• COMP5328 – Advanced Machine Learning
• CS 285. Deep Reinforcement Learning
• IE 3186 – APPROXIMATE DYNAMIC PROGRAMMING， University of Pittsburgh
• CS 7642: Reinforcement Learning | OMSCS – Georgia Tech

## reweighting method

The configurations generated in a Monte Carlo simulation contain a huge amount of information, from which we usually distill a couple of numbers.

It would be a shame to waste all that information. Reweighting is a method which allows us to “expand” the results from the original simulation, performed at inverse temperature $\beta_0$, say, to any other $\beta$ sufficiently close to the simulation point without performing any additional simulations.

The simplest form of the reweighting is based on the fact that the canonical probability of a configuration $\phi$ at inverse temperature $\beta, p_\beta(\phi)$, can be easily related to the distribution at other temperature $\beta^{\prime}$ :
$$p_{\beta^{\prime}}(\phi) \propto e^{-\beta^{\prime} E_\phi}=C e^{-\left(\beta^{\prime}-\beta\right) E_\phi} p_\beta(\phi),$$
where $C$ is a proportionality constant (which depends on $\beta$ and $\beta^{\prime}$, and will remain undetermined). Thus, the expectation value of an operator $O(\phi)$ at temperature $\beta^{\prime}$ can be written as
$$\langle O\rangle_{\beta^{\prime}} \equiv \frac{1}{Z_{\beta^{\prime}}} \int d \phi O(\phi) p_{\beta^{\prime}}(\phi)=\frac{C}{Z_{\beta^{\prime}}} \int d \phi O(\phi) e^{-\left(\beta^{\prime}-\beta\right) E_\phi} p_\beta(\phi)=\frac{Z_\beta}{Z_{\beta^{\prime}}} C\left\langle O e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta,$$
where in the last step the expectation value is evaluated at temperature $\beta$. In order to get the ratio of the partition functions, we can set $O=1$, which implies
$$\frac{Z_{\beta^{\prime}}}{Z_\beta}=C\left\langle e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta .$$

Thus, finally we obtain the desired result:
$$\langle O\rangle_{\beta^{\prime}}=\frac{\left\langle O e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta}{\left\langle e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta} .$$

This implies that the expectation value of any observable at any inverse temperature $\beta^{\prime}$ can be obtained in terms of expectation values evaluated at $\beta$.

This appears to indicate that it should be possible to do a simulation at one value of $\beta$, and use the above formula to obtain results at any other temperature. In reality the situation is not so simple, due to the finite statistics in realistic simulations. This will be discussed below.

Now the standard estimate of the expectation value is:
$$\langle O\rangle_\beta \equiv \frac{1}{Z} \sum_{{\phi}} O(\phi) \exp \left[-\beta E_\phi\right] \approx \frac{1}{N} \sum_{i=1}^N O_i$$
where the first sum goes over full configuration space, and the second over the Monte Carlo configurations/measurements.

Thus, the reweighting formula becomes
$$\langle O\rangle_{\beta^{\prime}}=\frac{\sum_i O_i e^{-\left(\beta^{\prime}-\beta\right) E_i}}{\sum_i e^{-\left(\beta^{\prime}-\beta\right) E_i}}=\frac{\left\langle O e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta}{\left\langle e^{-\left(\beta^{\prime}-\beta\right) E}\right\rangle_\beta}$$

Note: $\sum_i$ goes over measurements, and $E_i, O_i$ must be measured from the same configuration.

In practice: perform a Monte Carlo simulation at $\beta$, and during the simulation write down all measurements $E_i$ and various desired operators $O_i$ in a file. After the run, use the equation above to calculate $\langle O\rangle_\beta$.

Example: 2d Ising, $V=16^2 \ldots 128^2$. Specific heat (susceptibility of $E$ )
\begin{aligned} C_V & =-\frac{1}{V} \frac{\partial\langle E\rangle}{\partial \beta} \ & =\frac{1}{V}\left\langle(E-\langle E\rangle)^2\right\rangle \end{aligned}

Susceptibilities diverges with some critical exponent at the critical point. In this case, $C_V \sim L^{\alpha / \nu}$ when $L \rightarrow \infty$ and we are at $\beta=\beta_c$ :

Points: simulated values, curves: reweighted data. Dashed lines show the error band for $32^2$ (for clarity, not shown for others). We will return to errors below.

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### CSE 365LR Introduction to Computer Security

View ScheduleCSE 365LR Introduction to Computer SecurityLecture

This is an undergraduate-level course intended for junior and senior-level students and will teach them introductory concepts of computer security. The main foci of this course will be network, web security, and application security. Part of the work will be dedicated to ethical aspects of security, and online privacy. The course will be heavily hands-on, as opposed to theoretical teaching.Credits: 4