Entropy-regularized NPG converges linearly with linear function approximation.
problem Analyzing convergence of entropy-regularized NPG with function approximation.
method Established finite-time convergence analyses with entropy regularization and linear function approximation.
result Entropy-regularized NPG achieves linear convergence up to a function approximation error.
AR-DAE approximates entropy gradient for machine learning models.
problem Intractable computation of entropy gradient for continuous distributions.
method Amortized residual denoising autoencoder (AR-DAE) to approximate entropy gradient.
result AR-DAE provides an unbiased gradient approximation for entropy.
Theoretical analysis of entropy approximation for Gaussian mixtures.
problem Lack of theoretical guarantees for entropy approximation of Gaussian mixtures.
method Theoretical analysis of the error between true and approximate entropy.
result The error converges to zero as the ratios of means to variances tend to infinity, providing a guarantee for high-dimensional problems.
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary w…
New entropy-based objective for sparse coding improves learning.
problem Sparse coding with probabilistic priors and non-Gaussian observables.
method Derive a solely entropy-based learning objective for sparse coding parameters.
result Fully analytical ELBO objective for sparse coding with non-trivial posterior approximations.
The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entro…
Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
Efficient approximations reduce computation of matrix-based Renyi's entropy.
problem High computational complexity of matrix-based Renyi's entropy.
method Taylor, Chebyshev, and Lanczos approximations to reduce complexity.
result Reduced complexity to significantly less than O(n2) with negligible accuracy loss. The paper examines how randomness in forex returns increases during financial crises.
problem Measuring randomness in forex returns during financial crises.
method Approximate Entropy and Sample Entropy statistics.
result Randomness in forex returns decreases during financial crises, as evidenced by lower entropy values.
This paper optimizes matrix-based Renyi's entropy computation for large datasets.
problem Efficiently calculating matrix-based Renyi's entropy for large-scale applications.
method Develops randomized approximations for matrix-based Renyi's entropy with arbitrary α orders.
result Achieves a significant reduction in time complexity from O(n^3) to O(n^2sm), where s, m << n.
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is…
The paper analyzes the convergence rates of Q-learning with entropy regularization and linear function approximation.
problem Analyzing the convergence rates of Q-learning with entropy regularization and linear function approximation.
method The paper derives rates of convergence using the high-dimensional central limit theorem, linearization of the soft Bellman recursion, and Gaussian approximation for the leading martingale term.
result The algorithm's last iterate satisfies high-order moment bounds, with a Gaussian approximation bound of order n−1/4. This paper studies an entropy-based multi-objective Bayesian optimization (MBO). The entropy search is successful approach to Bayesian optimization. However, for MBO, existing entropy-based methods ignore trade-off among objectives or introduce unreliable approximations. We propose a novel entropy-based MBO called Pare…
This paper uses normalizing flows to approximate transport maps between densities.
problem Approximating transport maps between given densities.
method Construct time-dependent controls using normalizing flows.
result Provides bounds on the number of switches for piecewise constant approximations.
In this paper, we present a probability one convergence proof, under suitable conditions, of a certain class of actor-critic algorithms for finding approximate solutions to entropy-regularized MDPs using the machinery of stochastic approximation. To obtain this overall result, we prove the convergence of policy evaluat…
Algorithm learns Nash equilibria in stochastic games using entropy-regularized policies.
problem Learning Nash equilibria in zero-sum stochastic games is computationally expensive.
method Entropy-regularized soft policies for Q-function updates.
result Algorithm converges to Nash equilibrium under certain conditions.
Tree-AMP simplifies inference in complex tree-structured models.
problem Inference in high-dimensional tree-structured models.
method Approximate Message Passing algorithms for various machine learning tasks.
result Theoretical performance predictions and automated entropy estimation.
We approximate differential entropy for efficient Bayesian experimental design.
problem Efficiently estimating expected information gain in large-scale inference problems.
method Approximate differential entropy using Monte Carlo or quasi-Monte Carlo surrogates.
result Our approach achieves comparable or better convergence rates than state-of-the-art methods.
Researchers use Gaussian processes to approximate Lagrange multipliers for Maximum-Entropy distributions.
problem Finding Lagrange multipliers for Maximum-Entropy distributions is computationally challenging.
method Employed Gaussian processes to approximate the Lagrange multipliers as a map of moments. Optimized hyperparameters by maximizing log-likelihood.
result Data-driven Maximum-Entropy closure performs well in approximating non-equilibrium distributions.
The paper analyzes how factorized Gaussian approximations underestimate uncertainty in variational inference.
problem Underestimation of uncertainty in variational inference using factorized Gaussian approximations.
method Examined the trade-off between shrinkage and delinking in approximating a Gaussian with a diagonal covariance matrix.
result Entropy of the factorized Gaussian approximation underestimates both componentwise variance and entropy of the original Gaussian.
This paper improves Bayesian inference for predictive models with limited data.
problem Effective uncertainty quantification for training predictive models with limited data.
method Entropy-regularized gradient estimators to approximate the Bayesian posterior.
result The method generates diverse samples from the posterior distribution efficiently.
We study approximations of non-Gaussian stationary processes having long range correlations with microcanonical models. These models are conditioned by the empirical value of an energy vector, evaluated on a single realization. Asymptotic properties of maximum entropy microcanonical and macrocanonical processes and the…
Efficient approximation lies at the heart of large-scale machine learning problems. In this paper, we propose a novel, robust maximum entropy algorithm, which is capable of dealing with hundreds of moments and allows for computationally efficient approximations. We showcase the usefulness of the proposed method, its eq…
Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using …
In this paper we study the consistency of an empirical minimum error entropy (MEE) algorithm in a regression setting. We introduce two types of consistency. The error entropy consistency, which requires the error entropy of the learned function to approximate the minimum error entropy, is shown to be always true if the…
The paper analyzes ESS metrics and their connections to entropy families.
problem Improving the accuracy of effective sample size approximations.
method Examined Rényi and Tsallis entropy families and their relationship to ESS.
result Proved that ESS functions in the Huggins-Roy family meet theoretical conditions.
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
New algorithms reduce complexity for learning in MDPs with entropy regularization.
problem Efficient learning for MDPs with large or continuous state and action spaces.
method Multilevel Monte Carlo (MLMC) algorithms integrating fixed-point iteration and stochastic approximation of the Bellman operator.
result MLMC with unbiased approximation of the Bellman operator achieves polynomial sample complexity.
Study shows depth improves generalization in deep learning models.
problem Understanding why and when depth improves generalization in deep learning.
method Implementation-agnostic state-transition model to analyze depth and generalization.
result Identifies geometric and semigroup mechanisms that keep entropy contribution saturated or polynomial, clarifying depth's statistical advantage.
The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…
MaxEnt Model Correction improves reinforcement learning model accuracy.
problem Improving reinforcement learning model accuracy and convergence.
method MaxEnt Model Correction (MoCo) procedure to correct model's next-state distributions.
result MoCoVI and MoCoDyna algorithms converge faster and effectively use approximate models.
Parameter estimation in Markov random fields (MRFs) is a difficult task, in which inference over the network is run in the inner loop of a gradient descent procedure. Replacing exact inference with approximate methods such as loopy belief propagation (LBP) can suffer from poor convergence. In this paper, we provide a d…
Adaptive approximations improve variational inference for complex models.
problem Efficiently approximate marginal distributions and partition functions in complex probabilistic models.
method Two classes of adaptive approximations that include Bethe, tree-reweighted, and convex free energies.
result Proposed approximations automatically adapt to a given model and outperform existing methods.
Develops weak PINNs for efficient manifold solutions of hyperbolic equations.
problem Challenges in approximating weak solutions of nonlinear hyperbolic equations on manifolds.
method Introduces a novel weak PINN (wPINN) formulation on manifolds leveraging well-posedness theory.
result Demonstrates efficient approximation of entropy solutions on manifolds with a complexity independent of ambient space dimension.
JES optimizes expensive functions by considering joint entropy over input and output spaces.
problem Optimizing expensive functions with limited evaluations.
method Joint Entropy Search (JES) considers joint entropy over input and output spaces.
result JES outperforms other information-theoretic methods in Bayesian optimization.
We introduce an algorithm to locate contours of functions that are expensive to evaluate. The problem of locating contours arises in many applications, including classification, constrained optimization, and performance analysis of mechanical and dynamical systems (reliability, probability of failure, stability, etc.).…
To study the singularities that appear in mean curvature flow, one must understand self-shrinkers, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In 1989, Angenent constructed the first nontrivial example of a self-shrinker, a torus. A key…
In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the orthogonality condition of the atoms, yielding overcomplete dictionaries with an exten…
Two log-linear approximations speed up optimal transport for deep learning applications.
problem Computing optimal transport in high dimensions is computationally expensive.
method Locality-sensitive hashing (LSH) and Nyström approximation with LSH-based sparse corrections.
result Log-linear time algorithms for entropy-regularized OT perform well in high-dimensional spaces.
We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [Kristoufek & Vosvrda (2013), Physica A 392]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock marke…
Sharp lower bounds on shallow neural networks' approximation rates are derived.
problem The efficiency of shallow neural networks in approximating functions.
method Lower bounding the L2-metric entropy and Kolmogorov n-widths of the convex hull of neural network basis functions. result Sharp lower bounds on the approximation rates for shallow neural networks are provided.
McDiarmid's inequality under dependence via approximate tensorization of entropy
problem Dependent versions of McDiarmid's inequality
method Approximate tensorization of entropy (ATE)
result Derives McDiarmid's inequality for non-isotropic Gaussian random vectors
TES optimizes black-box functions efficiently with minimal approximations.
problem Efficient Bayesian optimization with minimal approximations and generalization to batch BO.
method TES acquisition function measures information gain on trusted maximizers.
result TES achieves state-of-the-art performance with minimal approximations.
Profile entropy measures learnability and compressibility of discrete distributions.
problem Understanding the learnability and compressibility of discrete distributions.
method Investigates profile entropy, showing its role in estimation, inference, and compression.
result Profile entropy is a fundamental measure unifying estimation, inference, and compression.
Paper characterizes embeddability of function spaces into Lp-type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into Lp-type RKBS. method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into Lp-type RKBS. LogDet estimator improves entropy estimation in neural networks.
problem Inconsistent observations and diversified interpretation in neural networks.
method Proposes LogDet estimator for reliable entropy approximation.
result LogDet estimator overcomes distributional diversity issues.
A new method for multi-objective Bayesian optimization using entropy search and variational lower bound maximization.
problem Efficiently optimizing multiple objectives in continuous domains.
method Approximates the Pareto-frontier using a mixture distribution and optimizes the balance through variational lower bound maximization.
result Demonstrated effectiveness especially with many objective functions.
We study the problem of identifying the causal relationship between two discrete random variables from observational data. We recently proposed a novel framework called entropic causality that works in a very general functional model but makes the assumption that the unobserved exogenous variable has small entropy in t…