EVI improves variational inference for non-Gaussian models.
problem Infeasibility of finding analytically tractable solutions for non-Gaussian statistical models.
method Extended Variational Inference (EVI) using lower-bound approximation to the variational objective function.
result Convergence of EVI depends on lower-bound approximation strategy.
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.
In this paper we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there i…
Mean-field variational inference is a method for approximate Bayesian posterior inference. It approximates a full posterior distribution with a factorized set of distributions by maximizing a lower bound on the marginal likelihood. This requires the ability to integrate a sum of terms in the log joint likelihood using …
General lower bounds on neural network approximation in L^p norm.
problem Fundamental limits of neural network expressivity.
method General lower bound proof on approximation in L^p norm, applied to feed-forward neural networks.
result Neural networks can't approximate certain functions as well as previously thought.
New algorithm optimally evaluates policies with linear approximations.
problem Policy evaluation with linear function approximation.
method Accelerated, variance-reduced fast temporal difference algorithm (VRFTD).
result VRFTD matches both deterministic and stochastic lower bounds.
The paper sets lower bounds for sampling non-log-concave distributions using Fisher information.
problem Understanding the complexity of sampling non-log-concave distributions.
method Proves two lower bounds using Fisher information in the context of sampling.
result Lower bounds on the complexity of sampling non-log-concave distributions, ruling out high-accuracy algorithms.
New algorithm finds approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex stochastic optimization.
method Design of an algorithm using O(ε−3) stochastic gradient and Hessian-vector products. result Optimal rate of O(ε−3) for finding ε-approximate stationary points, matching lower bounds. The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is…
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.
Study on size and depth of neural networks for approximating benign functions, showing barriers and explicit results.
problem Understanding how size and depth of neural networks affect their ability to approximate benign functions.
method Analyzing ReLU networks for benign functions, proving barriers and explicit results.
result Explicit benign functions that cannot be approximated by networks of certain sizes or depths, showing barriers to size and depth separation.
Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise naturally when working with factor graphs. Belief propagation is a widely deplo…
Randomly initialized ReLU networks of depth two can approximate smooth functions well.
problem Approximation power of two-layer networks of random ReLUs.
method Harmonic analysis and ridgelet representation theory for upper bounds, dimensionality arguments for lower bounds.
result Near-matching upper and lower bounds for L2-approximation and Sobolev norms. A new method simplifies variational inference for complex models.
problem Challenges in exact Bayesian inference due to intractable integrals.
method Rewriting the lower bound on model log-likelihood using a finite sample of Gaussian latent variables.
result Demonstrated effectiveness on synthetic and real-world examples.
Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.
problem Sparse tensor best rank-1 approximation.
method Four approximation algorithms exploiting multilinearity and sparsity.
result Theoretical worst-case approximation lower bounds for all algorithms.
This paper establishes lower bounds for smooth nonconvex finite-sum optimization.
problem Understanding the complexity of finding optimal solutions in nonconvex finite-sum optimization.
method Proving tight lower bounds for the complexity of finding ε-suboptimal points and ε-approximate stationary points.
result Existing algorithms achieve optimal IFO complexity up to logarithmic factors.
Paper finds how many neurons are needed to approximate histogram distributions.
problem How many neurons are needed to approximate a target probability distribution?
method Examined for uniform input distribution and histogram target distributions, using efficient neural net construction.
result Obtained a new upper bound on the number of required neurons, strictly better than previous bounds.
The study finds a positive lower bound for the extra field in Yang-Mills equations on closed 4-manifolds.
problem Analyzing the boundedness of solutions to complex Yang-Mills equations on 4-manifolds.
method Investigates the analytical properties of solutions under specific conditions on a closed Riemannian four-manifold.
result Establishes a positive lower bound for the extra field when the metric is 'good' and the connection is an approximate ASD connection.
The paper bounds neural networks' approximation error and applies it to regression and GANs.
problem Bounding the approximation error of norm-constrained neural networks.
method Proved upper and lower bounds on approximation error using Rademacher complexity.
result Obtained convergence rates for over-parameterized neural networks and optimal GAN learning rates.
Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The original approach to approximate inference in these models used variational compressio…
We introduce a new class of lower bounds on the log partition function of a Markov random field which makes use of a reversed Jensen's inequality. In particular, our method approximates the intractable distribution using a linear combination of spanning trees with negative weights. This technique is a lower-bound count…
The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.
problem Estimating intersection numbers of psi-classes on Deligne-Mumford's moduli spaces.
method Approximates intersection numbers by closed-form expressions and proves a uniform lower bound.
result Proves a lower bound for intersection numbers in terms of approximating expressions and an explicit factor.
Study finds the minimum number of finite Gaussian mixtures for best approximation.
problem Finding the minimum number of finite Gaussian mixtures for best approximation.
method Local moment matching for upper bound and spectral analysis for lower bound.
result Corrects a previous lower bound in the case of Gaussian mixing distributions.
Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.
problem Expressing high-dimensional Lipschitz functions with shallow ReLU networks.
method Established lower bounds on shallow network complexity for polynomial approximation.
result Shallow ReLU networks suffer from the curse of dimensionality for Lipschitz functions.
TADDAA improves accuracy diagnostics for variational approximations.
problem Challenges in evaluating the accuracy of variational approximations.
method Uses many short parallel MCMC chains to obtain lower bounds on the error of each posterior functional of interest.
result Validates the practical utility and computational efficiency of TADDAA on various models.
Study shows the number of attention heads affects transformer performance.
problem Understanding how the number of attention heads impacts transformer performance.
method Introduced a generalized D-retrieval task, established upper and lower bounds on parameter complexity, and validated with experiments. result Transformers with many heads can efficiently approximate functions, while few heads require a large number of parameters.
New lower bounds improve logistic log-likelihood optimization and inference.
problem Designing computationally tractable lower bounds for logistic log-likelihoods.
method Developed a piece-wise quadratic lower bound that uniformly improves tangent quadratic minorizers.
result Improves the speed of convergence and accuracy of variational Bayes approximations.
This tutorial derives the VAE loss function under Gaussian assumptions.
problem Computational intractability of posterior distributions in Bayesian machine learning.
method Derives the variational lower bound loss function of a standard VAE.
result The Kullback-Leibler divergence has a closed form solution under Gaussian assumptions.
Computing the partition function Z of a discrete graphical model is a fundamental inference challenge. Since this is computationally intractable, variational approximations are often used in practice. Recently, so-called gauge transformations were used to improve variational lower bounds on Z. In this paper, we pro…
We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.
Improves GP models with known bounds for sampling and optimization.
problem Functions with known upper and lower bounds.
method Transforms GP models with bounds for posterior sampling and BO.
result Bounded entropy search (BES) selects points satisfying constraints.
Sharp bounds for approximating Sobolev functions by ridge functions and networks.
problem Approximating Sobolev functions with multivariate ridge functions and networks.
method Proving sharp upper and lower bounds for approximation order.
result Order of approximation asymptotically behaves as n−r/(d−ℓ). Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.
problem Approximating functions with depth 2 networks in high dimensions.
method Lower bound proof using worst-to-average-case random self-reducibility.
result Proves depth 2 networks can't approximate certain functions as well as depth 3 networks, resolving an open problem.
We provide bounds for kernel matrices and new approximations for high-dimensional data.
problem Approximating high-dimensional empirical kernel matrices.
method Decoupling results for U-statistics and non-commutative Khintchine inequality.
result New tighter approximations for inner-product kernel matrices.
WiSE-ALE improves VAEs by learning a flexible aggregate posterior.
problem Learning compact latent representations from large datasets.
method Derives a new variational lower bound and uses it to place a prior on the entire dataset.
result WiSE-ALE achieves excellent reconstruction quality with a smooth, compact representation.
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…
Variational inference is a powerful tool for approximate inference. However, it mainly focuses on the evidence lower bound as variational objective and the development of other measures for variational inference is a promising area of research. This paper proposes a robust modification of evidence and a lower bound for…
Current approaches in approximate inference for Bayesian neural networks minimise the Kullback-Leibler divergence to approximate the true posterior over the weights. However, this approximation is without knowledge of the final application, and therefore cannot guarantee optimal predictions for a given task. To make mo…
Lower bounds found for nonconvex-strongly-concave min-max optimization problems.
problem Finding stationary points in nonconvex-strongly-concave min-max optimization.
method Provided lower bounds for first-order oracle complexity.
result Lower bounds of Ω(√κε⁻²) for deterministic oracles and Ω(√κε⁻² + κ¹/₃ε⁻⁴) for stochastic oracles.
Paper shows existence of solutions for inverse mean curvature flow on manifolds with Ricci lower bounds.
problem Existence of solutions for inverse mean curvature flow on manifolds with Ricci lower bounds.
method Approximation via p-Laplace equation and new gradient and decay estimates for p-harmonic capacity potentials. result Sharp estimates for the growth of solutions and mean curvature of level sets, well-behaved under Gromov-Hausdorff convergence.
The Fisher information approximation (FIA) is an implementation of the minimum description length principle for model selection. Unlike information criteria such as AIC or BIC, it has the advantage of taking the functional form of a model into account. Unfortunately, FIA can be misleading in finite samples, resulting i…
New RL algorithm tackles nonstationary MDPs with linear approximations and varying rewards.
problem Nonstationary reinforcement learning with evolving reward and state transition functions.
method Developed a new algorithm LSVI-UCB-Restart with periodic restart, and parameter-free Ada-LSVI-UCB-Restart for unknown variation budgets.
result First minimax dynamic regret lower bound for nonstationary linear MDPs and linear MDPs lower bound.
Study shows limits on deep and shallow neural networks for approximating compact sets.
problem Understanding the limitations of deep and shallow neural networks in approximating compact sets.
method Proved Carl's type inequalities for approximation error, using Lipschitz widths.
result Lower bounds on approximation error for neural network outputs.
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.
Sharp lower bound on GHHs' representation power of CPWL functions.
problem Proving the minimum number of nestings for GHHs to represent arbitrary CPWL functions.
method Using a key lemma about finite sums of periodic functions, proving necessity of n nestings.
result Proving necessity of n nestings for GHHs to achieve universal representation power.
This work improves bounds on Bayesian coreset quality.
problem Limitations of existing theoretical analysis of Bayesian coresets.
method Develops general upper and lower bounds on KL divergence.
result Demonstrates flexibility of new theoretical bounds in various models.
In this paper, we consider the problem of estimating the underlying graph associated with an Ising model given a number of independent and identically distributed samples. We adopt an \emph{approximate recovery} criterion that allows for a number of missed edges or incorrectly-included edges, in contrast with the widel…
The paper explores the tradeoff between fairness and accuracy in regression models.
problem Characterizing the tradeoff between fairness and accuracy in regression models.
method Provided a lower bound on the error of any fair regressor and extended the result to joint error using Wasserstein distance.
result Lower bounds on the error of fair regressors and their connection to Wasserstein distance.