CS250: Python for Data Science

Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is an important mode of learning in people, the machine is forced to build a compact internal representation of its world and then generate imaginative content from it. In contrast to supervised learning, where data is tagged by an expert, e.g. as a "ball" or "fish", unsupervised methods exhibit self-organization that captures patterns as probability densities or a combination of neural feature preferences. The other levels in the supervision spectrum are reinforcement learning, where the machine is given only a numerical performance score as guidance, and semi-supervised learning, where a smaller portion of the data is tagged.

Two broad methods in unsupervised learning are neural networks and probabilistic methods.

Neural networks

Tasks vs. methods

Tendency for a task to employ Supervised vs. Unsupervised methods. Task names straddling circle boundaries is intentional. It shows that the classical division of imaginative tasks (left) employing unsupervised methods is blurred in today's learning schemes.

Neural network tasks are often categorized as discriminative (recognition) or generative (imagination). Often but not always, discriminative tasks use supervised methods, and generative tasks use unsupervised ones. However, the separation is very hazy. For example, object recognition favors supervised learning, but unsupervised learning can also cluster objects into groups. Furthermore, as progress marches onward, some tasks employ both methods, and some tasks swing from one to another. For example, image recognition started off as heavily supervised but became a hybrid by employing unsupervised pre-training and then moved towards supervision again with the advent of dropout, relu, and adaptive learning rates.

Training

During the learning phase, an unsupervised network tries to mimic the data it's given and uses the error in its mimicked output to correct itself (ie. correct its weights & biases). Sometimes the error is expressed as a low probability that the erroneous output occurs, or it might be expressed as an unstable high-energy state in the network.

In contrast to the supervised method's dominant use of backpropagation, unsupervised learning also employs other methods, including Hopfield learning rule, Boltzmann learning rule, Contrastive Divergence, Wake Sleep, Variational Inference, Maximum Likelihood, Maximum A Posteriori, Gibbs Sampling, and backpropagating reconstruction errors or hidden state reparameterizations. See the table below for more details.

Energy

An energy function is a macroscopic measure of a network's activation state. In Boltzmann machines, it plays the role of the Cost function. This analogy with physics is inspired by Ludwig Boltzmann's analysis of a gas' macroscopic energy from the microscopic probabilities of particle motion $p \propto e^{E/kT}$, where k is the Boltzmann constant and T is temperature. In the RBM network the relation is $p = e−E / Z$, where p & E vary over every possible activation pattern and $\displaystyle \sum _{All Patterns} e ^{-E(pattern)}]$. To be more precise, $p(a) = e-E(a) / Z$, where a is an activation pattern of all neurons (visible and hidden). Hence, early neural networks bear the name Boltzmann Machine. Paul Smolensky calls -E the Harmony. A network seeks low energy, which is high Harmony.

Networks

This table shows connection diagrams of various unsupervised networks, the details of which will be given in the section Comparison of Network. Circles are neurons, and the edges between them are connection weights. As network design changes, features are added to enable new capabilities or removed to make learning faster. For instance, neurons changes between deterministic (Hopfield) and stochastic (Boltzmann) to allow robust output, weights are removed within a layer (RBM) to hasten learning, or connections are allowed to become asymmetric (Helmholtz).

Hopfield		A network based on magnetic domains in iron with a single self-connected layer. It can be used as a content addressable memory.
Boltzmann		Network is separated into 2 layers (hidden vs. visible), but still using symmetric 2-way weights. Following Boltzmann's thermodynamics, individual probabilities give rise to macroscopic energies.
RBM		Restricted Boltzmann Machine. This is a Boltzmann machine where lateral connections within a layer are prohibited to make analysis tractable.
Stacked Boltzmann		This network has multiple RBMs to encode a hierarchy of hidden features. After a single RBM is trained, another blue hidden layer (see left RBM) is added, and the top 2 layers are trained as a red & blue RBM. Thus the middle layers of an RBM act as hidden or visible, depending on the training phase it's in.
Helmholtz		Instead of the bidirectional symmetric connection of the stacked Boltzmann machines, we have separate one-way connections to form a loop. It does both generation and discrimination.
Autoencoder		A feed-forward network that aims to find a good middle layer representation of its input world. This network is deterministic, so it's not as robust as its successor, the VAE.
VAE		Applies Variational Inference to the Autoencoder. The middle layer is a set of means & variances for Gaussian distributions. The stochastic nature allows for more robust imagination than the deterministic autoencoder.

Of the networks bearing people's names, only Hopfield worked directly with neural networks. Boltzmann and Helmholtz came before artificial neural networks, but their work in physics and physiology inspired the analytical methods that were used.

History

Specific Networks

Hopfield Network

Ferromagnetism inspired Hopfield networks. A neuron corresponds to an iron domain with binary magnetic moments Up and Down, and neural connections correspond to the domain's influence on each other. Symmetric connections enable a global energy formulation. During inference, the network updates each state using the standard activation step function. Symmetric weights and the right energy functions guarantee convergence to a stable activation pattern. Asymmetric weights are difficult to analyze. Hopfield nets are used as Content Addressable Memories (CAM).

Boltzmann Machine

These are stochastic Hopfield nets. Their state value is sampled from this pdf as follows: suppose a binary neuron fires with the Bernoulli probability p(1) = 1/3 and rests with p(0) = 2/3. One samples from it by taking a UNIFORMLY distributed random number y and plugging it into the inverted cumulative distribution function, which in this case is the step function thresholded at 2/3. The inverse function = { 0 if x <= 2/3, 1 if x > 2/3 }

Sigmoid Belief Net

Introduced by Radford Neal in 1992, this network applies ideas from probabilistic graphical models to neural networks. A key difference is that nodes in graphical models have pre-assigned meanings, whereas Belief Net neurons' features are determined after training. The network is a sparsely connected directed acyclic graph composed of binary stochastic neurons. The learning rule comes from Maximum Likelihood on p(X): Δw_ij ∝ s_j * (s_i - p_i), where p_i = 1 / ( 1 + e^{weighted inputs into neuron i} ). s_js are activations from an unbiased sample of the posterior distribution, and this is problematic due to the Explaining Away problem raised by Judea Perl. Variational Bayesian methods use a surrogate posterior and blatantly disregard this complexity.

Deep Belief Network

Introduced by Hinton, this network is a hybrid of RBM and Sigmoid Belief Network. The top 2 layers are an RBM, and the second layer, downwards, forms a sigmoid belief network. One trains it by the stacked RBM method and then throws away the recognition weights below the top RBM. As of 2009, 3-4 layers seem to be the optimal depth.

Helmholtz machine

These are early inspirations for the Variational Auto Encoders. It is 2 networks combined into one - forward weights operate recognition, and backward weights implement imagination. It is perhaps the first network to do both. Helmholtz did not work in machine learning, but he inspired the view of a "statistical inference engine whose function is to infer probable causes of sensory input" (3). the stochastic binary neuron outputs a probability that its state is 0 or 1. The data input is normally not considered a layer, but in the Helmholtz machine generation mode, the data layer receives input from the middle layer and has separate weights for this purpose, so it is considered a layer. Hence this network has 3 layers.

Variational autoencoder

Comparison of networks

Hebbian Learning, ART, SOM

Among neural network models, the self-organizing map (SOM) and adaptive resonance theory (ART) are commonly used in unsupervised learning algorithms. The SOM is a topographic organization in which nearby locations on the map represent inputs with similar properties. The ART model allows the number of clusters to vary with problem size and lets the user control the degree of similarity between members of the same clusters by means of a user-defined constant called the vigilance parameter. ART networks are used for many pattern recognition tasks, such as automatic target recognition and seismic signal processing.

Probabilistic methods

A central application of unsupervised learning is in the field of density estimation in statistics, though unsupervised learning encompasses many other domains involving summarizing and explaining data features. It can be contrasted with supervised learning by saying that whereas supervised learning intends to infer a conditional probability distribution conditioned on the label of input data, unsupervised learning intends to infer an a priori probability distribution.

Approaches

• Clustering methods include: hierarchical clustering, k-means, mixture models, DBSCAN, and OPTICS algorithm
• Anomaly detection methods include: Local Outlier Factor and Isolation Forest
• Approaches for learning latent variable models such as Expectation–maximization algorithm (EM), Method of moments, and Blind signal separation techniques (Principal component analysis, Independent component analysis, Non-negative matrix factorization, Singular value decomposition)

Method of moments

Source: Wikipedia, https://en.wikipedia.org/wiki/Unsupervised_learning
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