Amongst deep studying practitioners, Kullback-Leibler divergence (KL divergence) is probably greatest identified for its position in coaching variational autoencoders (VAEs). To be taught an informative latent house, we don’t simply optimize for good reconstruction. Slightly, we additionally impose a previous on the latent distribution, and purpose to maintain them shut – typically, by minimizing KL divergence.

On this position, KL divergence acts like a watchdog; it’s a constraining, regularizing issue, and if anthropomorphized, would appear stern and extreme. If we depart it at that, nevertheless, we’ve seen only one aspect of its character, and are lacking out on its complement, an image of playfulness, journey, and curiosity. On this publish, we’ll check out that different aspect.

Whereas being impressed by a collection of tweets by Simon de Deo, enumerating purposes of KL divergence in an unlimited variety of disciplines,

we don’t aspire to offer a complete write-up right here – as talked about within the preliminary tweet, the subject might simply fill an entire semester of research.

The far more modest targets of this publish, then, are

1. to rapidly recap the position of KL divergence in coaching VAEs, and point out similar-in-character purposes;
2. as an example that extra playful, adventurous “different aspect” of its character; and
3. in a not-so-entertaining, however – hopefully – helpful method, differentiate KL divergence from associated ideas equivalent to cross entropy, mutual data, or free power.

Earlier than although, we begin with a definition and a few terminology.

KL divergence in a nutshell

KL divergence is the anticipated worth of the logarithmic distinction in chances in line with two distributions, (p) and (q). Right here it’s in its discrete-probabilities variant:

[begin{equation}
D_{KL}(p||q) = sumlimits_{x} p(x) log(frac{p(x)}{q(x)})
tag{1}
end{equation}]

Notably, it’s uneven; that’s, (D_{KL}(p||q)) just isn’t the identical as (D_{KL}(q||p)). (Which is why it’s a divergence, not a distance.) This facet will play an essential position in part 2 devoted to the “different aspect.”

To emphasize this asymmetry, KL divergence is typically referred to as relative data (as in “data of (p) relative to (q)”), or data achieve. We agree with one among our sources that due to its universality and significance, KL divergence would in all probability have deserved a extra informative title; equivalent to, exactly, data achieve. (Which is much less ambiguous pronunciation-wise, as nicely.)

KL divergence, “villain”

In lots of machine studying algorithms, KL divergence seems within the context of variational inference. Usually, for reasonable information, precise computation of the posterior distribution is infeasible. Thus, some type of approximation is required. In variational inference, the true posterior (p^*) is approximated by a less complicated distribution, (q), from some tractable household.
To make sure we now have a superb approximation, we decrease – in concept, at the least – the KL divergence of (q) relative to (p^*), thus changing inference by optimization.

In follow, once more for causes of intractability, the KL divergence minimized is that of (q) relative to an unnormalized distribution (widetilde{p})

[begin{equation}
J(q) = D_{KL}(q||widetilde{p})
tag{2}
end{equation}]

the place (widetilde{p}) is the joint distribution of parameters and information:

[begin{equation}
widetilde{p}(mathbf{x}) = p(mathbf{x}, mathcal{D}) = p^*(mathbf{x}) p(mathcal{D})
tag{3}
end{equation}]

and (p^*) is the true posterior:

[begin{equation}
p^*(mathbf{x}) = p(mathbf{x}|mathcal{D})
tag{4}
end{equation}]

Equal to that formulation (eq. (2)) – for a derivation see (Murphy 2012) – is that this, which exhibits the optimization goal to be an higher sure on the unfavorable log-likelihood (NLL):

[begin{equation}
J(q) = D_{KL}(q||p^*) – log p(D)
tag{5}
end{equation}]

One more formulation – once more, see (Murphy 2012) for particulars – is the one we really use when coaching (e.g.) VAEs. This one corresponds to the anticipated NLL plus the KL divergence between the approximation (q) and the imposed prior (p):

[begin{equation}
J(q) = D_{KL}(q||p) – E_q[- log p(mathcal{D}|mathbf{x})]
tag{6}
finish{equation}]

Negated, this formulation can also be referred to as the ELBO, for proof decrease sure. Within the VAE publish cited above, the ELBO was written

[begin{equation}
ELBO = E[log p(x|z)] – KL(q(z)||p(z))
tag{7}
finish{equation}]

with (z) denoting the latent variables ((q(z)) being the approximation, (p(z)) the prior, typically a multivariate regular).

Past VAEs

Generalizing this “conservative” motion sample of KL divergence past VAEs, we will say that it expresses the standard of approximations. An essential space the place approximation takes place is (lossy) compression. KL divergence supplies a strategy to quantify how a lot data is misplaced after we compress information.

Summing up, in these and related purposes, KL divergence is “dangerous” – though we don’t need it to be zero (or else, why trouble utilizing the algorithm?), we actually wish to preserve it low. So now, let’s see the opposite aspect.

KL divergence, good man

In a second class of purposes, KL divergence just isn’t one thing to be minimized. In these domains, KL divergence is indicative of shock, disagreement, exploratory habits, or studying: This actually is the attitude of data achieve.

Shock

One area the place shock, not data per se, governs habits is notion. For instance, eyetracking research (e.g., (Itti and Baldi 2005)) confirmed that shock, as measured by KL divergence, was a greater predictor of visible consideration than data, measured by entropy. Whereas these research appear to have popularized the expression “Bayesian shock,” this compound is – I believe – not essentially the most informative one, as neither half provides a lot data to the opposite. In Bayesian updating, the magnitude of the distinction between prior and posterior displays the diploma of shock led to by the info – shock is an integral a part of the idea.

Thus, with KL divergence linked to shock, and shock rooted within the basic strategy of Bayesian updating, a course of that could possibly be used to explain the course of life itself, KL divergence itself turns into basic. We might get tempted to see it in every single place. Accordingly, it has been utilized in many fields to quantify unidirectional divergence.

For instance, (Zanardo 2017) have utilized it in buying and selling, measuring how a lot an individual disagrees with the market perception. Greater disagreement then corresponds to greater anticipated positive factors from betting towards the market.

Nearer to the world of deep studying, it’s utilized in intrinsically motivated reinforcement studying (e.g., (Solar, Gomez, and Schmidhuber 2011)), the place an optimum coverage ought to maximize the long-term data achieve. That is doable as a result of like entropy, KL divergence is additive.

Though its asymmetry is related whether or not you employ KL divergence for regularization (part 1) or shock (this part), it turns into particularly evident when used for studying and shock.

Asymmetry in motion

Wanting once more on the KL formulation

[begin{equation}
D_{KL}(p||q) = sumlimits_{x} p(x) log(frac{p(x)}{q(x)})
tag{1}
end{equation}]

the roles of (p) and (q) are basically totally different. For one, the expectation is computed over the primary distribution ((p) in (1)). This facet is essential as a result of the “order” (the respective roles) of (p) and (q) might should be chosen in line with tractability (which distribution can we common over).

Secondly, the fraction contained in the (log) signifies that if (q) is ever zero at some extent the place (p) isn’t, the KL divergence will “blow up.” What this implies for distribution estimation typically is properly detailed in Murphy (2012). Within the context of shock, it signifies that if I be taught one thing I used to assume had likelihood zero, I will likely be “infinitely shocked.”

To keep away from infinite shock, we will be certain our prior likelihood is rarely zero. However even then, the attention-grabbing factor is that how a lot data we achieve in anyone occasion is determined by how a lot data I had earlier than. Let’s see a easy instance.

Assume that in my present understanding of the world, black swans in all probability don’t exist, however they may … possibly 1 p.c of them is black. Put in another way, my prior perception of a swan, ought to I encounter one, being black is (q = 0.01).

Now in truth I do encounter one, and it’s black.
The knowledge I’ve gained is:

[begin{equation}
l(p,q) = 0 * log(frac{0}{0.99}) + 1 * log(frac{1}{0.01}) = 6.6 bits
tag{8}
end{equation}]

Conversely, suppose I’d been far more undecided earlier than; say I’d have thought the percentages have been 50:50.
On seeing a black swan, I get loads much less data:

[begin{equation}
l(p,q) = 0 * log(frac{0}{0.5}) + 1 * log(frac{1}{0.5}) = 1 bit
tag{9}
end{equation}]

This view of KL divergence, when it comes to shock and studying, is inspiring – it could lead on one to seeing it in motion in every single place. Nonetheless, we nonetheless have the third and remaining process to deal with: rapidly evaluate KL divergence to different ideas within the space.

Entropy

All of it begins with entropy, or uncertainty, or data, as formulated by Claude Shannon.
Entropy is the common log likelihood of a distribution:

[begin{equation}
H(X) = – sumlimits_{x=1}^n p(x_i) log(p(x_i))
tag{10}
end{equation}]

As properly described in (DeDeo 2016), this formulation was chosen to fulfill 4 standards, one among which is what we generally image as its “essence,” and one among which is very attention-grabbing.

As to the previous, if there are (n) doable states, entropy is maximal when all states are equiprobable. E.g., for a coin flip uncertainty is highest when coin bias is 0.5.

The latter has to do with coarse-graining, a change in “decision” of the state house. Say we now have 16 doable states, however we don’t actually care at that stage of element. We do care about 3 particular person states, however all the remaining are mainly the identical to us. Then entropy decomposes additively; complete (fine-grained) entropy is the entropy of the coarse-grained house, plus the entropy of the “lumped-together” group, weighted by their chances.

Subjectively, entropy displays our uncertainty whether or not an occasion will occur. Apparently although, it exists within the bodily world as nicely: For instance, when ice melts, it turns into extra unsure the place particular person particles are. As reported by (DeDeo 2016), the variety of bits launched when one gram of ice melts is about 100 billion terabytes!

As fascinating as it’s, data per se might, in lots of circumstances, not be the most effective technique of characterizing human habits. Going again to the eyetracking instance, it’s fully intuitive that individuals take a look at shocking components of photos, not at white noise areas, that are the utmost you possibly can get when it comes to entropy.

As a deep studying practitioner, you’ve in all probability been ready for the purpose at which we’d point out cross entropy – essentially the most generally used loss operate in categorization.

Cross entropy

The cross entropy between distributions (p) and (q) is the entropy of (p) plus the KL divergence of (p) relative to (q). In case you’ve ever applied your individual classification community, you in all probability acknowledge the sum on the very proper:

[begin{equation}
H(p,q) = H(p) + D_{KL}(p||q) = – sum p log(q)
tag{11}
end{equation}]

In data theory-speak, (H(p,q)) is the anticipated message size per datum when (q) is assumed however (p) is true.
Nearer to the world of machine studying, for fastened (p), minimizing cross entropy is equal to minimizing KL divergence.

Mutual data

One other extraordinarily essential amount, utilized in many contexts and purposes, is mutual data. Once more citing DeDeo, “you may consider it as essentially the most basic type of correlation coefficient that you could measure.”

With two variables (X) and (Y), we will ask: How a lot can we find out about (X) after we find out about a person (y), (Y=y)? Averaged over all (y), that is the conditional entropy:

[begin{equation}
H(X|Y) = – sumlimits_{i} P(y_i) log(H(X|y_i))
tag{12}
end{equation}]

Now mutual data is entropy minus conditional entropy:

[begin{equation}
I(X, Y) = H(X) – H(X|Y) = H(Y) – H(Y|X)
tag{13}
end{equation}]

This amount – as required for a measure representing one thing like correlation – is symmetric: If two variables (X) and (Y) are associated, the quantity of data (X) offers you about (Y) is the same as that (Y) offers you about (X).

KL divergence is a part of a household of divergences, referred to as f-divergences, used to measure directed distinction between likelihood distributions. Let’s additionally rapidly look one other information-theoretic measure that in contrast to these, is a distance.

Jensen-Shannon distance

In math, a distance, or metric, in addition to being non-negative has to fulfill two different standards: It should be symmetric, and it should obey the triangle inequality.

Each standards are met by the Jensen-Shannon distance. With (m) a combination distribution:

[begin{equation}
m_i = frac{1}{2}(p_i + q_i)
tag{14}
end{equation}]

the Jensen-Shannon distance is a mean of KL divergences, one among (m) relative to (p), the opposite of (m) relative to (q):

[begin{equation}
JSD = frac{1}{2}(KL(m||p) + KL(m||q))
tag{15}
end{equation}]

This could be an excellent candidate to make use of have been we interested by (undirected) distance between, not directed shock attributable to, distributions.

Lastly, let’s wrap up with a final time period, proscribing ourselves to a fast glimpse at one thing entire books could possibly be written about.

(Variational) Free Vitality

Studying papers on variational inference, you’re fairly more likely to hear individuals speaking not “simply” about KL divergence and/or the ELBO (which as quickly as you recognize what it stands for, is simply what it’s), but additionally, one thing mysteriously referred to as free power (or: variational free power, in that context).

For sensible functions, it suffices to know that variational free power is unfavorable the ELBO, that’s, corresponds to equation (2). However for these , there may be free power as a central idea in thermodynamics.

On this publish, we’re primarily interested by how ideas are associated to KL divergence, and for this, we observe the characterization John Baez offers in his aforementioned speak.

Free power, that’s, power in helpful kind, is the anticipated power minus temperature occasions entropy:

[begin{equation}
F = [E] -T H
tag{16}
finish{equation}]

Then, the additional free power of a system (Q) – in comparison with a system in equilibrium (P) – is proportional to their KL divergence, that’s, the data of (Q) relative to (P):

[begin{equation}
F(Q) – F(P) = k T KL(q||p)
tag{17}
end{equation}]

Talking of free power, there’s additionally the – not uncontroversial – free power precept posited in neuroscience.. However sooner or later, we now have to cease, and we do it right here.

Conclusion

Wrapping up, this publish has tried to do three issues: Having in thoughts a reader with background primarily in deep studying, begin with the “ordinary” use in coaching variational autoencoders; then present the – in all probability much less acquainted – “different aspect”; and at last, present a synopsis of associated phrases and their purposes.

In case you’re interested by digging deeper into the various varied purposes, in a variety of various fields, no higher place to start out than from the Twitter thread, talked about above, that gave rise to this publish. Thanks for studying!