Among deep learning practitioners, the Kullback-Leibler divergence may be best known for its role in training variational autoencoders (VAEs). To effectively analyze a latent house that conveys valuable information, we must consider factors beyond mere reconstruction quality. Moderately, we also impose a prior on the latent distribution, aiming to keep it closed – typically achieved by minimizing the Kullback-Leibler (KL) divergence.
The KL divergence functions as a vigilant sentinel, imposing strict bounds and regularization constraints, much like a stern and exacting authority figure if personified. If we scrutinize the situation closely, though, we’re overlooking another crucial aspect of this entity’s nature – a dichotomy comprising lightheartedness, exploration, and inquisitiveness that remains unseen if we only consider one facet. Let’s explore that different aspect.
As I was struck by the versatility of KL divergence, as illustrated by Simon de Deo’s insightful tweet series, which showcased its multifaceted applications across various fields.
Without intending to provide a comprehensive overview here – as previously discussed in our initial tweet, this topic has the potential to span an entire semester’s worth of study.
The primary, more modest objectives of this installation, therefore, are
- To summarize the role of KL divergence in training Variational Autoencoders (VAEs), its primary function is to encourage the learning process by regulating the trade-off between reconstruction accuracy and disentanglement of latent representations.
- as a playful, adventurous undertone that adds yet another captivating dimension to its personality.
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In a straightforward yet informative manner, distinguish KL divergence from related concepts such as cross-entropy, mutual information, and Kullback-Leibler energy.
Although earlier definitions may have existed, let us start by clarifying key terms to establish a solid foundation for our discussion.
KL divergence in a nutshell
The Kullback-Leibler (KL) divergence measures the expected value of the log-likelihood ratio between two probability distributions. The probabilistic model that captures uncertainty is indeed situated here in its discrete-probabilities variant.
However, This nuance being considered, that distinction between and is what makes this aspect so crucial for part two, which delves into the “other side.”
To highlight the disparity between these two distributions, the KL divergence is often referred to as the (or ) – a notion that underscores the difference between the data and its counterpart. Given the universality and significance of KL divergence, it is likely that an alternative title would be more informative if it were to reflect its importance. Is significantly clearer in its enunciation.
KL divergence, “villain”
In many machine learning algorithms, KL divergence appears within the framework of probability theory. Typically, calculating the exact posterior distribution is impractical when dealing with real-world data. Therefore, a certain level of approximation becomes necessary. During variational inference, the true posterior is approximated by a more manageable probability distribution, often drawn from a tractable family of distributions.
To accurately estimate the difference between two probability distributions, we reduce the Kullback-Leibler (KL) divergence from to , thereby modifying inference through optimization.
While observing the KL divergence minimization for intractable causes, the minimized version relates to an unnormalized distribution?
The place is where the joint distribution of parameters and information converges.
Is the true probability density of the unobserved variables given the observed data.
Equal to that formulation (eq. To minimize the negative log-likelihood (NLL), thereby achieving a more favourable optimization goal?
One additional formulation, explored in further detail elsewhere, is actually employed when training VAEs, for instance. The total loss consists of the expected negative log-likelihood (NLL) and the Kullback-Leibler (KL) divergence between the approximated distribution and the target or imposed distribution.
Negated, this formulation can be referred to as the logical fallacy, for it relies on an invalid assumption. The evidence lower bound objective within the variational autoencoder model under consideration was formulated.
with denoting the latent variables, representing the approximation and the prior distribution, typically a multivariate normal.
Past VAEs
By extrapolating the conservative motion of KL divergence within the context of VAEs, we can infer that this expression embodies a benchmark for approximation quality. A crucial aspect where approximations occur is in lossy compression. The Kullback-Leibler (KL) divergence offers a means of quantifying the extent to which data is misallocated when compressing information.
To ensure a robust model performance, it’s crucial that KL divergence remains low in these applications and related contexts, implying that we do not desire it to be zero, as this would render the algorithm ineffective; rather, our goal is to minimize its value. Let’s examine the other side of the coin.
KL divergence, good man
In secondary objectives, KL divergence does not serve as something to be minimized? Within the realms of information science, KL divergence serves as a metric for detecting shocks, disagreements, exploratory behaviors, and learning patterns.
Shock
While physical surroundings may influence habits, it’s notion that truly shapes our routines and behaviors. Studies employing eye-tracking technology have substantiated the notion that shock, quantifiable through KL divergence, exerts a stronger influence on visual attention than data, measurable via entropy. While “Bayesian shock” may have gained popularity in research circles, its constituent parts fail to convey meaningful information, lacking specificity and relevance. The extent to which the prior distribution differs from the posterior distribution measures the impact of new information on the original belief – this contrast is a fundamental aspect of Bayesian updating.
As the KL divergence is intricately linked to shock, which in turn finds its roots in the fundamental principle of Bayesian updating, it’s possible to construct a framework that might shed light on the trajectory of human existence itself – thereby elevating KL divergence to an elementary concept. Won’t we find ourselves constantly drawn to its presence? As a result, Nei’s genetic distance has been widely applied across various disciplines to measure uni-directional divergence.
By applying this concept, I’ve effectively utilized its potential in both buying and selling, gauging just how far an individual’s opinion diverges from that of the broader market. As the magnitude of divergence grows, so too do expectations of significant returns for contrarian bets against prevailing sentiment.
In the realm of deep learning, this technique is employed in intrinsic motivated reinforcement learning, where an optimal policy should aim to maximize the long-term acquisition of information. As a natural consequence of entropy’s properties, KL divergence exhibits additivity.
While its asymmetry is inherently linked to the use of KL divergence for regularization (as discussed in Part 1) or shock, this becomes strikingly apparent when employed for analysis and prediction.
Asymmetry in motion
What is driving your need to revisit the KL components again?
The roles of a CEO and an entrepreneur are inherently distinct. The expectation is computed over the primary distribution (i.e., within). The significance of this aspect lies in its ability to ensure a harmonious “order” between different elements, which necessitates selecting compatible “roles.” The tractability of the system depends on the distribution that can be commonly agreed upon.
Secondly, the constraint inherent in the expression suggests that if x is ever zero at any point, then y cannot be, as the KL divergence will “blow up.” This implication has significant implications for distribution estimation, which are thoroughly discussed in [insert reference]. In the realm of shock, it’s suggested that discovering even a single phenomenon previously thought to have zero probability can lead to an infinite sense of astonishment.
To preclude the prospect of infinite shocks, we must ensure that our prior probabilities are never zero. Despite this, the crucial factor driving our ability to accumulate vast amounts of data at any given time remains. Let’s see a easy instance.
It’s likely I’ve never observed a black swan, but there’s a minute chance – roughly one percent – that some black swans might actually exist. I must admit that, upon reconsidering my initial notion of a swan, the likelihood of encountering one with predominantly black plumage still appears quite remote.
As I honestly confess, I chanced upon a single crow, its plumage as dark as the night.
The knowledge I’ve gained is:
Originally, I would have entertained considerable doubt and uncertainty about the outcome, potentially weighing the likelihood at an even split of 50% to 50%.
On encountering a black swan, my access to data is significantly curtailed.
As a result of contemplating the concept of KL divergence through introspection and exploration, a profound realization emerges – one that has the potential to reveal this phenomenon’s omnipresence in everyday life. Notwithstanding this, our ultimate task remains: to swiftly assess KL divergence among distinct concepts within that realm.
Entropy
Entropy, or information, as formulated by Claude Shannon, all begins with.
Entropy is the logarithmic probability average of a statistical distribution.
This formulation was designed to meet four criteria, including the “essence” of what we commonly perceive and another that is quite captivating.
When all possible states have equal probability, entropy is maximized. At the point where coin bias equals 0.5, the uncertainty surrounding the outcome of a coin flip reaches its peak.
The recent development regarding the state house’s alteration in decision is reportedly tied to a specific aspect. We currently consider 16 possible states, although the level of granularity isn’t crucial at this stage. While we take a special interest in three specific individual states, the remainder of the entities share a striking similarity with our own. Entropy decomposition occurs additively, with total entropy consisting of the entropy of the system’s macrostates, summed with the entropy of the microstates, proportionally weighted by their probabilities.
Uncertainty surrounding the likelihood of a particular occurrence is exemplified by entropy’s subjective manifestation. Intriguingly, this concept also manifests in the physical realm: Take, for instance, the phenomenon of melting ice, which becomes increasingly uncertain about the whereabouts of individual particles as it transitions from a solid to a liquid state. When a gram of ice melts, approximately 100 billion terabytes of energy are released.
While data may seem intriguing, it’s crucial to recognize that it often falls short of being the most effective method for capturing the intricacies of human behavior. When revisiting the eyetracking instance, it’s logically expected that people tend to gaze at visually striking elements within images rather than focusing on uniform white noise areas, which represent the pinnacle of randomness.
As a seasoned practitioner of deep learning, you’re likely well-prepared to tackle the fundamental topic where we’ll focus on – arguably the most widely utilized loss function in classification tasks.
Cross entropy
The cross-entropy between distributions p and q is the entropy of p plus the Kullback-Leibler (KL) divergence of q relative to p. Whenever you’ve conducted a personal classification project, you probably recognize the value in being precise.
Given the probability of occurrence of a symbol in a stochastic process, the expected value of the number of symbols that make up a message of length n is denoted as E[n], where the message contains exactly one occurrence of each possible symbol.
In the realm of machine learning, minimizing cross-entropy loss is equivalent to minimizing Kullback-Leibler (KL) divergence.
Mutual data
One fundamental aspect, used extensively across various spheres and applications, is water. Notably, DeDeo suggests considering Pearson’s r as the most prevalent and measurable type of correlation coefficient.
By learning about a person, we can infer considerable information about , which ultimately depends on the nature of the relationship between and a specific individual. On average, that is what you can expect.
Mutual information between two random variables is formally defined as the difference between their joint entropy and the sum of their individual entropies.
This symmetry – characteristic of measures defining a relationship between two variables, such as correlation – suggests that if two variables are linked, the amount of information provided by one variable’s data about the other is equivalent to the amount of information provided by the other variable’s data about the first.
KL divergence is a fundamental member of the family of divergences, commonly employed to quantify the directed difference between probability distributions. This other information-theoretic measure, which contrasts with entropy and mutual information, is the **Kolmogorov complexity**.
Jensen-Shannon distance
In mathematics, a metric or distance function, denoted by ⍭ or d, must meet two fundamental criteria in addition to being non-negative. Firstly, it must satisfy symmetry, implying that the distance between two points is the same regardless of their order. Secondly, it must adhere to the triangle inequality, which states that the sum of any two sides of a triangle must be greater than or equal to its third side.
Standards are consistently met with precision. With a mix distribution:
The Jensen-Shannon distance is a mean of KL divergences, one calculated relative to p, and the other calculated relative to q:
We were intrigued by the possibility of exploring undirected distances between, rather than directed shocks from, the underlying distributions.
Let’s distill the essence of our journey into a concise, definitive statement.
(Variational) Free Power
While studying variational inference, you may frequently hear individuals discussing not only KL divergence and its well-known notation, but also something enigmatically referred to as the “Evidence Lower Bound” or ELBO.
For well-behaved functions, it is sufficient to recognize that minimizing the expected log-likelihood of evidence, or ELBO, which corresponds to Equation 1. However, at the heart of these concepts lies a central idea.
We’re primarily interested in exploring the connection between ideas and KL divergence, as characterized by John Baez in his preceding work.
Vitality, indeed, refers to a concept of useful energy, which can be described as the expected vitality minus the temperature times entropy.
Then, the excess free energy of a system – compared to one at equilibrium – is directly proportional to its Kullback-Leibler divergence from the equilibrium distribution.
While discussing unbridled energy, another concept gaining traction in neuroscientific circles is the notion that… At an unspecified point in the future, our work comes to a close – and we’re doing it right here, today.
Conclusion
To wrap up, this overview has endeavored to tackle three distinct tasks: conceptualizing a reader with a background primarily rooted in deep learning, starting with the “pedestrian” application of variational autoencoders; next, introducing the potentially lesser-known “other side”; finally, providing a concise summary of related terminology and their functionalities.
What drives inquiry into diverse applications across multiple disciplines? It’s best to start with the foundation laid out earlier, which has led to this exploration. Thanks for studying!
DeDeo, Simon. 2016.
Murphy, Kevin. 2012. . MIT Press.
Zanardo, Enrico. 2017. In.
