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?

Effectively, in our endeavor to function the assorted capabilities of TensorFlow Chance (TFP) / tfprobability, we began displaying examples of how one can match hierarchical fashions, utilizing one in all TFP’s joint distribution courses and HMC. The technical facets being complicated sufficient in themselves, we by no means gave an introduction to the “math aspect of issues.” Right here we try to make up for this.

Seeing how it’s inconceivable, in a brief weblog put up, to supply an introduction to Bayesian modeling and Markov Chain Monte Carlo usually, and the way there are such a lot of glorious texts doing this already, we are going to presuppose some prior information. 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 potential.
In that spirit, welcome to a “glossary with a story.”

So what’s it for?

Sampling, or Monte Carlo, strategies usually are used once we wish to produce samples from, or statistically describe a distribution we don’t have a closed-form formulation of. Generally, we’d actually have an interest within the samples; generally 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, we’ve got a mannequin, a joint distribution, which is meant to explain some actuality. Ranging from probably the most primary state of affairs, it’d appear like this:

[
x sim mathcal{Poisson}(lambda)
]

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

We now wish to decide the parameter, (lambda), of the Poisson that make these knowledge most doubtless. Up to now, we’re not even being Bayesian but: There isn’t 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’ve got three parameters to find out: (lambda), (alpha) and (beta).
And what we’re taken with is the posterior distribution of the parameters given the information.

Now, relying on the distributions concerned, we often can’t calculate the posterior distributions in closed kind. As an alternative, we’ve got to make use of sampling strategies 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 neglect what’s it that we’re sampling. Attempt to at all times remember that what we’re sampling isn’t the information, it’s parameters: the parameters of the posterior distributions we’re taken with.

Sampling

Sampling strategies usually 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 we’ve got measured one thing and at the moment are in search of a mechanism that explains these measurements – the latter needs to be simpler: We “simply” want to find out the probability of the information beneath these hypothetical mannequin parameters. However how will we provide you with solutions to begin with?

In idea, easy(-ish) strategies exist that may very well be used to generate samples from an unknown (in closed kind) distribution – so long as their unnormalized possibilities could be evaluated, and the issue is (very) low-dimensional. (For concise portraits of these strategies, reminiscent of uniform sampling, significance sampling, and rejection sampling, see(MacKay 2002).) These are usually not 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, usually exemplified by good tales –, and we refer the reader to the go-to references, reminiscent of (McElreath 2016) and (Kruschke 2010). Each have been proven to be much less environment friendly than HMC, the principle subject of this put up, as a consequence of their random-walk conduct: Each proposal relies on the present place in state area, which means that samples could also be extremely correlated and state area exploration proceeds slowly.

HMC

So HMC is fashionable 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 (no less than) three languages to precise an algorithm: Math; code (together with pseudo-code, which can or might 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 completely different from most different circumstances in that though I discover the conceptual explanations fascinating, they lead to much less “perceived understanding” than both the equations or the code. For individuals with backgrounds in physics, statistical mechanics and/or differential geometry it will in all probability be completely different!

In any case, bodily analogies make for the very best 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 information beneath the mannequin parameters. Alternatively we will say, we wish to reduce the adverse 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 will visualize the dynamics as that of a frictionless puck that slides over a floor of various top. The state of this method consists of the place of the puck, given by a 2D vector q, and the momentum of the puck (its mass instances its velocity), given by a 2D vector p.

Now once you hear “momentum” (and on condition that I’ve primed you to consider deep studying) it’s possible you’ll really feel that sounds acquainted, however though the respective analogies are associated the affiliation doesn’t assist that a lot. In deep studying, momentum is often 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 received’t concentrate on 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 may not bear in mind from college physics, power is available in two kinds: 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 wish to know – the adverse 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 particularly properly defined in Ben Lambert’s video.)

Following his exposition and notation, right here we’ve got 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 observe? 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 unbiased), and
• remedy 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 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 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 (Okay(p)) is the kinetic power as a operate of momentum (known as (Okay(m)) above).

The partial derivatives of the Hamiltonian decide how (p) and (q) change over time, (t), in accordance 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 remedy this method of partial differential equations? The essential workhorse in numerical integration is Euler’s technique, the place time (or the unbiased variable, usually) is superior by a step of dimension (epsilon), and a brand new worth of the dependent variable is computed by taking the (partial) spinoff 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 could 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 technique.

Leapfrog algorithm

So after Hamiltonian, we’ve hit the second magic phrase: leapfrog. In contrast to Hamiltonian nonetheless, there’s much less thriller right here. The leapfrog technique is “simply” a extra environment friendly solution to carry out the numerical integration.

It consists of three steps, principally 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 possibly can see, every step makes use of the corresponding variable-to-differentiate’s worth computed within the previous step. In observe, 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 could be accepted or not. The best way that call is taken is just not explicit 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 a better probability, they’re accepted; if not, they’re accepted solely with a sure likelihood that is dependent upon the ratio between previous and new likelihoods.
In idea, power staying fixed in a Hamiltonian system, proposals ought to at all times be accepted; in observe, lack of precision as a consequence of numerical integration might yield an acceptance charge lower than 1.

HMC in just a few traces of code

We’ve talked about ideas, and we’ve seen the mathematics, however between analogies and equations, it’s simple to lose monitor of the general algorithm. Properly, Radford Neal’s paper (Neal 2012) has some code, too! Right here it’s reproduced, with only a few extra feedback added (many feedback have 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
  # unbiased commonplace regular variates
  p <- rnorm(size(q), 0, 1)  
  # Make a half step for momentum originally
  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 by means of but? Effectively, 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 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 lately added to TFP.

NUTS being extra of algorithm within the pc 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).

Easy methods to keep away from crashes

Bétancourt’s article is an superior learn, and a paragraph specializing in a single level made within the paper could 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 shall be.
A product of quantity and density, the one vital likelihood 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, nevertheless it will get an increasing number of tough to seek out it (and keep there). Now as we noticed above, HMC makes use of gradient info 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 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 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 area, 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 completely different views of the identical mathematical system, they are going to 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 area 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, nonetheless, offers a right away answer: 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 area, for instance, then the gravitational attraction shall 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 shall 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 useful – except you knew all of it (or extra) beforehand, through which case you in all probability wouldn’t have learn this put up 🙂

Thanks for studying!