Related papers: High-Dimensional Function Approximation: Breaking …
In this work we detail the application of a fast convolution algorithm computing high dimensional integrals to the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of…
As the size of engineered systems grows, problems in reliability theory can become computationally challenging, often due to the combinatorial growth in the cut sets. In this paper we demonstrate how Multilevel Monte Carlo (MLMC) - a…
I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time,…
Recent advances in quasi-Monte Carlo integration demonstrate that the median of linearly scrambled digital net estimators achieves near-optimal convergence rates for high-dimensional integrals without requiring a priori knowledge of the…
We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded…
When a Monte Carlo algorithm is used to evaluate a physical observable A, it is possible to slightly modify the algorithm so that it evaluates simultaneously A and the derivatives $\partial$ $\varsigma$ A of A with respect to each…
Constraints can be interpreted in a broad sense as any kind of explicit restriction over the parameters. While some constraints are defined directly on the parameter space, when they are instead defined by known behaviour on the model,…
We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the…
Monte Carlo and Quasi-Monte Carlo methods present a convenient approach for approximating the expected value of a random variable. Algorithms exist to adaptively sample the random variable until a user defined absolute error tolerance is…
We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by…
Computing the Jacobian of the solution of an optimization problem is a central problem in machine learning, with applications in hyperparameter optimization, meta-learning, optimization as a layer, and dataset distillation, to name a few.…
In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles.…
Approximating functions of a large number of variables poses particular challenges often subsumed under the term ``Curse of Dimensionality'' (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be…
In data-driven optimization, sample average approximation (SAA) is known to suffer from the so-called optimizer's curse that causes an over-optimistic evaluation of the solution performance. We argue that a special type of distributionallly…
We study the optimal sample complexity in large-scale Reinforcement Learning (RL) problems with policy space generalization, i.e. the agent has a prior knowledge that the optimal policy lies in a known policy space. Existing results show…
We investigate the properties of a sequential Monte Carlo method where the particle weight that appears in the algorithm is estimated by a positive, unbiased estimator. We present broadly-applicable convergence results, including a central…
It has been found empirically that quasi-Monte Carlo methods are often efficient for very high-dimensional problems, that is, with dimension in the hundreds or even thousands. The common explanation for this surprising fact is that those…
The celebrated Monte Carlo method estimates an expensive-to-compute quantity by random sampling. Bandit-based Monte Carlo optimization is a general technique for computing the minimum of many such expensive-to-compute quantities by adaptive…
Model counting of Disjunctive Normal Form (DNF) formulas is a critical problem in applications such as probabilistic inference and network reliability. For example, it is often used for query evaluation in probabilistic databases. Due to…
We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $\l$, where the weight of an independent set $I$ is $\l^{|I|}$. A recent result has established that Gibbs…