Questions tagged [stochastic-approximation]

This tag is for questions about stochastic approximation which are a family of methods of iterative stochastic optimization algorithms that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations.

72 questions
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SAA method (sample average approximation)

SAA method (sample average approximation) is just for continuous scenarios? I have a problem where scenarios are discretely defined. Can I use SAA?
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Proof for convergence of stochastic gradient descent to a local optimum for non convex functions

Let's say I have a (multivariable) function $F(x) : \mathbb{R}^n \rightarrow \mathbb{R}$, which I would like to minimize. There are no assumptions made on $F$, besides it being differentiable and ...
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Stochastic Optimization and Monte Carlo

Assume we are in a Brownian filtration where I denote $W$ the Brownian motion. My problem is to numerically compute $$\min_X E (\int^1_0 X^2_tdt),\ \ \ \ (*)$$ where $X$ is adapted to the filtration ...
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How to prove or disprove $\mathbb P[X\in[\mathbb E[X]-a\sigma(X),\mathbb E[X]+a\sigma(X)]]\leq\frac{1}{a^2}$? [closed]

$X$ is a random variable in $\mathcal{L^2}$, $a>0$ and $\sigma(X)$ is the standard deviation of $X$.
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Generalized polynomial chaos with periodic basis function?

In generalized polynomial chaos expansion, polynomial orthogonal to the distribution of uncertain (random) parameter is used. For example, for normally distributed random variable Hermite polynomial ...
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How to obtain the stationary distribution of a Markovian Jump Process from its diffusion approximation

If we have a Markovian Jump Process $x(t)$, we let $\varepsilon > 0$ be small, we scale jump sizes by $\varepsilon^{1/2}$ and the jump intensity by $\varepsilon^{-1/2}$. I.e. we make the process ...
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Euler Maruyama Estimate

Consider a SDE $dX_t=f(X_t)dt+\sigma(X_t)dW_t$ $(t\in [0,T], T<\infty)$ where $W$ is a Wiener Process. Let $X$ be its solution and $Y^{\Delta_N}$ ($N \in \mathbb{N})$ its approximation by Euler-...
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Does two-stage stochastic programming involve 2 decision variables?

I thought I was fairly confident in the formulation for “two-stage stochastic program with recourse” until I read the Wikipedia page. I am used to seeing the problem formulated with one decision ...
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Why does the stochastic sum is not converging to Ito's Formula?

This could be asked in Computational Science Stack Exchange, but since i don't think the mistake is in computing, i tought about asking here. (If you prefer that this question goes to CS, just notify ...
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The use of dt in brownian motion simulations.

Let's say that I have daily data over three years of some stochastic process. I use Maximum likelihood method to estimate the process. Let's Say its a Ornstein Uhlenbeck Process, now I have often ...
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Stochastic optimization vs stochastic programming

How should I think about the differences between stochastic optimization (SO) and stochastic programming (SP)? From Wikipedia, it seems that SO is a framework that uses randomness to solve a pre-...
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Maximum Likelihood Estimation of Multivariate Gaussian Density, where the number of samples is smaller than the unknown parameters

If we want to estimate the $p\times p$ (full rank) covariance matrix $\Sigma$ of multivariate normal density, using $n$ sample vectors $\mathbf{x}_1, \ldots, \mathbf{x}_n$, then the empirical ...
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Optimize Expected value of noisy data (stochastic aproximation)

I have a simulation that gives me noisy/stochastic data for every vector of parameters I put into it. So for the simulation data we can consider a function: $$F(\theta ,\xi )$$ where $\theta$ is ...
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Understanding stochastic approximation for a function.

I am trying to understand the paper https://arxiv.org/pdf/1606.06988.pdf. Currently I am stuck with the part Let us recall that, in order to construct a stochastic algorithm, which approximates the ...
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How likely are two events to occur at the same time?

Let's think of two events $1$ and $2$. Both events happen randomly $n_1$/$n_2$-times during a given time $T$ and last for a time of $t_1$/$t_2$. What is probability $P$, that both events happen ...
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time step for Brownian motion

Given a stochastic differential equation: $$dx=a(x,t)dt+b(x,t)dW(t)$$ where $x$ is a random variable, $a(x,t)$ is a function of $x$ and $t$; called drift term, and $b(x,t)$ is the diffusion term. ...
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What is the difference between optimum and robust optimum

While reading some research papers, I come to know that author has explicitly mentioned robust optimum and non-robust optimum terms. Is there any difference? If yes, can you please explain with an ...
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Ornstein-Uhlenbeck Process simulation bug

I think I found a bug in a programm somebody posted but I can't fix it. It is about the simulation of an Ornstein-Uhlenbeck Process. The problem is from this [article] & and from wikipedia from ...
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Approximations in Stochastic Differential Equations

Given a general SDE: $dX_t=b(X_t,t)dt+\sigma (X_t,t)dB_t$ , $X_0 =x$ and a solution $X^x_t$ . Where $b|x-y|+\sigma |x-y|\leq |x-y|$ . Prove that: $E(X^x_t)^2\leq L exp(Lt)$ for some none ...
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Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
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Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
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Central Limit Theorem and Normal Approximation

having started 'learning' all that is related to the Central Limit Theorem just one day ago, I am already a bit confused - maybe you can help me seeing through the cloud of misunderstanding. Let's ...
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Deriving the E-Step and M-Steps of the EM-Algorithm?

Insects of a certain species were exposed to cold temperature and how long the insects survived was recorded. The survival times of 9 of the 10 insects, in hours, are given below. 0.8, 0.6, ...
I simulate random walk on a divide difference grid to solve heat equation 1D. I want to prove numerically that this method has $Ν^{-1/2}$ error accuracy. My problem is that I don't know which norm ...