A really first conceptual introduction to Hamiltonian Monte Carlo

Why a very (that means: VERY!) first conceptual introduction to Hamiltonian Monte Carlo (HMC) on this weblog?

Nicely, in our endeavor to function the varied capabilities of TensorFlow Likelihood (TFP) / tfprobability, we began displaying examples of the right way to match hierarchical fashions, utilizing considered one of TFP’s joint distribution lessons and HMC. The technical elements being advanced sufficient in themselves, we by no means gave an introduction to the “math aspect of issues.” Right here we are attempting to make up for this.

Seeing how it’s unattainable, in a brief weblog publish, to supply an introduction to Bayesian modeling and Markov Chain Monte Carlo typically, and the way there are such a lot of glorious texts doing this already, we are going to presuppose some prior data. Our particular focus then is on the most recent and best, the magic buzzwords, the well-known incantations: Hamiltonian Monte Carlo, leapfrog steps, NUTS – as at all times, attempting to demystify, to make issues as comprehensible as doable.
In that spirit, welcome to a “glossary with a story.”

So what’s it for?

Sampling, or Monte Carlo, methods typically are used once we need to produce samples from, or statistically describe a distribution we don’t have a closed-form formulation of. Typically, we’d actually have an interest within the samples; typically we simply need them so we will compute, for instance, the imply and variance of the distribution.

What distribution? In the kind of purposes we’re speaking about, now we have a mannequin, a joint distribution, which is meant to explain some actuality. Ranging from essentially the most primary state of affairs, it would appear to be this:

[
x sim mathcal{Poisson}(lambda)
]

This “joint distribution” solely has a single member, a Poisson distribution, that’s imagined to mannequin, say, the variety of feedback in a code evaluate. We even have information on precise code evaluations, like this, say:

We now need to decide the parameter, (lambda), of the Poisson that make these information most probably. Up to now, we’re not even being Bayesian but: There isn’t any prior on this parameter. However in fact, we need to be Bayesian, so we add one – think about mounted priors on its parameters:

[
x sim mathcal{Poisson}(lambda)
lambda sim gamma(alpha, beta)
alpha sim […]
beta sim […]
]

This being a joint distribution, now we have three parameters to find out: (lambda), (alpha) and (beta).
And what we’re eager about is the posterior distribution of the parameters given the information.

Now, relying on the distributions concerned, we normally can’t calculate the posterior distributions in closed type. As an alternative, now we have to make use of sampling methods to find out these parameters. What we’d prefer to level out as a substitute is the next: Within the upcoming discussions of sampling, HMC & co., it’s very easy to overlook what’s it that we’re sampling. Attempt to at all times understand that what we’re sampling isn’t the information, it’s parameters: the parameters of the posterior distributions we’re eager about.

Sampling

Sampling strategies typically encompass two steps: producing a pattern (“proposal”) and deciding whether or not to maintain it or to throw it away (“acceptance”). Intuitively, in our given state of affairs – the place now we have measured one thing and at the moment are on the lookout for a mechanism that explains these measurements – the latter must be simpler: We “simply” want to find out the probability of the information underneath these hypothetical mannequin parameters. However how will we give you ideas to start out with?

In concept, simple(-ish) strategies exist that might be used to generate samples from an unknown (in closed type) distribution – so long as their unnormalized possibilities might be evaluated, and the issue is (very) low-dimensional. (For concise portraits of these strategies, similar to uniform sampling, significance sampling, and rejection sampling, see(MacKay 2002).) These will not be utilized in MCMC software program although, for lack of effectivity and non-suitability in excessive dimensions. Earlier than HMC grew to become the dominant algorithm in such software program, the Metropolis and Gibbs strategies had been the algorithms of alternative. Each are properly and understandably defined – within the case of Metropolis, typically exemplified by good tales –, and we refer the reader to the go-to references, similar to (McElreath 2016) and (Kruschke 2010). Each had been proven to be much less environment friendly than HMC, the principle matter of this publish, attributable to their random-walk conduct: Each proposal is predicated on the present place in state area, that means that samples could also be extremely correlated and state area exploration proceeds slowly.

HMC

So HMC is in style as a result of in comparison with random-walk-based algorithms, it’s a lot extra environment friendly. Sadly, it’s also much more troublesome to “get.” As mentioned in Math, code, ideas: A 3rd highway to deep studying, there appear to be (not less than) three languages to precise an algorithm: Math; code (together with pseudo-code, which can or is probably not on the verge to math notation); and one I name conceptual which spans the entire vary from very summary to very concrete, even visible. To me personally, HMC is completely different from most different instances in that despite the fact that I discover the conceptual explanations fascinating, they end in much less “perceived understanding” than both the equations or the code. For individuals with backgrounds in physics, statistical mechanics and/or differential geometry this can most likely be completely different!

In any case, bodily analogies make for the perfect begin.

Bodily analogies

The traditional bodily analogy is given within the reference article, Radford Neal’s “MCMC utilizing Hamiltonian dynamics” (Neal 2012), and properly defined in a video by Ben Lambert.

So there’s this “factor” we need to maximize, the loglikelihood of the information underneath the mannequin parameters. Alternatively we will say, we need to reduce the unfavorable loglikelihood (like loss in a neural community). This “factor” to be optimized can then be visualized as an object sliding over a panorama with hills and valleys, and like with gradient descent in deep studying, we would like it to finish up deep down in some valley.

In Neal’s personal phrases

In two dimensions, we will visualize the dynamics as that of a frictionless puck that slides over a floor of various top. The state of this technique consists of the place of the puck, given by a 2D vector q, and the momentum of the puck (its mass occasions its velocity), given by a 2D vector p.

Now once you hear “momentum” (and provided that I’ve primed you to consider deep studying) it’s possible you’ll really feel that sounds acquainted, however despite the fact that the respective analogies are associated the affiliation doesn’t assist that a lot. In deep studying, momentum is usually praised for its avoidance of ineffective oscillations in imbalanced optimization landscapes.
With HMC nonetheless, the main target is on the idea of power.

In statistical mechanics, the likelihood of being in some state (i) is inverse-exponentially associated to its power. (Right here (T) is the temperature; we gained’t deal with this so simply think about it being set to 1 on this and subsequent equations.)

[P(E_i) sim e^{frac{-E_i}{T}} ]

As you would possibly or won’t bear in mind from faculty physics, power is available in two types: potential power and kinetic power. Within the sliding-object state of affairs, the thing’s potential power corresponds to its top (place), whereas its kinetic power is expounded to its momentum, (m), by the method

[K(m) = frac{m^2}{2 * mass} ]

Now with out kinetic power, the thing would slide downhill at all times, and as quickly because the panorama slopes up once more, would come to a halt. By way of its momentum although, it is ready to proceed uphill for some time, simply as if, going downhill in your bike, you choose up pace it’s possible you’ll make it over the following (quick) hill with out pedaling.

In order that’s kinetic power. The opposite half, potential power, corresponds to the factor we actually need to know – the unfavorable log posterior of the parameters we’re actually after:

[U(theta) sim – log (P(x | theta) P(theta))]

So the “trick” of HMC is augmenting the state area of curiosity – the vector of posterior parameters – by a momentum vector, to enhance optimization effectivity. Once we’re completed, the momentum half is simply thrown away. (This side is very properly defined in Ben Lambert’s video.)

Following his exposition and notation, right here now we have the power of a state of parameter and momentum vectors, equaling a sum of potential and kinetic energies:

[E(theta, m) = U(theta) + K(m)]

The corresponding likelihood, as per the connection given above, then is

[P(E) sim e^{frac{-E}{T}} = e^{frac{- U(theta)}{T}} e^{frac{- K(m)}{T}}]

We now substitute into this equation, assuming a temperature (T) of 1 and a mass of 1:

[P(E) sim P(x | theta) P(theta) e^{frac{- m^2}{2}}]

Now on this formulation, the distribution of momentum is simply a regular regular ((e^{frac{- m^2}{2}}))! Thus, we will simply combine out the momentum and take (P(theta)) as samples from the posterior distribution:

[
begin{aligned}
& P(theta) =
int ! P(theta, m) mathrm{d}m = frac{1}{Z} int ! P(x | theta) P(theta) mathcal{N}(m|0,1) mathrm{d}m
& P(theta) = frac{1}{Z} int ! P(x | theta) P(theta)
end{aligned}
]

How does this work in apply? At each step, we

• pattern a brand new momentum worth from its marginal distribution (which is identical because the conditional distribution given (U), as they’re impartial), and
• clear up for the trail of the particle. That is the place Hamilton’s equations come into play.

Hamilton’s equations (equations of movement)

For the sake of much less confusion, do you have to determine to learn the paper, right here we change to Radford Neal’s notation.

Hamiltonian dynamics operates on a d-dimensional place vector, (q), and a d-dimensional momentum vector, (p). The state area is described by the Hamiltonian, a operate of (p) and (q):

[H(q, p) =U(q) +K(p)]

Right here (U(q)) is the potential power (known as (U(theta)) above), and (Ok(p)) is the kinetic power as a operate of momentum (known as (Ok(m)) above).

The partial derivatives of the Hamiltonian decide how (p) and (q) change over time, (t), in line with Hamilton’s equations:

[
begin{aligned}
& frac{dq}{dt} = frac{partial H}{partial p}
& frac{dp}{dt} = – frac{partial H}{partial q}
end{aligned}
]

How can we clear up this technique of partial differential equations? The fundamental workhorse in numerical integration is Euler’s methodology, the place time (or the impartial variable, typically) is superior by a step of dimension (epsilon), and a brand new worth of the dependent variable is computed by taking the (partial) by-product and including it to its present worth. For the Hamiltonian system, doing this one equation after the opposite appears like this:

[
begin{aligned}
& p(t+epsilon) = p(t) + epsilon frac{dp}{dt}(t) = p(t) − epsilon frac{partial U}{partial q}(q(t))
& q(t+epsilon) = q(t) + epsilon frac{dq}{dt}(t) = q(t) + epsilon frac{p(t)}{m})
end{aligned}
]

Right here first a brand new place is computed for time (t + 1), making use of the present momentum at time (t); then a brand new momentum is computed, additionally for time (t + 1), making use of the present place at time (t).

This course of might be improved if in step 2, we make use of the new place we simply freshly computed in step 1; however let’s instantly go to what’s really utilized in up to date software program, the leapfrog methodology.

Leapfrog algorithm

So after Hamiltonian, we’ve hit the second magic phrase: leapfrog. Not like Hamiltonian nonetheless, there may be much less thriller right here. The leapfrog methodology is “simply” a extra environment friendly strategy to carry out the numerical integration.

It consists of three steps, mainly splitting up the Euler step 1 into two elements, earlier than and after the momentum replace:

[
begin{aligned}
& p(t+frac{epsilon}{2}) = p(t) − frac{epsilon}{2} frac{partial U}{partial q}(q(t))
& q(t+epsilon) = q(t) + epsilon frac{p(t + frac{epsilon}{2})}{m}
& p(t+ epsilon) = p(t+frac{epsilon}{2}) − frac{epsilon}{2} frac{partial U}{partial q}(q(t + epsilon))
end{aligned}
]

As you’ll be able to see, every step makes use of the corresponding variable-to-differentiate’s worth computed within the previous step. In apply, a number of leapfrog steps are executed earlier than a proposal is made; so steps 3 and 1 (of the next iteration) are mixed.

Proposal – this key phrase brings us again to the higher-level “plan.” All this – Hamiltonian equations, leapfrog integration – served to generate a proposal for a brand new worth of the parameters, which might be accepted or not. The way in which that call is taken shouldn’t be specific to HMC and defined intimately within the above-mentioned expositions on the Metropolis algorithm, so we simply cowl it briefly.

Acceptance: Metropolis algorithm

Underneath the Metropolis algorithm, proposed new vectors (q*) and (p*) are accepted with likelihood

[
min(1, exp(−H(q∗, p∗) +H(q, p)))
]

That’s, if the proposed parameters yield the next probability, they’re accepted; if not, they’re accepted solely with a sure likelihood that is determined by the ratio between outdated and new likelihoods.
In concept, power staying fixed in a Hamiltonian system, proposals ought to at all times be accepted; in apply, lack of precision attributable to numerical integration might yield an acceptance price lower than 1.

HMC in a number of strains of code

We’ve talked about ideas, and we’ve seen the mathematics, however between analogies and equations, it’s simple to lose observe of the general algorithm. Properly, Radford Neal’s paper (Neal 2012) has some code, too! Right here it’s reproduced, with just some further feedback added (many feedback had been preexisting):

# U is a operate that returns the potential power given q
# grad_U returns the respective partial derivatives
# epsilon stepsize
# L variety of leapfrog steps
# current_q present place

# kinetic power is assumed to be sum(p^2/2) (mass == 1)
HMC <- operate (U, grad_U, epsilon, L, current_q) {
  q <- current_q
  # impartial commonplace regular variates
  p <- rnorm(size(q), 0, 1)  
  # Make a half step for momentum at first
  current_p <- p 
  # Alternate full steps for place and momentum
  p <- p - epsilon * grad_U(q) / 2 
  for (i in 1:L) {
    # Make a full step for the place
    q <- q + epsilon * p
    # Make a full step for the momentum, besides at finish of trajectory
    if (i != L) p <- p - epsilon * grad_U(q)
    }
  # Make a half step for momentum on the finish
  p <- p - epsilon * grad_U(q) / 2
  # Negate momentum at finish of trajectory to make the proposal symmetric
  p <- -p
  # Consider potential and kinetic energies at begin and finish of trajectory 
  current_U <- U(current_q)
  current_K <- sum(current_p^2) / 2
  proposed_U <- U(q)
  proposed_K <- sum(p^2) / 2
  # Settle for or reject the state at finish of trajectory, returning both
  # the place on the finish of the trajectory or the preliminary place
  if (runif(1) < exp(current_U-proposed_U+current_K-proposed_K)) {
    return (q)  # settle for
  } else {
    return (current_q)  # reject
  }
}

Hopefully, you discover this piece of code as useful as I do. Are we via but? Nicely, up to now we haven’t encountered the final magic phrase: NUTS. What, or who, is NUTS?

NUTS

NUTS, added to Stan in 2011 and a couple of month in the past, to TensorFlow Likelihood’s grasp department, is an algorithm that goals to bypass one of many sensible difficulties in utilizing HMC: The selection of variety of leapfrog steps to carry out earlier than making a proposal. The acronym stands for No-U-Flip Sampler, alluding to the avoidance of U-turn-shaped curves within the optimization panorama when the variety of leapfrog steps is chosen too excessive.

The reference paper by Hoffman & Gelman (Hoffman and Gelman 2011) additionally describes an answer to a associated issue: selecting the step dimension (epsilon). The respective algorithm, twin averaging, was additionally not too long ago added to TFP.

NUTS being extra of algorithm within the laptop science utilization of the phrase than a factor to elucidate conceptually, we’ll go away it at that, and ask the reader to learn the paper – and even, seek the advice of the TFP documentation to see how NUTS is carried out there. As an alternative, we’ll spherical up with one other conceptual analogy, Michael Bétancourts crashing (or not!) satellite tv for pc (Betancourt 2017).

Tips on how to keep away from crashes

Bétancourt’s article is an superior learn, and a paragraph specializing in a single level made within the paper might be nothing than a “teaser” (which is why we’ll have an image, too!).

To introduce the upcoming analogy, the issue begins with excessive dimensionality, which is a given in most real-world issues. In excessive dimensions, as normal, the density operate has a mode (the place the place it’s maximal), however essentially, there can’t be a lot quantity round it – identical to with k-nearest neighbors, the extra dimensions you add, the farther your nearest neighbor will probably be.
A product of quantity and density, the one vital likelihood mass resides within the so-called typical set, which turns into increasingly slim in excessive dimensions.

So, the everyday set is what we need to discover, nevertheless it will get increasingly troublesome to search out it (and keep there). Now as we noticed above, HMC makes use of gradient data to get close to the mode, but when it simply adopted the gradient of the log likelihood (the place) it might go away the everyday set and cease on the mode.

That is the place momentum is available in – it counteracts the gradient, and each collectively make sure that the Markov chain stays on the everyday set. Now right here’s the satellite tv for pc analogy, in Bétancourt’s personal phrases:

For instance, as a substitute of attempting to motive a couple of mode, a gradient, and a typical set, we will equivalently motive a couple of planet, a gravitational discipline, and an orbit (Determine 14). The probabilistic endeavor of exploring the everyday set then turns into a bodily endeavor of putting a satellite tv for pc in a steady orbit across the hypothetical planet. As a result of these are simply two completely different views of the identical mathematical system, they’ll endure from the identical pathologies. Certainly, if we place a satellite tv for pc at relaxation out in area it is going to fall within the gravitational discipline and crash into the floor of the planet, simply as naive gradient-driven trajectories crash into the mode (Determine 15). From both the probabilistic or bodily perspective we’re left with a catastrophic final result.

The bodily image, nonetheless, offers a direct resolution: though objects at relaxation will crash into the planet, we will keep a steady orbit by endowing our satellite tv for pc with sufficient momentum to counteract the gravitational attraction. We have now to watch out, nonetheless, in how precisely we add momentum to our satellite tv for pc. If we add too little momentum transverse to the gravitational discipline, for instance, then the gravitational attraction will probably be too sturdy and the satellite tv for pc will nonetheless crash into the planet (Determine 16a). Then again, if we add an excessive amount of momentum then the gravitational attraction will probably be too weak to seize the satellite tv for pc in any respect and it’ll as a substitute fly out into the depths of area (Determine 16b).

And right here’s the image I promised (Determine 16 from the paper):

And with this, we conclude. Hopefully, you’ll have discovered this beneficial – except you knew all of it (or extra) beforehand, wherein case you most likely wouldn’t have learn this publish 🙂

Thanks for studying!