Deep networks can learn functions approximated by shallow networks, but not all functions.
problem The learnability of functions by deep neural networks and the approximation capacity of simpler classes.
method Study the connection between learnability and approximation capacity of functions by deep neural networks and simpler classes.
result A necessary condition for a function to be learnable by deep neural networks is to be approximable by shallow networks.
New methods stabilize Q-learning with linear approximations.
problem Stabilizing Q Q Q -learning with linear function approximation. method Target network and truncation.
result Provably stable Q Q Q -learning with linear function approximation. New method optimizes neural network learning by adjusting random parameters to target function features.
problem Difficulty in setting optimal random parameters for neural network learning.
method Adjusts sigmoid slopes and positions to target function features in a randomized learning method.
result Significantly better approximation of complex target functions compared to standard methods.
Proposes a new method for estimating non-pathwise differentiable functional parameters.
problem Estimating dose-response curves for continuous exposure.
method Targeted Highly Adaptive Lasso (HAL) for non-pathwise differentiable functional parameters.
result The Targeted HAL-MLE achieves dimension-free rates up to log(n) factors and outperforms other methods in simulations.
Bayesian optimization with binary auxiliary info for faster target function optimization.
problem Optimizing target functions with expensive binary auxiliary information.
method Mixed-type Gaussian process (MOGP) and information-based acquisition functions (MT-ES, MT-PES).
result Efficient approximation of mixed-type predictive ES via random features.
In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is li…
Generative adversarial networks (GAN) approximate a target data distribution by jointly optimizing an objective function through a "two-player game" between a generator and a discriminator. Despite their empirical success, however, two very basic questions on how well they can approximate the target distribution remain…
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
A new method for target propagation using iterative approximations converges fast and is more biologically plausible.
problem Improving target propagation methods for neural networks.
method Iterative approximate inverses and local auto-encoders.
result The method converges exponentially fast under certain conditions.
AMCI improves Monte Carlo integration by amortizing over both datasets and target functions.
problem Inefficiency in approximating expectations for known target functions using current approaches.
method Introduces AMCI, a method for amortizing Monte Carlo integration directly, producing three distinct amortized proposals.
result AMCI can theoretically produce arbitrarily small errors for any integrable target function using only a single sample from each proposal at runtime.
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
The mixture of experts (MoE) model is a popular neural network architecture for nonlinear regression and classification. The class of MoE mean functions is known to be uniformly convergent to any unknown target function, assuming that the target function is from Sobolev space that is sufficiently differentiable and tha…
Sparse neural networks can match dense models on Lipschitz functions.
problem Sparse networks are more efficient but lack theoretical guarantees.
method Formal model of sparse networks, LSH-based routing function, Lipschitz function approximation.
result Sparse networks can approximate dense networks on Lipschitz functions.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
The paper analyzes an actor-critic algorithm with target networks for deep reinforcement learning.
problem Lack of theoretical understanding of target networks in actor-critic methods.
method Proposes a theoretical analysis of an online target-based actor-critic algorithm with linear function approximation.
result Establishes asymptotic convergence results and finite-time analysis for both critic and actor.
Enhances RL with function approximation, improving regret bounds.
problem Improving exploration in reinforcement learning with function approximation.
method Prior-dependent Bayesian regret bound for PSRL with linear mixture MDPs, using value-targeted model learning and variance reduction.
result Established an upper bound of O ( d H 3 T log T ) {\mathcal{O}}(d\sqrt{H^3 T \log T}) O ( d H 3 T log T ) for PSRL. 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.
Bayesian optimization technique scaled using Vecchia approximations.
problem Scalability issue with Gaussian process surrogate models in Bayesian optimization.
method Adapted Vecchia approximation from spatial statistics to Gaussian processes, developed improvements and extensions.
result Methods compared favorably to state-of-the-art on various test functions and reinforcement learning problems.
Directly approximates functions on unknown data manifolds without complex computations.
problem Function approximation on unknown data-defined manifolds with conservative results from traditional methods.
method Direct approach using graph Laplacian and local approximation techniques without eigen-decomposition or atlas.
result Universal estimates for smooth functions without prior knowledge of the target function.
Survey examines deep neural networks' ability to approximate functions.
problem Approximation of target functions by deep neural networks.
method Examination of feed-forward and residual architectures, focusing on optimization problems in regression and classification.
result Deep neural networks can approximate functions effectively, especially with ReLU activation functions.
We propose a new framework for Hamiltonian Monte Carlo (HMC) on truncated probability distributions with smooth underlying density functions. Traditional HMC requires computing the gradient of potential function associated with the target distribution, and therefore does not perform its full power on truncated distribu…
This paper analyzes how periodic and soft target updates stabilize linear Q-learning.
problem Theoretical explanation of stabilization mechanisms for linear Q-learning.
method Exact analysis using switched linear system dynamics and the joint spectral radius.
result Periodic and soft target updates can guarantee convergence to the exact projected Q-Bellman solution under specific conditions.
VA-OPE improves OPE by incorporating variance information, achieving tighter error bounds.
problem Estimating value function of a target policy from offline data collected by a behavior policy.
method Proposes VA-OPE, an algorithm that reweights Bellman residual using estimated variance of the value function.
result Achieves a tighter error bound than the best-known result.
Proposes a convergent TD algorithm for off-policy RL.
problem Learning value function from different policies in RL.
method Convergent on-policy TD algorithm with linear function approximation.
result Proposes a convergent TD algorithm for off-policy RL.
New method stabilizes FQE by reweighting Bellman targets.
problem Stability guarantees for FQE often rely on Bellman completeness, which can fail with function approximation.
method Proposes stationary-weighted FQE, reweighting Bellman targets by stationary target-to-behavior density ratio.
result Proves finite-sample linear convergence to stationary projected Bellman fixed point without Bellman completeness.
Neural networks cannot approximate certain functions in Sobolev spaces, leading to unbounded parameter growth.
problem Non-closedness of sets of neural networks in Sobolev spaces.
method Construction of sequences of neural networks whose realizations converge to functions not realizable by neural networks.
result Sets of realized neural networks are not closed in order- ( m − 1 ) (m-1) ( m − 1 ) Sobolev spaces W m − 1 , p W^{m-1,p} W m − 1 , p for p ∈ [ 1 , ∞ ] p \in [1,\infty] p ∈ [ 1 , ∞ ] . New framework for scalable approximate inference tackles high-dimensional models and large datasets.
problem Efficient approximate inference for high-dimensional probability models and large datasets.
method Combines Stein's method with adaptive importance sampling and gradient-based sampling.
result Proposed algorithms improve upon existing methods in terms of efficiency and applicability.
New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
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.
Study bond market making with hit-ratio target using optimal control and HJB equations.
problem Optimizing bond market making with hit-ratio target in OTC markets.
method Stochastic optimal control approach, dualizing hit-ratio target, HJB equation, Riccati equation, linearization.
result Explicit quote decompositions into riskless spread, inventory-risk correction, and hit-ratio correction.
EigenVI uses orthogonal function expansions for efficient variational inference.
problem Efficiently approximate complex distributions in variational inference.
method EigenVI constructs variational approximations using orthogonal function expansions, minimizing Fisher divergence.
result EigenVI provides more accurate approximations than existing methods for Gaussian BBVI.
Paper proves GDL models can approximate any continuous function on non-Euclidean data.
problem Processing non-Euclidean data with universal feedforward models.
method Introduces geometric deep learning framework for differentiable manifold geometries.
result GDL models can uniformly approximate any continuous function on compact sets.
Deep, wide ConvResNets can approximate functions and their smoothness.
problem Function approximation and smoothness in deep networks.
method Analyzing ConvResNets, proving their ability to approximate functions and their smoothness.
result Large ConvResNets can approximate functions and exhibit sufficient first-order smoothness.
We prove a negative result for the approximation of functions defined on compact subsets of R d \mathbb{R}^d R d (where d ≥ 2 d \geq 2 d ≥ 2 ) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to…
Paper proves bounds for shallow networks without dimensionality issues.
problem Avoiding curse of dimensionality and estimating approximation degree.
method Abstract theorem for G G G -networks on compact metric measure spaces. result Dimension independent bounds for approximation by shallow networks.
The paper analyzes the sample complexities for policy evaluation with linear function approximation.
problem Policy evaluation with linear function approximation in discounted infinite horizon Markov decision processes.
method Investigates sample complexities for two policy evaluation algorithms: TD and TDC.
result Establishes high-probability sample complexity bounds for policy evaluation algorithms.
ESNs trained with Tikhonov least squares approximate ergodic dynamical systems in L2(μ) norm.
problem Approximating ergodic dynamical systems using ESNs.
method Tikhonov least squares regression on ESNs trained on observations from an ergodic dynamical system.
result ESNs trained with Tikhonov least squares approximate the target function in the L2(μ) norm.
The key idea of Bayesian optimization is replacing an expensive target function with a cheap surrogate model. By selection of an acquisition function for Bayesian optimization, we trade off between exploration and exploitation. The acquisition function typically depends on the mean and the variance of the surrogate mod…
New framework estimates target functions from incomplete data.
problem Estimating target functions from partially observed data.
method IF-learning framework using influence functions.
result Two learning algorithms developed for estimation.
This paper optimizes off-policy evaluation in reinforcement learning with function approximation.
problem Estimating cumulative value of a new policy from logged data generated by an unknown policy.
method Regression-based fitted Q iteration method, equivalent to estimating conditional mean embedding of transition operator.
result The method is minimax-optimal, with nearly minimal estimation error.
Proves existence of optimal shallow neural networks with ReLU activation.
problem Proving the existence of optimal shallow feedforward networks with ReLU activation.
method Proves existence of global minima in the loss landscape for continuous target functions using shallow feedforward neural networks with ReLU activation.
result Existence of global minima in the loss landscape for shallow feedforward networks with ReLU activation.
KALE flow approximates KL divergence for distributions with disjoint support.
problem Approximating KL divergence for distributions with disjoint support.
method Relaxed KL gradient flow using RKHS, continuously interpolating between KL and MMD.
result Global convergence of KALE flow under sufficient smoothness assumptions.
New algorithm improves knowledge transfer in dynamic decision-making.
problem Utilizing data from existing ventures to improve decision-making in new ventures.
method Proposes Transferred Fitted Q Q Q -Iteration algorithm for estimating optimal action-state function Q ∗ Q^* Q ∗ . result Significantly improved final learning error of Q ∗ Q^* Q ∗ function. 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.
The paper explores transferring functions from one data space to another.
problem Approximating a function on a new data set using a learned function from an old data set.
method Transfer learning from one data space to another, focusing on subsets of the target data space.
result Local smoothness of the function and its lifting are related.
Paper proposes f-DPG for aligning language models with preferences.
problem Aligning language models with user preferences.
method Uses f-divergence to approximate target distributions and minimizes a forward KL from it using DPG.
result Jensen-Shannon divergence often outperforms forward KL divergence, leading to significant improvements.
Feature selection problems arise in a variety of applications, such as microarray analysis, clinical prediction, text categorization, image classification and face recognition, multi-label learning, and classification of internet traffic. Among the various classes of methods, forward feature selection methods based on …
The paper tackles singularities in diffusion models on submanifolds.
problem Analyzing singularities in diffusion models on lower-dimensional submanifolds.
method Small-time approximations of the Green's function and derivation of a new target function.
result The new target function remains bounded for singular data distributions.