A really first conceptual introduction to Hamiltonian Monte Carlo

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

Nicely, in our endeavor to characteristic the varied capabilities of TensorFlow Chance (TFP) / tfprobability, we began exhibiting examples of how one can match hierarchical fashions, utilizing one in all TFP’s joint distribution lessons and HMC. The technical features 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 not possible, in a brief weblog put up, 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 newest and biggest, the magic buzzwords, the well-known incantations: Hamiltonian Monte Carlo, leapfrog steps, NUTS – as all the time, making an attempt 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, strategies typically are used after we wish to produce samples from, or statistically describe a distribution we don’t have a closed-form formulation of. Typically, we would actually have an interest within the samples; generally we simply need them so we are able to compute, for instance, the imply and variance of the distribution.

What distribution? In the kind of purposes we’re speaking about, we now have a mannequin, a joint distribution, which is meant to explain some actuality. Ranging from essentially the most fundamental 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 alleged to mannequin, say, the variety of feedback in a code evaluate. We even have knowledge on precise code critiques, like this, say:

We now wish to decide the parameter, (lambda), of the Poisson that make these knowledge most possible. To this point, we’re not even being Bayesian but: There is no such thing as a prior on this parameter. However in fact, we wish 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, we now have three parameters to find out: (lambda), (alpha) and (beta).
And what we’re involved in is the posterior distribution of the parameters given the info.

Now, relying on the distributions concerned, we normally can not calculate the posterior distributions in closed kind. As an alternative, we now have to make use of sampling strategies to find out these parameters. What we’d prefer to level out as an alternative is the next: Within the upcoming discussions of sampling, HMC & co., it’s very easy to neglect what’s it that we’re sampling. Attempt to all the time remember that what we’re sampling isn’t the info, it’s parameters: the parameters of the posterior distributions we’re involved in.

Sampling

Sampling strategies typically include 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 we now have measured one thing and at the moment are in search of a mechanism that explains these measurements – the latter ought to be simpler: We “simply” want to find out the chance of the info beneath these hypothetical mannequin parameters. However how will we give you strategies to start out with?

In idea, simple(-ish) strategies exist that might be used to generate samples from an unknown (in closed kind) distribution – so long as their unnormalized chances might be evaluated, and the issue is (very) low-dimensional. (For concise portraits of these strategies, akin to uniform sampling, significance sampling, and rejection sampling, see(MacKay 2002).) These aren’t utilized in MCMC software program although, for lack of effectivity and non-suitability in excessive dimensions. Earlier than HMC turned the dominant algorithm in such software program, the Metropolis and Gibbs strategies have 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, akin to (McElreath 2016) and (Kruschke 2010). Each have been proven to be much less environment friendly than HMC, the primary subject of this put up, attributable to their random-walk conduct: Each proposal relies on the present place in state house, which means that samples could also be extremely correlated and state house exploration proceeds slowly.

HMC

So HMC is widespread as a result of in comparison with random-walk-based algorithms, it’s a lot extra environment friendly. Sadly, it’s also much more tough to “get.” As mentioned in Math, code, ideas: A 3rd street to deep studying, there appear to be (a minimum of) three languages to precise an algorithm: Math; code (together with pseudo-code, which can or will not be 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 totally different from most different instances in that regardless 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 may most likely be totally 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 wish to maximize, the loglikelihood of the info beneath the mannequin parameters. Alternatively we are able to say, we wish to decrease the damaging 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 wish it to finish up deep down in some valley.

In Neal’s personal phrases

In two dimensions, we are able to 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 whenever 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 regardless 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 nevertheless, the main focus is on the idea of vitality.

In statistical mechanics, the chance of being in some state (i) is inverse-exponentially associated to its vitality. (Right here (T) is the temperature; we received’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 may or won’t keep in mind from faculty physics, vitality is available in two types: potential vitality and kinetic vitality. Within the sliding-object state of affairs, the thing’s potential vitality corresponds to its top (place), whereas its kinetic vitality is expounded to its momentum, (m), by the method

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

Now with out kinetic vitality, the thing would slide downhill all the time, and as quickly because the panorama slopes up once more, would come to a halt. Via its momentum although, it is ready to proceed uphill for some time, simply as if, going downhill in your bike, you choose up velocity it’s possible you’ll make it over the subsequent (brief) hill with out pedaling.

In order that’s kinetic vitality. The opposite half, potential vitality, corresponds to the factor we actually wish to know – the damaging 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 house 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 particularly properly defined in Ben Lambert’s video.)

Following his exposition and notation, right here we now have the vitality 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 chance, 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 an ordinary regular ((e^{frac{- m^2}{2}}))! Thus, we are able to 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 similar because the conditional distribution given (U), as they’re impartial), and
• resolve 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, must you determine to learn the paper, right here we swap to Radford Neal’s notation.

Hamiltonian dynamics operates on a d-dimensional place vector, (q), and a d-dimensional momentum vector, (p). The state house 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 vitality (referred to as (U(theta)) above), and (Okay(p)) is the kinetic vitality as a operate of momentum (referred to as (Okay(m)) above).

The partial derivatives of the Hamiltonian decide how (p) and (q) change over time, (t), in response to 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 resolve this technique of partial differential equations? The essential 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 seems 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 immediately 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. In contrast to Hamiltonian nevertheless, there may be much less thriller right here. The leapfrog methodology is “simply” a extra environment friendly approach 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 may 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 following 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 best way that call is taken just isn’t 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 chance

[
min(1, exp(−H(q∗, p∗) +H(q, p)))
]

That’s, if the proposed parameters yield the next chance, they’re accepted; if not, they’re accepted solely with a sure chance that relies on the ratio between outdated and new likelihoods.
In idea, vitality staying fixed in a Hamiltonian system, proposals ought to all the time be accepted; in apply, lack of precision attributable to numerical integration might yield an acceptance price lower than 1.

HMC in just a few 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 have been preexisting):

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

# kinetic vitality is assumed to be sum(p^2/2) (mass == 1)
HMC <- operate (U, grad_U, epsilon, L, current_q) {
  q <- current_q
  # impartial normal regular variates
  p <- rnorm(size(q), 0, 1)  
  # Make a half step for momentum in the beginning
  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, thus far 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 Chance’s grasp department, is an algorithm that goals to avoid 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 problem: selecting the step dimension (epsilon). The respective algorithm, twin averaging, was additionally just lately added to TFP.

NUTS being extra of algorithm within the laptop science utilization of the phrase than a factor to clarify 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 applied there. As an alternative, we’ll spherical up with one other conceptual analogy, Michael Bétancourts crashing (or not!) satellite tv for pc (Betancourt 2017).

How one can 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 ordinary, the density operate has a mode (the place the place it’s maximal), however essentially, there can’t be a lot quantity round it – similar to with k-nearest neighbors, the extra dimensions you add, the farther your nearest neighbor can be.
A product of quantity and density, the one vital chance mass resides within the so-called typical set, which turns into an increasing number of slender in excessive dimensions.

So, the everyday set is what we wish to discover, however it will get an increasing number of tough 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 chance (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 be certain 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 an alternative of making an attempt to motive a couple of mode, a gradient, and a typical set, we are able to equivalently motive a couple of planet, a gravitational subject, and an orbit (Determine 14). The probabilistic endeavor of exploring the everyday set then turns into a bodily endeavor of inserting a satellite tv for pc in a steady orbit across the hypothetical planet. As a result of these are simply two totally different views of the identical mathematical system, they may endure from the identical pathologies. Certainly, if we place a satellite tv for pc at relaxation out in house it’ll fall within the gravitational subject 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 end result.

The bodily image, nevertheless, supplies a right away resolution: though objects at relaxation will crash into the planet, we are able to preserve a steady orbit by endowing our satellite tv for pc with sufficient momentum to counteract the gravitational attraction. Now we have to watch out, nevertheless, in how precisely we add momentum to our satellite tv for pc. If we add too little momentum transverse to the gravitational subject, for instance, then the gravitational attraction can be too robust 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 can be too weak to seize the satellite tv for pc in any respect and it’ll as an alternative fly out into the depths of house (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, through which case you most likely wouldn’t have learn this put up 🙂

Thanks for studying!