDPs), the task of statistical inference needs to accommodate the online nature of the sampling mechanism. Temporal Difference (TD) learning (Sutton (1988)) is arguably the most widely used algorithm designed for this purpose. TD learning, which is an instance of stochastic approximation (SA) method (Robbins and Monro (1951)), approximates the value function of a given policy in an iterative manner. Towards understanding the non-asymptotic convergence rate of TD to the target value function, extensive recent efforts have been made (see, e.g.

The softmax gating function is arguably the most popular choice in mixture of experts modeling. Despite its widespread use in practice, softmax gating may lead to unnecessary competition among experts, potentially causing the undesirable phenomenon of representation collapse due to its inherent structure. In response, the sigmoid gating function has been recently proposed as an alternative and has been demonstrated empirically to achieve superior performance. However, a rigorous examination of the sigmoid gating function is lacking in current literature. In this paper, we verify theoretically that sigmoid gating, in fact, enjoys a higher sample efficiency than softmax gating for the statistical task of expert estimation. Towards that goal, we consider a regression framework in which the unknown regression function is modeled as a mixture of experts, and study the rates of convergence of the least squares estimator in the over-specified case in which the number of experts fitted is larger than the true value. We show that two gating regimes naturally arise and, in each of them, we formulate identifiability conditions for the expert functions and derive the corresponding convergence rates. In both cases, we find that experts formulated as feed-forward networks with commonly used activation such as $\mathrm{ReLU}$ and $\mathrm{GELU}$ enjoy faster convergence rates under sigmoid gating than softmax gating. Furthermore, given the same choice of experts, we demonstrate that the sigmoid gating function requires a smaller sample size than its softmax counterpart to attain the same error of expert estimation and, therefore, is more sample efficient.

Mixture of experts (MoE) model is a statistical machine learning design that aggregates multiple expert networks using a softmax gating function in order to form a more intricate and expressive model. Despite being commonly used in several applications owing to their scalability, the mathematical and statistical properties of MoE models are complex and difficult to analyze. As a result, previous theoretical works have primarily focused on probabilistic MoE models by imposing the impractical assumption that the data are generated from a Gaussian MoE model. In this work, we investigate the performance of the least squares estimators (LSE) under a deterministic MoE model where the data are sampled according to a regression model, a setting that has remained largely unexplored. We establish a condition called strong identifiability to characterize the convergence behavior of various types of expert functions. We demonstrate that the rates for estimating strongly identifiable experts, namely the widely used feed forward networks with activation functions $\mathrm{sigmoid}(\cdot)$ and $\tanh(\cdot)$, are substantially faster than those of polynomial experts, which we show to exhibit a surprising slow estimation rate. Our findings have important practical implications for expert selection.

Sequential change detection is a classical problem with a variety of applications. However, the majority of prior work has been parametric, for example, focusing on exponential families. We develop a fundamentally new and general framework for sequential change detection when the pre- and post-change distributions are nonparametrically specified (and thus composite). Our procedures come with clean, nonasymptotic bounds on the average run length (frequency of false alarms). In certain nonparametric cases (like sub-Gaussian or sub-exponential), we also provide near-optimal bounds on the detection delay following a changepoint. The primary technical tool that we introduce is called an \emph{e-detector}, which is composed of sums of e-processes -- a fundamental generalization of nonnegative supermartingales -- that are started at consecutive times. We first introduce simple Shiryaev-Roberts and CUSUM-style e-detectors, and then show how to design their mixtures in order to achieve both statistical and computational efficiency. Our e-detector framework can be instantiated to recover classical likelihood-based procedures for parametric problems, as well as yielding the first change detection method for many nonparametric problems. As a running example, we tackle the problem of detecting changes in the mean of a bounded random variable without i.i.d. assumptions, with an application to tracking the performance of a basketball team over multiple seasons.

This paper is concerned with the problem of policy evaluation with linear function approximation in discounted infinite horizon Markov decision processes. We investigate the sample complexities required to guarantee a predefined estimation error of the best linear coefficients for two widely-used policy evaluation algorithms: the temporal difference (TD) learning algorithm and the two-timescale linear TD with gradient correction (TDC) algorithm. In both the on-policy setting, where observations are generated from the target policy, and the off-policy setting, where samples are drawn from a behavior policy potentially different from the target policy, we establish the first sample complexity bound with high-probability convergence guarantee that attains the optimal dependence on the tolerance level. We also exhihit an explicit dependence on problem-related quantities, and show in the on-policy setting that our upper bound matches the minimax lower bound on crucial problem parameters, including the choice of the feature maps and the problem dimension.

Recent empirical and theoretical analyses of several commonly used prediction procedures reveal a peculiar risk behavior in high dimensions, referred to as double/multiple descent, in which the asymptotic risk is a non-monotonic function of the limiting aspect ratio of the number of features or parameters to the sample size. To mitigate this undesirable behavior, we develop a general framework for risk monotonization based on cross-validation that takes as input a generic prediction procedure and returns a modified procedure whose out-of-sample prediction risk is, asymptotically, monotonic in the limiting aspect ratio. As part of our framework, we propose two data-driven methodologies, namely zero- and one-step, that are akin to bagging and boosting, respectively, and show that, under very mild assumptions, they provably achieve monotonic asymptotic risk behavior. Our results are applicable to a broad variety of prediction procedures and loss functions, and do not require a well-specified (parametric) model. We exemplify our framework with concrete analyses of the minimum $\ell_2$, $\ell_1$-norm least squares prediction procedures. As one of the ingredients in our analysis, we also derive novel additive and multiplicative forms of oracle risk inequalities for split cross-validation that are of independent interest.