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Alessandro Rinaldo

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22 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

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

Fanqi Yan
Huy Nguyen
Le Dung
Pedram Akbarian
Nhat Ho
Alessandro Rinaldo

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.

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ICML Conference 2024 Conference Paper

On Least Square Estimation in Softmax Gating Mixture of Experts

Huy Nguyen
Nhat Ho
Alessandro Rinaldo

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.

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NeurIPS Conference 2024 Conference Paper

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

Huy Nguyen
Nhat Ho
Alessandro Rinaldo

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 $\mathrm{ReLU}$ and $\mathrm{GELU}$ 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.

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ICML Conference 2023 Conference Paper

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

Wanshan Li
Daren Wang
Alessandro Rinaldo

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.

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NeurIPS Conference 2022 Conference Paper

Detecting Abrupt Changes in Sequential Pairwise Comparison Data

Wanshan Li
Alessandro Rinaldo
Daren Wang

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.

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ICML Conference 2022 Conference Paper

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

Heejong Bong
Alessandro Rinaldo

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.

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UAI Conference 2022 Conference Paper

ℓ ∞ -Bounds of the MLE in the BTL Model under General Comparison Graphs

Wanshan Li
Shamindra Shrotriya
Alessandro Rinaldo

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 Erd{ö}s-R{é}nyi 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.

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NeurIPS Conference 2021 Conference Paper

Lattice partition recovery with dyadic CART

OSCAR HERNAN MADRID PADILLA
Yi Yu
Alessandro Rinaldo

We study piece-wise constant signals corrupted by additive Gaussian noise over a $d$-dimensional lattice. Data of this form naturally arise in a host of applications, and the tasks of signal detection or testing, de-noising and estimation have been studied extensively in the statistical and signal processing literature. In this paper we consider instead the problem of partition recovery, i. e. ~of estimating the partition of the lattice induced by the constancy regions of the unknown signal, using the computationally-efficient dyadic classification and regression tree (DCART) methodology proposed by \citep{donoho1997cart}. We prove that, under appropriate regularity conditions on the shape of the partition elements, a DCART-based procedure consistently estimates the underlying partition at a rate of order $\sigma^2 k^* \log (N)/\kappa^2$, where $k^*$ is the minimal number of rectangular sub-graphs obtained using recursive dyadic partitions supporting the signal partition, $\sigma^2$ is the noise variance, $\kappa$ is the minimal magnitude of the signal difference among contiguous elements of the partition and $N$ is the size of the lattice. Furthermore, under stronger assumptions, our method attains a sharper estimation error of order $\sigma^2\log(N)/\kappa^2$, independent of $k^*$, which we show to be minimax rate optimal. Our theoretical guarantees further extend to the partition estimator based on the optimal regression tree estimator (ORT) of \cite{chatterjee2019adaptive} and to the one obtained through an NP-hard exhaustive search method. We corroborate our theoretical findings and the effectiveness of DCART for partition recovery in simulations.

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ICML Conference 2020 Conference Paper

On Conditional Versus Marginal Bias in Multi-Armed Bandits

Jaehyeok Shin
Aaditya Ramdas
Alessandro Rinaldo

The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the conditional inference setting in which the sample means are computed only if some specific conditions are satisfied. In this paper, we characterize the sign of the conditional bias of monotone functions of the rewards, including the sample mean. Our results hold for arbitrary conditioning events and leverage natural monotonicity properties of the data collection policy. We further demonstrate, through several examples from sequential testing and best arm identification, that the sign of the conditional and marginal bias of the sample mean of an arm can be different, depending on the conditioning event. Our analysis offers new and interesting perspectives on the subtleties of assessing the bias in data adaptive settings.

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NeurIPS Conference 2019 Conference Paper

Are sample means in multi-armed bandits positively or negatively biased?

Jaehyeok Shin
Aaditya Ramdas
Alessandro Rinaldo

It is well known that in stochastic multi-armed bandits (MAB), the sample mean of an arm is typically not an unbiased estimator of its true mean. In this paper, we decouple three different sources of this selection bias: adaptive \emph{sampling} of arms, adaptive \emph{stopping} of the experiment, and adaptively \emph{choosing} which arm to study. Through a new notion called ``optimism'' that captures certain natural monotonic behaviors of algorithms, we provide a clean and unified analysis of how optimistic rules affect the sign of the bias. The main takeaway message is that optimistic sampling induces a negative bias, but optimistic stopping and optimistic choosing both induce a positive bias. These results are derived in a general stochastic MAB setup that is entirely agnostic to the final aim of the experiment (regret minimization or best-arm identification or anything else). We provide examples of optimistic rules of each type, demonstrate that simulations confirm our theoretical predictions, and pose some natural but hard open problems.

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JMLR Journal 2019 Journal Article

DBSCAN: Optimal Rates For Density-Based Cluster Estimation

Daren Wang
Xinyang Lu
Alessandro Rinaldo

We study the problem of optimal estimation of the density cluster tree under various smoothness assumptions on the underlying density. Inspired by the seminal work of Chaudhuri et al. (2014), we formulate a new notion of clustering consistency which is better suited to smooth densities, and derive minimax rates for cluster tree estimation under Hölder smooth densities of arbitrary degree. We present a computationally efficient, rate optimal cluster tree estimator based on simple extensions of the popular DBSCAN algorithm of Ester et al. (1996). Our procedure relies on kernel density estimators and returns a sequence of nested random geometric graphs whose connected components form a hierarchy of clusters. The resulting optimal rates for cluster tree estimation depend on the degree of smoothness of the underlying density and, interestingly, match the minimax rates for density estimation under the sup-norm loss. Our results complement and extend the analysis of the DBSCAN algorithm in Sriperumbudur and Steinwart (2012). Finally, we consider level set estimation and cluster consistency for densities with jump discontinuities. We demonstrate that the DBSCAN algorithm attains the minimax rate in terms of the jump size and sample size in this setting as well. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

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NeurIPS Conference 2019 Conference Paper

Statistical Analysis of Nearest Neighbor Methods for Anomaly Detection

Xiaoyi Gu
Leman Akoglu
Alessandro Rinaldo

Nearest-neighbor (NN) procedures are well studied and widely used in both supervised and unsupervised learning problems. In this paper we are concerned with investigating the performance of NN-based methods for anomaly detection. We first show through extensive simulations that NN methods compare favorably to some of the other state-of-the-art algorithms for anomaly detection based on a set of benchmark synthetic datasets. We further consider the performance of NN methods on real datasets, and relate it to the dimensionality of the problem. Next, we analyze the theoretical properties of NN-methods for anomaly detection by studying a more general quantity called distance-to-measure (DTM), originally developed in the literature on robust geometric and topological inference. We provide finite-sample uniform guarantees for the empirical DTM and use them to derive misclassification rates for anomalous observations under various settings. In our analysis we rely on Huber's contamination model and formulate mild geometric regularity assumptions on the underlying distribution of the data.

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ICML Conference 2019 Conference Paper

Uniform Convergence Rate of the Kernel Density Estimator Adaptive to Intrinsic Volume Dimension

Jisu Kim
Jaehyeok Shin
Alessandro Rinaldo
Larry A. Wasserman

We derive concentration inequalities for the supremum norm of the difference between a kernel density estimator (KDE) and its point-wise expectation that hold uniformly over the selection of the bandwidth and under weaker conditions on the kernel and the data generating distribution than previously used in the literature. We first propose a novel concept, called the volume dimension, to measure the intrinsic dimension of the support of a probability distribution based on the rates of decay of the probability of vanishing Euclidean balls. Our bounds depend on the volume dimension and generalize the existing bounds derived in the literature. In particular, when the data-generating distribution has a bounded Lebesgue density or is supported on a sufficiently well-behaved lower-dimensional manifold, our bound recovers the same convergence rate depending on the intrinsic dimension of the support as ones known in the literature. At the same time, our results apply to more general cases, such as the ones of distribution with unbounded densities or supported on a mixture of manifolds with different dimensions. Analogous bounds are derived for the derivative of the KDE, of any order. Our results are generally applicable but are especially useful for problems in geometric inference and topological data analysis, including level set estimation, density-based clustering, modal clustering and mode hunting, ridge estimation and persistent homology.

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JMLR Journal 2018 Journal Article

Robust Topological Inference: Distance To a Measure and Kernel Distance

Frédéric Chazal
Brittany Fasy
Fabrizio Lecci
Bertrand Michel
Alessandro Rinaldo
Larry Wasserman

Let $P$ be a distribution with support $S$. The salient features of $S$ can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point $x$ to $S$). Given a sample from $P$ we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by \cite{chazal2011geometric}, and the kernel distance, introduced by \cite{phillips2014goemetric}, are smooth functions that provide useful topological information but are robust to noise and outliers. \cite{massart2014} derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

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NeurIPS Conference 2017 Conference Paper

A Sharp Error Analysis for the Fused Lasso, with Application to Approximate Changepoint Screening

Kevin Lin
James Sharpnack
Alessandro Rinaldo
Ryan Tibshirani

In the 1-dimensional multiple changepoint detection problem, we derive a new fast error rate for the fused lasso estimator, under the assumption that the mean vector has a sparse number of changepoints. This rate is seen to be suboptimal (compared to the minimax rate) by only a factor of $\log\log{n}$. Our proof technique is centered around a novel construction that we call a lower interpolant. We extend our results to misspecified models and exponential family distributions. We also describe the implications of our error analysis for the approximate screening of changepoints.

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NeurIPS Conference 2016 Conference Paper

Statistical Inference for Cluster Trees

Jisu Kim
Yen-Chi Chen
Sivaraman Balakrishnan
Alessandro Rinaldo
Larry Wasserman

A cluster tree provides an intuitive summary of a density function that reveals essential structure about the high-density clusters. The true cluster tree is estimated from a finite sample from an unknown true density. This paper addresses the basic question of quantifying our uncertainty by assessing the statistical significance of different features of an empirical cluster tree. We first study a variety of metrics that can be used to compare different trees, analyzing their properties and assessing their suitability for our inference task. We then propose methods to construct and summarize confidence sets for the unknown true cluster tree. We introduce a partial ordering on cluster trees which we use to prune some of the statistically insignificant features of the empirical tree, yielding interpretable and parsimonious cluster trees. Finally, we provide a variety of simulations to illustrate our proposed methods and furthermore demonstrate their utility in the analysis of a Graft-versus-Host Disease (GvHD) data set.

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ICML Conference 2015 Conference Paper

Subsampling Methods for Persistent Homology

Frédéric Chazal
Brittany Terese Fasy
Fabrizio Lecci
Bertrand Michel
Alessandro Rinaldo
Larry A. Wasserman

Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nature of the existing algorithms. We propose to compute the persistent homology of several subsamples of the data and then combine the resulting estimates. We study the risk of two estimators and we prove that the subsampling approach carries stable topological information while achieving a great reduction in computational complexity.

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JMLR Journal 2014 Journal Article

Statistical Analysis of Metric Graph Reconstruction

Fabrizio Lecci
Alessandro Rinaldo
Larry Wasserman

A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph embedded in $\mathbb{R}^D$ from a random sample. We derive lower and upper bounds on the minimax risk for the noiseless case and tubular noise case. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012). [abs] [ pdf ][ bib ] &copy JMLR 2014. ( edit, beta )

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NeurIPS Conference 2013 Conference Paper

Cluster Trees on Manifolds

Sivaraman Balakrishnan
Srivatsan Narayanan
Alessandro Rinaldo
Aarti Singh
Larry Wasserman

We investigate the problem of estimating the cluster tree for a density $f$ supported on or near a smooth $d$-dimensional manifold $M$ isometrically embedded in $\mathbb{R}^D$. We study a $k$-nearest neighbor based algorithm recently proposed by Chaudhuri and Dasgupta. Under mild assumptions on $f$ and $M$, we obtain rates of convergence that depend on $d$ only but not on the ambient dimension $D$. We also provide a sample complexity lower bound for a natural class of clustering algorithms that use $D$-dimensional neighborhoods.

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JMLR Journal 2013 Journal Article

Differential Privacy for Functions and Functional Data

Rob Hall
Alessandro Rinaldo
Larry Wasserman

Differential privacy is a rigorous cryptographically-motivated characterization of data privacy which may be applied when releasing summaries of a database. Previous work has focused mainly on methods for which the output is a finite dimensional vector, or an element of some discrete set. We develop methods for releasing functions while preserving differential privacy. Specifically, we show that adding an appropriate Gaussian process to the function of interest yields differential privacy. When the functions lie in the reproducing kernel Hilbert space (RKHS) generated by the covariance kernel of the Gaussian process, then the correct noise level is established by measuring the “sensitivity” of the function in the RKHS norm. As examples we consider kernel density estimation, kernel support vector machines, and functions in RKHSs. [abs] [ pdf ][ bib ] &copy JMLR 2013. ( edit, beta )

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JMLR Journal 2012 Journal Article

Stability of Density-Based Clustering

Alessandro Rinaldo
Aarti Singh
Rebecca Nugent
Larry Wasserman

High density clusters can be characterized by the connected components of a level set L(λ) = {x: p(x)>λ} of the underlying probability density function p generating the data, at some appropriate level λ ≥ 0. The complete hierarchical clustering can be characterized by a cluster tree T= ∪ λ L(λ). In this paper, we study the behavior of a density level set estimate L̂(λ) and cluster tree estimate T̂ based on a kernel density estimator with kernel bandwidth h. We define two notions of instability to measure the variability of L̂(λ) and T̂ as a function of h, and investigate the theoretical properties of these instability measures. [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

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NeurIPS Conference 2011 Conference Paper

Minimax Localization of Structural Information in Large Noisy Matrices

Mladen Kolar
Sivaraman Balakrishnan
Alessandro Rinaldo
Aarti Singh

We consider the problem of identifying a sparse set of relevant columns and rows in a large data matrix with highly corrupted entries. This problem of identifying groups from a collection of bipartite variables such as proteins and drugs, biological species and gene sequences, malware and signatures, etc is commonly referred to as biclustering or co-clustering. Despite its great practical relevance, and although several ad-hoc methods are available for biclustering, theoretical analysis of the problem is largely non-existent. The problem we consider is also closely related to structured multiple hypothesis testing, an area of statistics that has recently witnessed a flurry of activity. We make the following contributions: i) We prove lower bounds on the minimum signal strength needed for successful recovery of a bicluster as a function of the noise variance, size of the matrix and bicluster of interest. ii) We show that a combinatorial procedure based on the scan statistic achieves this optimal limit. iii) We characterize the SNR required by several computationally tractable procedures for biclustering including element-wise thresholding, column/row average thresholding and a convex relaxation approach to sparse singular vector decomposition.

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