arXiv subjects

Alessandro Rinaldo

Publications and source records attributed to Alessandro Rinaldo.

At least 37 records · Page 2Linked to original sources

arXiv2025

Time-uniform concentration bounds for iterative algorithms

We develop a new framework for deriving time-uniform concentration bounds for the output of stochastic sequential algorithms satisfying certain recursive inequalities akin to those defining the almost-supermartingale processes introduced by \cite{robbins1971convergence}. Our approach is of wide applicability, and can be deployed in settings in which exponential supermartingale processes, required by prevailing methodologies for anytime-valid concentration inequalities, are not readily available. Our results can be viewed as quantitative versions of the classical Robbins-Siegmund Lemma. We demonstrate the effectiveness of our method by providing new and optimal time-uniform concentration bounds for Oja's algorithm for streaming PCA, stochastic gradient descent, and stochastic approximations.

math.ST↗

arXiv2025

On Minimax Estimation of Parameters in Softmax-Contaminated Mixture of Experts

The softmax-contaminated mixture of experts (MoE) model is deployed when a large-scale pre-trained model, which plays the role of a fixed expert, is fine-tuned for learning downstream tasks by including a new contamination part, or prompt, functioning as a new, trainable expert. Despite its popularity and relevance, the theoretical properties of the softmax-contaminated MoE have remained unexplored in the literature. In the paper, we study the convergence rates of the maximum likelihood estimator of gating and prompt parameters in order to gain insights into the statistical properties and potential challenges of fine-tuning with a new prompt. We find that the estimability of these parameters is compromised when the prompt acquires overlapping knowledge with the pre-trained model, in the sense that we make precise by formulating a novel analytic notion of distinguishability. Under distinguishability of the pre-trained and prompt models, we derive minimax optimal estimation rates for all the gating and prompt parameters. By contrast, when the distinguishability condition is violated, these estimation rates become significantly slower due to their dependence on the prompt convergence rate to the pre-trained model. Finally, we empirically corroborate our theoretical findings through several numerical experiments.

stat.ML↗

arXiv2025

Dual Induction CLT for High-dimensional m-dependent Data

We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of $m$-dependent random vectors over the class of hyper-rectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as non-degenerate covariances and finite third moments, and exhibit an optimal sample complexity of order $m^{(q-1)/(q-2)}/\sqrt{n}$. Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel inductive relationship between anti-concentration inequalities and Berry--Esseen bounds inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data that may be of independent interest.

math.PR↗

arXiv2025

Sigmoid Self-Attention has Lower Sample Complexity than Softmax Self-Attention: A Mixture-of-Experts Perspective

At the core of the popular Transformer architecture is the self-attention mechanism, which dynamically assigns softmax weights to each input token so that the model can focus on the most salient information. However, the softmax structure slows down the attention computation due to its row-wise nature, and it inherently introduces competition among tokens: as the weight assigned to one token increases, the weights of others decrease. This competitive dynamic may narrow the focus of self-attention to a limited set of features, potentially overlooking other informative characteristics. Recent experimental studies have shown that using the element-wise sigmoid function helps eliminate token competition and reduce the computational overhead. Despite these promising empirical results, a rigorous comparison between sigmoid and softmax self-attention mechanisms remains absent in the literature. This paper closes this gap by theoretically demonstrating that sigmoid self-attention is more sample-efficient than its softmax counterpart. Toward that goal, we represent the self-attention matrix as a mixture of experts and show that ``experts'' in sigmoid self-attention require significantly less data to achieve the same approximation error as those in softmax self-attention.

cs.LG↗

arXiv2025

Convergence Rates for Softmax Gating Mixture of Experts

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.

stat.ML↗

arXiv2024

Sigmoid Gating is More Sample Efficient than Softmax Gating in Mixture of Experts

The softmax gating function is arguably the most popular choice in mixture of experts modeling. Despite its widespread use in practice, the 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 the sigmoid gating, in fact, enjoys a higher sample efficiency than the 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 under the over-specified case in which the number of fitted experts is larger than the true value. We show that two gating regimes naturally arise and, in each of them, we formulate an identifiability condition 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 ReLU and GELU enjoy faster convergence rates under the sigmoid gating than those under 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.

stat.ML↗

arXiv2024

On Least Square Estimation in Softmax Gating Mixture of Experts

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.

stat.ML↗

arXiv2024

Multilayer random dot product graphs: Estimation and online change point detection

We study the multilayer random dot product graph (MRDPG) model, an extension of the random dot product graph to multilayer networks. To estimate the edge probabilities, we deploy a tensor-based methodology and demonstrate its superiority over existing approaches. Moving to dynamic MRDPGs, we formulate and analyse an online change point detection framework. At every time point, we observe a realization from an MRDPG. Across layers, we assume fixed shared common node sets and latent positions but allow for different connectivity matrices. We propose efficient tensor algorithms under both fixed and random latent position cases to minimize the detection delay while controlling false alarms. Notably, in the random latent position case, we devise a novel nonparametric change point detection algorithm based on density kernel estimation that is applicable to a wide range of scenarios, including stochastic block models as special cases. Our theoretical findings are supported by extensive numerical experiments, with the code available online https://github.com/MountLee/MRDPG.

stat.ME↗

arXiv2024

High-probability sample complexities for policy evaluation with linear function approximation

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.

stat.ML↗

arXiv2024

Inference for Projection Parameters in Linear Regression: beyond $d = o(n^{1/2})$

We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of a smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator. Precisely, we provide an explicit finite sample Berry Esseen bound on the Normal approximation to the law of the linear contrasts of the proposed estimator normalized by the sandwich standard error estimate. Our bound, under only finite moment conditions on covariates and errors, tends to 0 as long as $d = o(n^{2/3})$ up to the polylogarithmic factors. Furthermore, we leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference and that leads to a sharper miscoverage control compared to Wald's. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.

math.ST↗

arXiv2023

E-detectors: a nonparametric framework for sequential change detection

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.

stat.ME↗

arXiv2023

On the estimation of persistence intensity functions and linear representations of persistence diagrams

The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.

math.ST↗

arXiv2023

Divide and Conquer Dynamic Programming: An Almost Linear Time Change Point Detection Methodology in High Dimensions

We develop a novel, general and computationally efficient framework, called Divide and Conquer Dynamic Programming (DCDP), for localizing change points in time series data with high-dimensional features. DCDP deploys a class of greedy algorithms that are applicable to a broad variety of high-dimensional statistical models and can enjoy almost linear computational complexity. We investigate the performance of DCDP in three commonly studied change point settings in high dimensions: the mean model, the Gaussian graphical model, and the linear regression model. In all three cases, we derive non-asymptotic bounds for the accuracy of the DCDP change point estimators. We demonstrate that the DCDP procedures consistently estimate the change points with sharp, and in some cases, optimal rates while incurring significantly smaller computational costs than the best available algorithms. Our findings are supported by extensive numerical experiments on both synthetic and real data.

stat.ME↗

arXiv2023

A Sequential Test for Log-Concavity

On observing a sequence of i.i.d.\ data with distribution $P$ on $\mathbb{R}^d$, we ask the question of how one can test the null hypothesis that $P$ has a log-concave density. This paper proves one interesting negative and positive result: the non-existence of test (super)martingales, and the consistency of universal inference. To elaborate, the set of log-concave distributions $\mathcal{L}$ is a nonparametric class, which contains the set $\mathcal G$ of all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales for $\mathcal G$ (a process that is simultaneously a nonnegative supermartingale for every distribution in $\mathcal G$), and hence also for its superset $\mathcal{L}$. Due to this negative result, we turn our attention to constructing an e-process -- a process whose expectation at any stopping time is at most one, under any distribution in $\mathcal{L}$ -- which yields a level-$α$ test by simply thresholding at $1/α$. We take the approach of universal inference, which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of our knowledge, there is no other e-process or sequential test for $\mathcal{L}$.

math.ST↗

arXiv2022

High-dimensional Berry-Esseen Bound for $m$-Dependent Random Samples

In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.

math.PR↗

arXiv2022

Detecting Abrupt Changes in Sequential Pairwise Comparison Data

The Bradley-Terry-Luce (BTL) model is a classic and very popular statistical approach for eliciting a global ranking among a collection of items using pairwise comparison data. In applications in which the comparison outcomes are observed as a time series, it is often the case that data are non-stationary, in the sense that the true underlying ranking changes over time. In this paper we are concerned with localizing the change points in a high-dimensional BTL model with piece-wise constant parameters. We propose novel and practicable algorithms based on dynamic programming that can consistently estimate the unknown locations of the change points. We provide consistency rates for our methodology that depend explicitly on the model parameters, the temporal spacing between two consecutive change points and the magnitude of the change. We corroborate our findings with extensive numerical experiments and a real-life example.

stat.ME↗

arXiv2022

$\ell_{\infty}$-Bounds of the MLE in the BTL Model under General Comparison Graphs

The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model parameters in the $\ell_{\infty}$-loss. The difficulty of this task depends crucially on the topology of the pairwise comparison graph over the given items. However, beyond very few well-studied cases, such as the complete and Erdös-Rényi comparison graphs, little is known about the performance of the maximum likelihood estimator MLE) of the BTL model parameters in the $\ell_{\infty}$-loss under more general graph topologies. In this paper, we derive novel, general upper bounds on the $\ell_{\infty}$ estimation error of the BTL MLE that depend explicitly on the algebraic connectivity of the comparison graph, the maximal performance gap across items and the sample complexity. We demonstrate that the derived bounds perform well and in some cases are sharper compared to known results obtained using different loss functions and more restricted assumptions and graph topologies. We carefully compare our results to Yan et al. (2012), which is closest in spirit to our work. We further provide minimax lower bounds under $\ell_{\infty}$-error that nearly match the upper bounds over a class of sufficiently regular graph topologies. Finally, we study the implications of our $\ell_{\infty}$-bounds for efficient (offline) tournament design. We illustrate and discuss our findings through various examples and simulations.

math.ST↗

arXiv2022

Generalized Results for the Existence and Consistency of the MLE in the Bradley-Terry-Luce Model

Ranking problems based on pairwise comparisons, such as those arising in online gaming, often involve a large pool of items to order. In these situations, the gap in performance between any two items can be significant, and the smallest and largest winning probabilities can be very close to zero or one. Furthermore, each item may be compared only to a subset of all the items, so that not all pairwise comparisons are observed. In this paper, we study the performance of the Bradley-Terry-Luce model for ranking from pairwise comparison data under more realistic settings than those considered in the literature so far. In particular, we allow for near-degenerate winning probabilities and arbitrary comparison designs. We obtain novel results about the existence of the maximum likelihood estimator (MLE) and the corresponding $\ell_2$ estimation error without the bounded winning probability assumption commonly used in the literature and for arbitrary comparison graph topologies. Central to our approach is the reliance on the Fisher information matrix to express the dependence on the graph topologies and the impact of the values of the winning probabilities on the estimation risk and on the conditions for the existence of the MLE. Our bounds recover existing results as special cases but are more broadly applicable.

math.ST↗