I'm trying to understand the physical meaning of the proposal step generating function in Metropolis algorithm.

In the original paper, and most derivations I found, it seems that it's not much important as long as it's symmetric (for detailed-balance) and allows a good exploration of the configuration space (e.g. $\sim 0.5$ acceptance ratio). Usually it's either a uniform or a normal distribution.

Now when studying Simulated Annealing, most papers assume a common use of a normal distribution with $\sigma = T$.

Given simulated annealing is deeply rooted in physical analogy, if we put the temperature there, there must be some physical sense we're trying to model. I mean, we could use a fixed uniform distribution with the proper width and still get good enough parameter space exploration and boltzmann distribution in the result, so why we use a gaussian and why the temperature dependence?

Also, when Cauchy distribution is introduced (e.g. by Szu and Hartley, 1987) they say the temperature used as a control parameter is kind of artificial, implying that the one in the Gaussian was not.

So what's the physical phenomenon with Gaussian, temperature dependent, steps we're modeling?

I know it all can be better explained with markov chains, priors, posteriors and likelihood, but that way you lose the physical analogy...

  • $\begingroup$ I know nothing about physics, but the temperature $T$ in the target $\pi_T$ and the proposal are two different things: the decreasing temperature induces the target $\pi_T$ to become more and more peaked, hence the maxima to become more and more attractive and sticky. The proposal is used to walk around the current target $\pi_T$ with a reasonable chance of success (e.g., an acceptance probability of $0.234$). $\endgroup$ – Xi'an Feb 13 '16 at 8:22
  • $\begingroup$ @Xi'an that's the role of the temperature in the target distribution, which is usually a Boltzmann one, the lower the temperature, the higher the rejection rate for distant steps. What I cannot understand though, is the role of the temperature in the proposal distribution which is usually something like $g(x) \simeq \exp(-x^2/T^2)$. Guess it comes from some thermal diffusion model, but I'd like to see a more formal derivation... $\endgroup$ – filippo Feb 13 '16 at 8:34
  • $\begingroup$ Again, I cannot give you any physical reason. From a probabilistic perspective, the scale of the random walk, whether normal or not, must adapt to the scale of the target $\pi_T$, according to theoretical developments (Roberts et al., 1997) that show the optimal variance of the random walk is twice the variance of the target. Hence the presence of $T$ in the Gaussian density. (I am unsure it is $1/T^2$ though.) The fact that it does not seem to matter for the Cauchy random walk is that the Cauchy distribution has no variance. $\endgroup$ – Xi'an Feb 13 '16 at 8:40
  • $\begingroup$ Will look into the paper, thanks. $1/T^2$ is taken from e.g. Szu and Hartely, 1989 and Ingber, 1989 $\endgroup$ – filippo Feb 13 '16 at 8:49
  • $\begingroup$ It all depends on the variance of the target distribution, $\pi_T$, so $T^2$ or $2T^2$ is not always the optimal choice for the random walk. $\endgroup$ – Xi'an Feb 13 '16 at 9:07

I suppose one could draw a physical analogy, noting that the proposal length scale should be proportional to the kinetic energy of the particle, but as @Xi'an I too would argue that this is simply a practical choice to maintain sensible acceptance rates as the temperature goes down.

To see this, assume we want to optimize a typical statistical problem, where the target function is a likelihood, which is asymptotically normal. Assume for simplicity that the optimum is at 0, and the width (sd) is $\sigma_t$. We optimize the log Likelihood, which is proportional to

$$ LL \propto - x^2 / \sigma_t^2 $$

x being the distance from the maximum at 0. In the standard simulated annealing, our acceptance probability would thus be proportional to

$$ p_a \propto e^{- \frac{x^2} {\sigma_t^2 \cdot T } } $$

Now, consider that the expected value of the random variable x depends on the proposal function

$$ <x^2> \sim proposal $$

As this is tagged self-study, I'll leave it to you to see how you should scale the proposal function to get constant acceptance rates (I would argue that this is a sensible goal).

  • $\begingroup$ Thanks, I'll have to think about it, but it seems a promising and simple enough approach.About constant acceptance rates, I agree it should be a sensible goal but... Tipical rates start with a plateau at high temperatures, go roughly linearly as the annealing goes on, and reach a constant plateau $\sim 0$ at low temperatures. This is often used as a stopping criterion (IIRC Kirkpatrick used this very same criterion in his paper), so it doesn't seem a goal much shared in the Annealing literature. $\endgroup$ – filippo Feb 27 '16 at 15:15
  • $\begingroup$ Yes, OK, but the first part of the practical experience may also be because you haven't reached the optimum yet - in the code above I assume you are proposing to go away from the optimum. It's also my experience that acceptance goes to zero, but this may also be because people tune the proposal to shrink slower than the acceptance (which is what your $\sigma$ = T suggests to me). I wonder about the sense of that though, because there would be other ways to check convergence, so why would you waste your computation time on all these rejections. $\endgroup$ – Florian Hartig Feb 27 '16 at 17:38
  • $\begingroup$ In my practical experience with $\sigma = T$ the acceptance ratio grows again at small temperatures (small steps, small $\Delta E$, small $T$, proposal $\rightarrow 1$). Using $\sigma \propto T$ with a proper, big enough, constant, this effect is mitigated, but I didn't see any real gain (e.g. faster or better convergence) over constant $\sigma$ steps. If I may go a little off-topic, could you give me some pointers about this better methods of checking convergence? right now I'm monitoring energy variance but it's not that different in behaviour from acceptance rate. $\endgroup$ – filippo Feb 28 '16 at 8:04
  • $\begingroup$ better methods - not a big secret - just check if you still get better with time (by some epsilon), if not stop. $\endgroup$ – Florian Hartig Feb 28 '16 at 19:01
  • $\begingroup$ I finally had a little time to look into this again. $\sigma = \sqrt{T}$ actually leads to roughly constant acceptance rates as expected. This is not what Szu, Geman and Geman, and a lot of other literature on Bolzmann and Cauchy machines, use though: they use $\sigma = T$ with steps $\propto \exp{-x^2 / T^2}$ or $\propto T/(x^2+T^2)$. Same for Scipy annealing implementation (not that it really proves anything...) $\endgroup$ – filippo Mar 3 '16 at 9:29

Here's an analogy from physics - consider the distribution in $n$ dimensions to have many peaks or valleys throughout. In a MCMC sampling problem, the 'height' of the peaks would be how improbable the state was - we would want to go to the valleys. Now, let us imagine this same surface to be a the map of potential energy for a physical system. Then, since a particle or object would want to find a minimum for the potential, it will try to go 'downhill'.

Now, in a classical sense, the particle can never pass through or 'tunnel' through the potential barriers. These barriers are the peaks in our distribution. However, in quantum theory, these particles always have a nonzero probability of tunneling through these peaks, and the probability of this tunneling behavior is dependent on the size of the barrier and the amount of energy the particle has.

Essentially, if the particle has a lot of energy, it is moving fast. If it is moving fast, it can tunnel through the potential barrier. If a particle's aggregate phase of matter has a high temperature, each particle has a lot of kinetic energy, on average. Let's reconsider our analogy with simulated annealing. If we have particles with high temperature, we allow them to escape local minima by going through otherwise unfavorable portions of state space. They are, in this analogy, 'tunneling' through the regions of low probability.

Now, as to how this relates to the distributions, I'm not sure.

  • $\begingroup$ The thing is that as long as the temperature is non zero and the proposal distribution is such that the system is ergodic (symmetric uniform is enough) the particle will tunnel given enough time. Metropolis, at least in its original form, is good to describe the physics at equilibrium, how it gets there seems nothing more than a numeric method with little resemblance to the system dynamics. Reading Roberts et al, 1997 though it seems that Langevin diffusion is the key to physically understand the non-equilibrium dynamics, and other sources seem to confirm it. $\endgroup$ – filippo Feb 13 '16 at 16:55

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