Approximates bond option volatilities using affine short-rate models.
problem Calculating implied volatilities for bond options.
method Derive asymptotic approximation for bond option volatilities under affine short-rate dynamics.
result Accuracy of approximation validated through numerical experiments.
Paper studies Gaussian approximation in linear regression with rates derived.
problem Gaussian approximation in online linear regression.
method Derives rates for constant learning rate settings, analyzes dependence on d d d and design matrix. result Rate of normal approximation is log n / n \sqrt{\log{n}/n} log n / n for large n n n . Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
problem Analyzing convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
method Novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing length.
result First almost sure convergence rate and maximal concentration bound with exponential tails for contractive stochastic approximation algorithms with Markovian noise.
Study shows convergence rates for BSDEs approximated by compound Poisson processes.
problem Analyzing convergence rates of BSDEs driven by Lévy processes.
method Approximating Lévy processes by compound Poisson processes and studying BSDEs.
result Optimal convergence rates derived for BSDEs in L 2 \mathbb L^2 L 2 -norm and Wasserstein distance. Study shows rate of convergence for particle approximation of PDEs in Wasserstein space.
problem Analyzing convergence rates for particle approximations of PDEs in Wasserstein space.
method Backward stochastic differential equations techniques.
result Proved a rate of convergence of order 1/N for pathwise error and 1/sqrt(N) for L2-error on the derivative.
We analyze analytic approximation formulae for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical approximation due to Choi and Wir…
Paper studies shallow ReLU networks' approximation rates for Hölder functions.
problem Understanding shallow ReLU networks' efficiency in approximating Hölder functions.
method Analyzes rates of uniform approximation by ReLU shallow neural networks with m m m hidden neurons. result Shows ReLU shallow neural networks can uniformly approximate Hölder functions with rates close to optimal.
A new method automatically and dynamically sets learning rates in deep learning.
problem Determining the appropriate learning rate in deep learning tasks is challenging and often subjective.
method Local Quadratic Approximation (LQA) to automatically and dynamically set learning rates.
result The proposed method leads to nearly optimal learning rates in a computationally efficient way.
Paper shows deep neural networks can approximate Korobov functions nearly optimally.
problem Approximating Korobov functions with deep neural networks.
method Used deep neural networks and measured approximation rates with L p L_p L p and H 1 H^1 H 1 norms. result Achieved a super-convergence rate, outperforming traditional methods.
This paper optimizes ReLU networks for approximating Hölder continuous functions.
problem Optimizing the approximation rate of ReLU networks in terms of width and depth.
method Constructive proof of ReLU networks' approximation power with specific width and depth constraints.
result Optimal approximation rate of ReLU networks with width and depth constraints.
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.
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLU k ^k k neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
The paper analyzes convergence rates for stochastic approximation and reinforcement learning.
problem Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise.
method A novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the Moreau-envelope smoothing for contractive mappings.
result Almost sure convergence rates for specific learning rates are derived, with rates arbitrarily close to o ( n 1 − 2 η ) o(n^{1 - 2η}) o ( n 1 − 2 η ) and o ( n − 1 ) o(n^{-1}) o ( n − 1 ) . 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 L 2 L^2 L 2 -metric entropy and Kolmogorov n n 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.
Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.
problem Approximating functions from Sobolev spaces using shallow ReLU^k neural networks.
method Utilizing the Radon transform and discrepancy theory, we provide nearly optimal approximation rates.
result Optimal approximation rates for smoothness up to order s = k + (d+1)/2.
Wide neural networks can be closely approximated by Gaussian processes, with rates depending on the activation function's properties.
problem Approximating the behavior of wide neural networks using Gaussian processes.
method Established convergence rates for the central limit theorem in an infinite-dimensional functional space, using a transportation distance metric.
result Explicit convergence rates for neural networks approximated by Gaussian processes, varying based on the activation function's properties.
Deep ReLU networks can approximate and learn smooth functions efficiently.
problem Efficiently approximating and learning smooth functions using deep ReLU neural networks.
method Extending recent results to anisotropic and mixed smooth function classes, establishing approximation rates.
result Deep ReLU networks achieve minimax optimal rates up to logarithmic factors for various smooth function classes.
Study on CNNs' learning rates and approximation capacities.
problem Learning rates and approximation capacities of CNNs.
method New approximation bound and covering number analysis for CNNs.
result Derives minimax optimal convergence rates for CNNs in various learning problems.
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 n^{-1/4} n − 1/4 . 2D CNNs approximate Korobov functions with near-optimal rates.
problem Approximating Korobov functions using 2D CNNs.
method Constructive approach for 2D CNNs with ReLU activations and fully connected layers.
result 2D CNNs achieve near-optimal approximation rates for Korobov functions.
Paper derives convergence rates and confidence intervals for LSA with Markovian noise.
problem Analyzing convergence rates and constructing confidence intervals for LSA with Markovian noise.
method Derives non-asymptotic Berry-Esseen bounds and multiplier block bootstrap procedure.
result Provides O ( n − 1 / 4 ) \mathcal{O}(n^{-1/4}) O ( n − 1/4 ) convergence rates and guarantees consistent inference. Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.
problem Improving approximation of fully connected deep neural networks for optimal convergence rates.
method Deriving approximation bounds specifically for a narrower fully connected deep neural network.
result Achieves an optimal rate (up to a logarithmic factor) for fully connected deep neural networks.
Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
problem Approximating weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
method Used Euler type scheme with integrated kernels to study weak convergence rate.
result Obtained weak convergence rate of min ( 3 α − 1 , 1 ) \min(3α-1,1) min ( 3 α − 1 , 1 ) for discretised rough Ornstein-Uhlenbeck process and stochastic rough volatility model. Develops a fast variational approximation for high-dimensional empirical Bayes posteriors.
problem Optimal posterior computation in high-dimensional settings with prior tails effect.
method Variational approximation of empirical Bayes posterior with data-driven centers and thin-tailed conjugate priors.
result Retains optimal concentration rate properties and superior performance compared to existing methods.
In this paper, we propose a generic framework for devising an adaptive approximation scheme for value function approximation in reinforcement learning, which introduces multiscale approximation. The two basic ingredients are multiresolution analysis as well as tree approximation. Starting from simple refinable function…
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.
Study proves convergence of interest rate model approximations.
problem Investigating convergence of stochastic interest rate models.
method Developed analytical tools for true and truncated EM solutions, proving convergence in probability.
result True solution converges in probability to truncated EM solution as step size approaches zero.
Paper develops Gaussian approximations and bootstrap for federated LSA with trade-off bounds.
problem Analyzing convergence rates and trade-offs in federated linear stochastic approximation.
method Established Berry-Esseen-type bounds for federated LSA, developed multiplier bootstrap for inference.
result First federated Gaussian approximations with explicit trade-off terms and non-asymptotic validity guarantees.
Smooth activations enable optimal error rates in neural networks for Sobolev function classes.
problem Achieving optimal approximation and estimation error rates for neural networks in Sobolev function classes.
method Study of neural networks with smooth activations, proving optimal rates via approximation and statistical properties.
result Constant-depth networks with smooth activations achieve optimal rates of approximation and estimation, demonstrating smoothness adaptivity.
Paper provides an explicit formula for local volatility in Cheyette models.
problem Approximating local volatility in Cheyette interest rate models.
method Extended Dupire framework, perturbation methods, probabilistic techniques.
result Explicit analytical formula for local volatility in Cheyette models.
GANs learn distributions well from samples, with rates depending on intrinsic dimension.
problem Learning distributions from samples using GANs.
method Oracle inequality, Hölder functions approximation, neural network approximation, integral probability metrics.
result Convergence rates of GANs depend on intrinsic dimension, not ambient dimension.
Neural networks can approximate high-dimensional classifiers with ReLU networks under margin conditions.
problem Approximating high-dimensional discontinuous classifiers with neural networks.
method Using ReLU neural networks with three hidden layers, approximating a classifier with a Barron-regular decision boundary.
result High-dimensional discontinuous classifiers can be approximated with a rate of n − 1 n^{-1} n − 1 under strong margin conditions. Develops a method to approximate convexity adjustments for interest rate products.
problem Finding accurate convexity adjustments for interest rate products.
method Uses Malliavin calculus to develop an approximation method.
result Excellent numerical accuracy of the formulas for various interest rate products.
Deep ReLU networks can efficiently approximate Sobolev and Besov functions.
problem Approximating functions in Sobolev and Besov spaces using deep neural networks.
method Used deep ReLU neural networks with varied width and depth to approximate functions in Sobolev and Besov spaces.
result Generalized the approximation rate to hold under the Sobolev embedding condition.
We study the problem of approximate ranking from observations of pairwise interactions. The goal is to estimate the underlying ranks of n n n objects from data through interactions of comparison or collaboration. Under a general framework of approximate ranking models, we characterize the exact optimal statistical error …
Near-optimal rates for multi-task learning with shared representations.
problem Approximation and statistical complexity of learning multiple operators.
method Multiple Neural Operators (MNO) architecture and comparison with DeepONet.
result Near-optimal upper and lower bounds for approximation and generalization.
Given a function dictionary D \cal D D and an approximation budget N ∈ N + N\in\mathbb{N}^+ N ∈ N + , nonlinear approximation seeks the linear combination of the best N N N terms { T n } 1 ≤ n ≤ N ⊆ D \{T_n\}_{1\le n\le N}\subseteq{\cal D} { T n } 1 ≤ n ≤ N ⊆ D to approximate a given function f f f with the minimum approximation error\[\varepsilon_{L,f}:=\min_{\{g_n\}\subseteq{\ma…
In this survey paper we discuss recent advances on short interest rate models which can be formulated in terms of a stochastic differential equation for the instantaneous interest rate (also called short rate) or a system of such equations in case the short rate is assumed to depend also on other stochastic factors. Ou…
The paper clarifies the approximation of SGD with Ito SDEs for finite learning rates.
problem Theoretical justification and experimental verification of the Ito SDE approximation for finite learning rates in SGD.
method An efficient simulation algorithm SVAG and a necessary condition test for the SDE approximation.
result The Ito SDE approximation can meaningfully capture training and generalization properties of deep nets with finite learning rates.
Study on DiTs' rates of approximation and estimation under various data assumptions.
problem Investigating statistical rates of conditional diffusion transformers.
method Discretization and Taylor expansion of conditional diffusion score function under Hölder smooth data assumption.
result Establishes statistical limits for conditional and unconditional DiTs, offering practical guidance.
This paper improves neural network approximation for analytic functions with adjustable depth and width.
problem Approximating analytic functions using neural networks with depth and width parameters.
method Characterizes approximation rates as a joint function of width (N) and depth (L) for ReLU networks.
result Establishes upper bounds for analytic function approximation rates of O(N^(-CL^τ)) with τ influenced by N and L.
Three-hidden-layer neural networks can approximate Hölder continuous functions uniformly with exponential rate.
problem Approximating Hölder continuous functions with neural networks.
method Introduced Floor-Exponential-Step (FLES) networks with three hidden layers.
result Uniform approximation of Hölder continuous functions with an exponential rate.
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
Random feature approximation speeds up spectral methods and improves learning rates.
problem Improving the efficiency and generalization of spectral methods in large-scale algorithms.
method Combining random feature approximation with spectral regularization methods.
result Optimal learning rates for estimators over various regularity classes, including those not in the RKHS.
Collaborative filtering (CF) is a popular technique in today's recommender systems, and matrix approximation-based CF methods have achieved great success in both rating prediction and top-N recommendation tasks. However, real-world user-item rating matrices are typically sparse, incomplete and noisy, which introduce ch…
A new method approximates option pricing in stochastic interest rate markets.
problem Approximating option pricing in markets with stochastic interest rates.
method Gaussian moment matching technique applied to a conditional Black \& Scholes formula.
result The method performs remarkably well, even compared to other techniques.
The paper analyzes contraction rates for GP regression approximations.
problem Computational infeasibility of exact GP posterior in large-scale applications.
method Lanczos and conjugate gradient approximations of the posterior mean.
result Minimax contraction rates for these approximations in large-scale applications.