Efficient sampling reduces memory usage for Minimax distance analysis.
problem Quadratic memory requirement for existing Minimax distance methods.
method Proposes a novel sampling technique with linear space complexity.
result Demonstrates significant reduction in memory usage for Minimax distances.
We study the minimax optimal rate for estimating the Wasserstein-1 metric between two unknown probability measures based on n i.i.d. empirical samples from them. We show that estimating the Wasserstein metric itself between probability measures, is not significantly easier than estimating the probability measures u…
Develops a new minimax probability machine for imbalanced classification tasks.
problem Imbalanced classification tasks with non-decomposable performance measures.
method Derives an equivalent form of the MPMF model for solving linear and nonlinear classifiers.
result Demonstrates the effectiveness of the new model on real-world datasets.
We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerica…
A new method solves complex constrained minimax problems.
problem Solving constrained minimax optimization problems.
method First-order augmented Lagrangian method.
result Established an operation complexity of O(ε−4logε−1). We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on n independent samples from them. Curiously, we show that estimating the IPM itself between probability measures, is not significantly easier than estimating the probabili…
Preconditioned non-convex gradient descent improves noisy matrix estimation.
problem Estimating low-rank matrices from noisy measurements.
method Preconditioned non-convex gradient descent for noisy measurements.
result Preconditioned method converges to minimax optimal estimate at a linear rate.
In this paper we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope with respect to a family of absolutely continuous probability measures. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent…
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ-smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Choquet and minimax expectations are equivalent in European option pricing.
problem Pricing European options in incomplete markets.
method Comparing Choquet and minimax expectations in the context of European options.
result Choquet and minimax expectations are equal for European options.
This work establishes distribution-free upper and lower bounds on the minimax label complexity of active learning with general hypothesis classes, under various noise models. The results reveal a number of surprising facts. In particular, under the noise model of Tsybakov (2004), the minimax label complexity of active …
Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.
problem Empirical success of SGD in semi-discrete OT, but lack of theoretical guarantees.
method Averaged projected SGD with a minimax convergence rate of O(1/√n).
result SGD methods can estimate the OT map with a minimax convergence rate of O(1/√n).
Bayesian approach to robust risk measures under model uncertainty.
problem Representing robust risk measures as a single probability measure.
method Introducing two types of risk measures and analyzing their relation to robust risk measures.
result Robust risk measures can be represented by a mixture probability measure, a Bayesian approach.
Develops a minimax optimal estimator for system stability under distribution shift.
problem Ensuring system reliability under changes in the underlying environment.
method Minimax optimal estimation of stability defined in terms of acceptable performance degradation.
result Characterizes the minimax convergence rate and demonstrates practical utility.
The study establishes minimax bounds for estimating operators from noisy samples.
problem Estimating unknown operators between Hilbert spaces from noisy data.
method Developed a minimax theory for uniformly bounded Lipschitz operators, proving lower and upper bounds.
result Sharp characterizations of minimax risk for generic Lipschitz operators, showing a curse of sample complexity.
Paper characterizes minimax regret rates for online ranking with top-k feedback.
problem Analyzing online ranking with partial feedback.
method Developed techniques from partial monitoring to characterize minimax regret rates.
result Full characterization of minimax regret rates for Precision@n.
Bayesian histograms achieve optimal distribution estimation with minimal memory usage.
problem Efficiently estimating distributions with minimal memory footprint.
method Bayesian histograms for distribution estimation under Wasserstein distance.
result Bayesian histograms require fewer bins to achieve minimax optimality, reducing memory usage by a polynomial factor.
A new method solves a complex optimization problem efficiently.
problem Nonconvex-strongly-concave constrained minimax optimization.
method First-order augmented Lagrangian method with a first-order subproblem solver.
result Achieves improved operation complexity for finding solutions.
New method for sequential probability assignment reduces regret using contextual Shtarkov sums.
problem Minimizing regret in sequential probability assignment with arbitrary hypothesis classes.
method Introducing contextual Shtarkov sum and contextual Normalized Maximum Likelihood (cNML) algorithm.
result The contextual Shtarkov sum characterizes minimax regret and provides a minimax optimal strategy.
Transformers recall from long distributions with statistical guarantees.
problem Designing Transformers that can recall from arbitrarily long, distributional contexts.
method Recast associative memory as probability measures, decomposing the task into recall and prediction.
result A shallow measure-theoretic Transformer learns the recall-and-predict map under spectral assumptions.
The paper analyzes generalization bounds for NC-SC/NC-C stochastic minimax optimization.
problem Generalization analysis of nonconvex-(strongly)-concave stochastic minimax optimization.
method Established algorithm-agnostic and algorithm-dependent generalization bounds via uniform convergence and stability arguments.
result Sample complexities and generalization bounds for NC-SC and NC-C settings.
Paper improves risk bounds for nonconvex-strongly-concave minimax problems.
problem Achieving sharper risk bounds for nonconvex-strongly-concave minimax problems.
method Using uniform localized convergence to derive high probability generalization error bounds.
result Derives n times faster excess primal risk bounds for popular algorithms.
Paper estimates the order of vertices in random recursive trees.
problem Estimating the order of arrival of vertices in random recursive trees.
method Proposes an order estimator based on the Jordan centrality measure and defines risk measures.
result Establishes a nearly optimal estimator for the problem.
New research optimizes HSIC estimation rate for translation-invariant kernels.
problem Optimizing the rate of HSIC estimation for translation-invariant kernels.
method Proved minimax optimal rate of O(n−1/2) for HSIC estimation. result Optimality of various HSIC estimators proven.
Study dual representations for quasiconvex systemic risk measures.
problem Finding dual representations for quasiconvex systemic risk measures.
method Abstract infinite-dimensional setting, explicit formula for penalty function, nonstandard minimax inequality.
result Explicit formula for the penalty function of quasiconvex compositions.
Paper explores fair classification with bounded disparity using finite datasets.
problem Ensuring fairness in binary classification with protected groups.
method Minimax optimal approach with fairness constraints and demographic disparity control.
result Proposes FairBayes-DDP+ method that achieves minimax lower bound on fairness-aware excess risk.
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…
When eliciting judgements from humans for an unknown quantity, one often has the choice of making direct-scoring (cardinal) or comparative (ordinal) measurements. In this paper we study the relative merits of either choice, providing empirical and theoretical guidelines for the selection of a measurement scheme. We pro…
This paper analyzes neural networks for solving complex optimization problems.
problem Minimax optimization problems in infinite-dimensional function spaces.
method Mean-field analysis of stochastic gradient descent-ascent in neural networks.
result The algorithm converges to a stationary point at a sublinear rate.
State-of-the-art adversarial attacks are aimed at neural network classifiers. By default, neural networks use gradient descent to minimize their loss function. The gradient of a classifier's loss function is used by gradient-based adversarial attacks to generate adversarially perturbed images. We pose the question whet…
The paper studies PCA of probability measures with varying sample sizes and finds optimal convergence rates.
problem PCA of multiple probability measures with varying sample sizes.
method Double asymptotic regime analysis with convergence rates n−1/2+m−α for empirical covariance and PCA risk. result Optimal convergence rates for empirical covariance and PCA risk in the dense regime are proven.
SDP achieves optimal error in noisy phase synchronization.
problem Phase synchronization with noisy measurements.
method SDP relaxation of Maximum Likelihood Estimation (MLE).
result Achieves error bound of (1+o(1))2npσ2 under normalized squared ℓ2 loss, matching minimax lower bound. There is a rapidly increasing interest in crowdsourcing for data labeling. By crowdsourcing, a large number of labels can be often quickly gathered at low cost. However, the labels provided by the crowdsourcing workers are usually not of high quality. In this paper, we propose a minimax conditional entropy principle to…
Study nonparametric density estimation via measure transport, achieving optimal rates.
problem Nonparametric density estimation with optimal rates.
method Measure transport, penalized maximum likelihood, and sieved wavelet estimators.
result Achieve minimax optimal convergence rates over Hölder classes of densities.
Paper develops a new method for differential privacy sampling using Wasserstein distance.
problem Sampling from distributions under differential privacy constraints with geometric structure consideration.
method Develops a novel framework with Wasserstein Projection Mechanism (WPM) for minimax optimal mechanisms.
result Proposes efficient algorithms for approximate computation of the Wasserstein Projection Mechanism.
Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.
problem Online learning with set-valued feedback, where labels are sets rather than single labels.
method Introduced new combinatorial dimensions (Set Littlestone and Measure Shattering) to characterize learnability.
result Characterized deterministic and randomized online learnability, and established bounds for various learning settings.
Study stabilizes adversarial training in neural networks over infinite-dimensional spaces.
problem Stability issues in adversarial training of neural networks.
method Functional analysis of minimax optimization over infinite-dimensional spaces of continuous functions and probability measures.
result Convergence property of minimax problems under certain conditions, interpreted as stabilization techniques.
Paper reconciles minimax rates and optimal recovery rates for noisy observations.
problem Estimating a function from noisy observations.
method Develops NLA minimax rates for Besov classes in Lq-norms. result NLA minimax rates continuously depend on noise level and match optimal recovery rates as noise decreases.
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
Scalable algorithm for computing Wasserstein-2 barycenters without bias.
problem Computing Wasserstein-2 barycenters efficiently and accurately.
method Input convex neural networks and cycle-consistency regularization.
result Our approach avoids introducing bias and does not require minimax optimization.
In this paper, we consider low rank matrix estimation using either matrix-version Dantzig Selector A^λd or matrix-version LASSO estimator A^λL. We consider sub-Gaussian measurements, i.e., the measurements X1,…,Xn∈Rm×m have i.i.d. sub-Gaussian entries. Suppose $\textrm…
New protocol for online learning with partial feedback, extending classical methods.
problem Learning with partial feedback where only one acceptable label is observed per round.
method Introducing a collection version space to address the lack of direct extension of classical methods.
result Characterization of learnability in set-realizable regime using Partial-Feedback Littlestone dimension and Partial-Feedback Measure Shattering dimension.
It is generally believed that ensemble approaches, which combine multiple algorithms or models, can outperform any single algorithm at machine learning tasks, such as prediction. In this paper, we propose Bayesian convex and linear aggregation approaches motivated by regression applications. We show that the proposed a…
New methods estimate transport-growth pairs in unbalanced optimal transport.
problem Statistical guarantees for Monge-type estimation in unbalanced optimal transport remain limited.
method Developed two estimators for transport-growth pairs under different setups.
result Achieved minimax optimal rate for estimation of transport-growth pairs.
New algorithm ranks players from partial comparisons with optimal rate.
problem Ranking players from partial pairwise comparisons.
method Divide-and-conquer approach, local MLE within groups.
result Optimal ranking algorithm with minimax rate.
New algorithms ensure reproducibility and optimal convergence in convex optimization.
problem Trade-off between reproducibility and convergence rate in convex optimization.
method Regularization-based algorithms for smooth convex minimization and minimax optimization.
result Achieves optimal reproducibility and near-optimal gradient complexity for various oracle settings.
Measuring the similarity of two files is an important task in malware analysis, with fuzzy hash functions being a popular approach. Traditional fuzzy hash functions are data agnostic: they do not learn from a particular dataset how to determine similarity; their behavior is fixed across all datasets. In this paper, we …
The paper develops adaptive confidence intervals for Efron's Gaussian two-groups model with unknown contamination.
problem Developing robust uncertainty quantification for Efron's Gaussian two-groups model with unknown contamination fraction.
method The approach involves Fourier-based certification procedures to find minimax-optimal adaptive confidence intervals.
result The minimax-optimal length of adaptive confidence intervals is polynomially worse than when contamination fraction is known.