# High-Dimensional Function Approximation: Breaking the Curse with Monte Carlo Methods

@article{Kunsch2017HighDimensionalFA, title={High-Dimensional Function Approximation: Breaking the Curse with Monte Carlo Methods}, author={Robert J. Kunsch}, journal={arXiv: Numerical Analysis}, year={2017} }

In this dissertation we study the tractability of the information-based complexity $n(\varepsilon,d)$ for $d$-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $d$. For integration problems one can usually break the curse with the standard Monte Carlo method. For function approximation problems…

## 5 Citations

The difficulty of Monte Carlo approximation of multivariate monotone functions

- Mathematics, Computer ScienceJ. Approx. Theory
- 2019

Lower complexity bounds reveal a joint $(\varepsilon,d)-dependency and from which it is deduced that the L_1-approximation of d-variate monotone functions is not weakly tractable.

Breaking the curse for uniform approximation in Hilbert spaces via Monte Carlo methods

- Mathematics, Computer ScienceJ. Complex.
- 2018

It is shown that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness $r > 1/2$, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability.

A Framework for Solving Non-Linear DSGE Models

- MathematicsSSRN Electronic Journal
- 2019

We propose a framework to solve non-linear DSGE models combining approximation and estimation techniques. Instead of relying on a fixed grid, we use Monte Carlo methods to draw samples from the state…

Monte Carlo methods for uniform approximation on periodic Sobolev spaces with mixed smoothness

- Mathematics, Computer ScienceJ. Complex.
- 2018

Abstract We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness with integrability 1 p ∞…

Minimax optimal sequential hypothesis tests for Markov processes

- Mathematics
- 2018

Under mild Markov assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. First, the design of…

## References

SHOWING 1-10 OF 79 REFERENCES

Bernstein Numbers and Lower Bounds for the Monte Carlo Error

- Computer Science, MathematicsMCQMC
- 2014

It is found that for the \(L_{\infty}\) approximation of smooth functions from the class \(C^{\infty }([0,1]^d)\) with uniformly bounded partial derivatives, randomized algorithms suffer from the curse of dimensionality, as it is known for deterministic algorithms.

Tractability of Multivariate Problems

- Mathematics
- 2008

Multivariate problems occur in many applications. These problems are defined on spaces of d-variate functions and d can be huge – in the hundreds or even in the thousands. Some high-dimensional…

On the absolute constants in the Berry-Esseen-type inequalities

- Mathematics
- 2011

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\Delta_n\leq0.3328(\beta_3+0.429)/\sqrt{n}$ and $\Delta_n\leq0.33554(\beta_3+0.415)/\sqrt{n}$…

The information-based complexity of approximation problem by adaptive Monte Carlo methods

- Mathematics
- 2008

AbstractIn this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MWp,αr($$
\mathbb{T}^d
$$), 1 < p < ∞, in the…

Product rules are optimal for numerical integration in classical smoothness spaces

- Computer Science, MathematicsJ. Complex.
- 2017

This work proves explicit error bounds without hidden constants and shows that the optimal order of the error is min { 1, d n − r / d } , where now the hidden constant only depends on r, not on d.

Quasi-polynomial tractability

- Computer Science, MathematicsJ. Complex.
- 2011

The main purpose of this paper is to promote quasi-polynomial tractability, especially for the study of unweighted multivariate problems and for algorithms using arbitrary linear functionals or only function values.

Learning Monotone Decision Trees in Polynomial Time

- Mathematics, Computer ScienceSIAM J. Comput.
- 2007

This is the first algorithm that can learn arbitrary monotone Boolean functions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of a decision tree size of f.

The randomized complexity of indefinite integration

- Computer Science, MathematicsJ. Complex.
- 2011

Two algorithms are presented, one being of optimal order, the other up to logarithmic factors, that can be approximated by a randomized algorithm uniformly over [email protected]?[0,1]^d with the same rate n^-^1+^1^/^m^i^n^(^p^,^2^) as the optimal rate for a single integral.

Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions

- Computer Science, MathematicsJ. Complex.
- 2014

It is shown that renorming the space of infinitely differentiable functions in a suitable way allows weakly tractable uniform approximation by using only function values.

Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

- Mathematics
- 2015

Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for…