Amongst deep studying practitioners, *Kullback-Leibler divergence* (KL divergence) is maybe finest recognized for its position in coaching variational autoencoders (VAEs). To study an informative latent house, we don’t simply optimize for good reconstruction. Relatively, we additionally impose a previous on the latent distribution, and purpose to maintain them shut – usually, 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 submit, we’ll check out that different aspect.

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

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

The way more modest targets of this submit, then, are

- to rapidly recap the position of KL divergence in coaching VAEs, and point out similar-in-character purposes;
- as an example that extra playful, adventurous “different aspect” of its character; and
- in a not-so-entertaining, however – hopefully – helpful method, differentiate KL divergence from associated ideas similar 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 possibilities in keeping 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)) will not be 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 acquire*. We agree with one in every of our sources that due to its universality and significance, KL divergence would in all probability have deserved a extra informative identify; similar to, exactly, *data acquire*. (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*. Typically, for lifelike information, actual 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’ve approximation, we decrease – in principle, a minimum of – the KL divergence of (q) relative to (p^*), thus changing inference by optimization.

In observe, 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}]

Yet one more formulation – once more, see (Murphy 2012) for particulars – is the one we truly 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 submit 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, usually a multivariate regular).

### Past VAEs

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

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

## KL divergence, good man

In a second class of purposes, KL divergence will not be one thing to be minimized. In these domains, KL divergence is indicative of shock, disagreement, exploratory conduct, or studying: This actually is the attitude of *data acquire*.

### Shock

One area the place *shock*, not data per se, governs conduct 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 probably 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 means of Bayesian updating, a course of that might be used to explain the course of life itself, KL divergence itself turns into basic. We might get tempted to see it all over the 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 increased anticipated good points from betting towards the market.

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

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

### Asymmetry in motion

Trying once more on the KL system

[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 completely 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 need to be chosen in keeping with tractability (which distribution can we common over).

Secondly, the fraction contained in the (log) implies 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 usually is properly detailed in Murphy (2012). Within the context of shock, it implies that if I study one thing I used to suppose had chance zero, I shall be “infinitely stunned.”

To keep away from infinite shock, we are able to be certain our prior chance is rarely zero. However even then, the fascinating factor is that how a lot data we acquire 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 might … perhaps 1 % of them is black. Put in a different 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 data 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 way more undecided earlier than; say I’d have thought the chances had been 50:50.

On seeing a black swan, I get so much 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, by way of shock and studying, is inspiring – it may lead one to seeing it in motion all over the place. Nevertheless, we nonetheless have the third and closing 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 typical log chance 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 in every of which is what we generally image as its “essence,” and one in every of which is particularly fascinating.

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’ve 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 remainder 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 possibilities.

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 one of the best technique of characterizing human conduct. Going again to the eyetracking instance, it’s utterly intuitive that folks have a look at shocking elements of photographs, not at white noise areas, that are the utmost you possibly can get by way of entropy.

As a deep studying practitioner, you’ve in all probability been ready for the purpose at which we’d point out *cross entropy* – probably the most generally used loss perform 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 personal 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 mounted (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’ll be able to consider it as probably the most basic type of correlation coefficient you can measure.”

With two variables (X) and (Y), we are able to 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 knowledge (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 chance 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*, moreover 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 mix 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 in every of (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 may be a super candidate to make use of had been we fascinated about (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 complete books might be written about.

### (Variational) Free Power

Studying papers on variational inference, you’re fairly more likely to hear folks speaking not “simply” about KL divergence and/or the *ELBO* (which as quickly as you realize what it stands for, is simply what it’s), but in addition, 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’s *free power* as a central idea in thermodynamics.

On this submit, we’re primarily fascinated about how ideas are associated to KL divergence, and for this, we comply with the characterization John Baez offers in his aforementioned talk.

*Free* power, that’s, power in helpful kind, is the anticipated power minus temperature instances 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 knowledge 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 energy principle posited in neuroscience.. However in some unspecified time in the future, we’ve to cease, and we do it right here.

## Conclusion

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

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

DeDeo, Simon. 2016. “Info Idea for Clever Individuals.”

*Nature Critiques. Neuroscience*11 (February): 127–38. https://doi.org/10.1038/nrn2787.

*Advances in Neural Info Processing Methods 18 [Neural Information Processing Systems, NIPS 2005, December 5-8, 2005, Vancouver, British Columbia, Canada]*, 547–54. http://papers.nips.cc/paper/2822-bayesian-surprise-attracts-human-attention.

Murphy, Kevin. 2012. *Machine Studying: A Probabilistic Perspective*. MIT Press.

*CoRR*abs/1103.5708. http://arxiv.org/abs/1103.5708.

Zanardo, Enrico. 2017. “HOW TO MEASURE DISAGREEMENT ?” In.