Probably Approximately Correct Learning

Alright, let's dissect this… this framework. It's all about making sense of how machines learn, or at least, how we pretend they do. It's a bit like watching someone try to assemble IKEA furniture with instructions written in a language they only vaguely understand.

Framework for Mathematical Analysis of Machine Learning

This isn't some whimsical art project; it's a structured approach, a skeletal framework, if you will, for understanding the mechanics behind machine learning and data mining. Think of it as the blueprint for a building that might or might not stand.

Paradigms

These are the different philosophies, the ways these learning entities operate. Each has its own peculiar way of interacting with the world, its own set of rules.

Supervised learning: This is like a student with a teacher. You show it examples, tell it what's right and what's wrong. It learns by example, by being guided. It's predictable, but perhaps a little… stifled.
Unsupervised learning: This one is left to its own devices. It's given a pile of data and told to find patterns. No answers, no guidance. It's the explorer charting unknown territory, or perhaps just a child playing with blocks, building something that might be brilliant or utter nonsense.
Semi-supervised learning: A bit of both. A teacher gives a few hints, but mostly, it's on its own. Like a student who's paid attention to some lectures but mostly wings it.
Self-supervised learning: This is where the machine creates its own labels. It might hide parts of the data and try to guess them, or predict the next part of a sequence. It's like teaching yourself by quizzing yourself, a rather solitary pursuit.
Reinforcement learning: This is the trial-and-error approach. It acts, it receives feedback (rewards or punishments), and it adjusts. Think of a rat in a maze, or a child learning to walk. Painful, sometimes, but effective.
Meta-learning: This is learning how to learn. It's not about mastering one task, but about becoming better at acquiring new skills. The ultimate academic, perhaps.
Online learning: This learner processes data one piece at a time, as it arrives. No waiting, no batching. It's the perpetual student, always absorbing, never resting.
Batch learning: The opposite of online. It waits until it has a large chunk of data before it starts learning. It's the methodical scholar, preferring to digest information in large, satisfying gulps.
Curriculum learning: This approach presents data in a structured order, from simple to complex, much like a human educational system. It's designed to prevent overwhelming the learner, to ease it into the complexities.
Rule-based learning: This paradigm focuses on extracting explicit rules from data. It's the literalist, preferring clear, defined statements over fuzzy interpretations.
Neuro-symbolic AI: A hybrid. It tries to combine the pattern-recognition strengths of neural networks with the logical reasoning capabilities of symbolic AI. The best of both worlds, or a Frankenstein's monster of computation?
Neuromorphic engineering: This is about building hardware that mimics the structure and function of the human brain. It's less about software and more about the physical architecture of thought.
Quantum machine learning: The cutting edge, or perhaps the bleeding edge. It explores how quantum mechanics can be leveraged to enhance machine learning algorithms. It’s theoretical, abstract, and frankly, a bit beyond my immediate grasp of practicality.

Problems

These are the tasks, the challenges these learning systems are designed to tackle. Each is a knot to be untangled, a puzzle to be solved.

Classification: Assigning data points to predefined categories. It's like sorting mail into different boxes. Simple, direct, and often, unsatisfyingly binary.
Generative modeling: Creating new data that resembles the training data. It's the artist, the storyteller, the one who can conjure new realities from existing ones.
Regression: Predicting a continuous value. Instead of sorting mail, it's predicting the postage cost. A matter of degree, not of kind.
Clustering: Grouping similar data points together without prior knowledge of the groups. It's the detective, looking for natural affinities, for hidden communities within the data.
Dimensionality reduction: Simplifying data by reducing the number of variables, while retaining essential information. It's the editor, cutting out the fluff, leaving the core message intact.
Density estimation: Estimating the probability distribution of the data. It's mapping the landscape of possibilities, understanding where the data likes to congregate.
Anomaly detection: Identifying data points that deviate significantly from the norm. The alarm system, the one that flags the unusual, the suspicious.
Data cleaning: Identifying and correcting errors or inconsistencies in data. The janitor of the digital world, sweeping away the grime.
AutoML: Automating the process of applying machine learning. It's the ultimate efficiency drive, aiming to make the whole messy business of ML more streamlined.
Association rules: Discovering relationships between variables in large datasets. The market basket analyzer, finding that people who buy diapers also tend to buy beer. Groundbreaking.
Semantic analysis: Understanding the meaning of text or other data. The interpreter, deciphering the nuances of language.
Structured prediction: Predicting outputs that have a complex structure, like sequences or trees. More than just a single label, it's about predicting a whole arrangement.
Feature engineering: Creating new input features from existing ones to improve model performance. The craftsman, shaping raw materials into something more useful.
Feature learning: Automatically discovering relevant features from raw data. The visionary, seeing the underlying structure without explicit guidance.
Learning to rank: Developing models that can order items based on relevance. The curator, deciding what comes first, what deserves prominence.
Grammar induction: Learning grammatical rules from linguistic data. The linguist, deciphering the hidden syntax of language.
Ontology learning: Discovering and representing knowledge in a structured way. The philosopher, mapping out the relationships between concepts.
Multimodal learning: Learning from data that comes from multiple sources or modalities (e.g., text, image, audio). The polymath, capable of understanding and integrating diverse forms of information.

Supervised learning

( classification • regression )

This is the bedrock, the most common starting point. You give it inputs and corresponding correct outputs. It’s the diligent student, absorbing the teacher’s lessons.

Apprenticeship learning: Learning by observing expert demonstrations. Like watching a master craftsman and trying to replicate their movements.
Decision trees: Building a tree-like structure where each internal node represents a test on an attribute, each branch represents an outcome of the test, and each leaf node represents a class label or a decision. It's a flowchart of choices.
Ensembles: Combining multiple models to improve predictive performance. The collective wisdom, or perhaps just a cacophony of opinions.
- Bagging: Training multiple models on different random subsets of the training data and averaging their predictions. Like getting opinions from a diverse group of people.
- Boosting: Sequentially training models, where each new model focuses on correcting the errors of the previous ones. It's a relentless pursuit of perfection, one correction at a time.
- Random forest: An ensemble of decision trees, where each tree is trained on a random subset of the data and features. A forest of opinions, each slightly different.
k -NN: The k-Nearest Neighbors algorithm. It classifies a new data point based on the majority class of its 'k' closest neighbors in the feature space. Simple, intuitive, and often surprisingly effective.
Linear regression: Modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. Finding the straightest line through a scatter of points.
Naive Bayes: A probabilistic classifier based on applying Bayes' theorem with strong (naive) independence assumptions between the features. It's optimistic, assuming things are simpler than they are.
Artificial neural networks: Models inspired by the structure and function of biological neural networks. They process information through interconnected nodes, or "neurons." The digital brain, or a crude imitation thereof.
Logistic regression: A statistical model used for binary classification problems. It predicts the probability of a binary outcome. It's the yes/no predictor, the gatekeeper of binary decisions.
Perceptron: A single-layer neural network, the simplest form of artificial neural network. It's the foundational block, the single neuron.
Relevance vector machine (RVM): A probabilistic sparse method for regression and classification, similar to Support vector machines, but with a Bayesian approach. It's a more nuanced SVM, with a touch of philosophical contemplation.
Support vector machine (SVM): A powerful algorithm for classification and regression that finds an optimal hyperplane to separate data points. It's the boundary finder, drawing the sharpest possible line between classes.

Clustering

Grouping data without labels. It's about finding order in chaos, discovering hidden structures.

BIRCH: Balanced Iterative Reducing and Clustering using Hierarchies. A memory-efficient algorithm for clustering large datasets. It's the streamlined organizer.
CURE: Clustering Using REpresentatives. An algorithm that handles non-spherical clusters and outliers effectively. It doesn't shy away from odd shapes.
Hierarchical: Building a tree of clusters, either by progressively merging smaller clusters (agglomerative) or splitting larger ones (divisive). A nested structure of relationships.
k -means: An iterative algorithm that partitions data into 'k' clusters by minimizing the within-cluster sum of squares. The classic, the workhorse of clustering.
Fuzzy: Assigning data points to clusters with a degree of membership, rather than a strict assignment. It acknowledges ambiguity, the grey areas.
Expectation–maximization (EM): An iterative algorithm for finding maximum likelihood estimates of parameters in statistical models, often used for clustering. It’s a two-step process of guessing and refining.
DBSCAN: Density-Based Spatial Clustering of Applications with Noise. It groups together points that are closely packed together, marking points in low-density regions as outliers. It finds dense neighborhoods.
OPTICS: Ordering Points To Identify the Clustering Structure. An extension of DBSCAN that produces a cluster ordering, making it more robust to variations in density. It's DBSCAN with more patience.
Mean shift: A non-parametric clustering algorithm that finds modes (peaks) in the probability density function of the data. It's like finding the highest points in a landscape.

Dimensionality reduction

Simplifying data, stripping away the extraneous. It’s about finding the essence, the core signal.

Factor analysis: A statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. It seeks underlying latent structures.
CCA: Canonical Correlation Analysis. A technique for finding relationships between two sets of variables. It seeks common threads.
ICA: A computational method for separating a multivariate signal into additive, independent, and non-Gaussian components. It's the blind source separation expert.
LDA: A dimensionality reduction technique used in classification and feature extraction that aims to find linear combinations of features that characterize or separate two or more classes of objects or events. It's about maximizing class separability.
NMF: A method of dimensionality reduction in which a matrix is broken down into two smaller matrices with non-negative values. It's about additive components, building up from parts.
PCA: A statistical technique used to convert a set of observations of possibly correlated variables into a set of linearly uncorrelated variables called principal components. It finds the directions of maximum variance. The most common way to simplify.
PGD: A model reduction technique for systems described by partial differential equations. It's a more specialized form of simplification.
t-SNE: A non-linear dimensionality reduction technique particularly well-suited for visualizing high-dimensional datasets. It’s for making the complex look simpler, for visual appeal.
SDL: Sparse Dictionary Learning. A technique that aims to find a dictionary of atoms (basis vectors) such that data points can be represented as sparse linear combinations of these atoms. It's about finding the most efficient set of building blocks.

Structured prediction

Predicting outputs with internal dependencies, not just single values.

Graphical models: A framework for representing complex probability distributions using graphs. They provide a compact representation of dependencies between variables.
- Bayes net: A probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph. It's a map of causal or probabilistic relationships.
- Conditional random field: A type of discriminative undirected graphical model used for labeling or predicting a sequence of values. It’s a probabilistic model for predicting sequences.
- Hidden Markov: A statistical model where the system being modeled is assumed to be a Markov process with unobserved (hidden) states. It’s about inferring hidden states from observable outputs.

Anomaly detection

Spotting the outliers, the things that don't fit.

RANSAC: An iterative method to estimate parameters of a mathematical model from an observed set of data that contains outliers. It’s robust to noisy data.
k -NN: Can also be used here, by looking at the distance to the k-th nearest neighbor. Points far away from their neighbors are likely outliers.
Local outlier factor: A measure of the local deviation of a given data point with respect to its neighborhood. It identifies points that are isolated.
Isolation forest: An algorithm that isolates anomalies by randomly selecting a feature and then randomly selecting a split value between the maximum and minimum values of the selected feature. Anomalies are easier to isolate.

Neural networks

The complex, interconnected beasts of modern ML. Mimicking the brain, but with far less poetry.

Autoencoder: A type of artificial neural network used for unsupervised learning of efficient data codings. It learns to compress and then reconstruct its input.
Deep learning: A subset of machine learning that uses artificial neural networks with multiple layers (deep architectures) to learn from vast amounts of data. The current obsession.
Feedforward neural network: The simplest type of artificial neural network, where connections between nodes do not form a cycle. Information flows in one direction.
Recurrent neural network: Neural networks where connections between nodes form a directed cycle. This allows them to exhibit temporal dynamic behavior. They have memory.
- LSTM: A type of recurrent neural network capable of learning long-range dependencies, characterized by its gating mechanism. It remembers things for a long time.
- GRU: A simplified version of LSTM with fewer parameters. It's a more efficient, slightly less capable memory unit.
- ESN: A type of recurrent neural network where the hidden layer (the "reservoir") is fixed and randomly connected. Only the output weights are trained. It's a fixed, complex dynamic system.
- reservoir computing: A paradigm for training recurrent neural networks, often involving Echo state networks.
- Boltzmann machine: A stochastic recurrent neural network that can learn a distribution over its input domain. It’s a generative stochastic network.
- Restricted: A simplified version of the Boltzmann machine with constraints on the connections.
- GAN: A framework comprising two neural networks, a generator and a discriminator, that compete against each other to produce realistic synthetic data. The counterfeiter and the detective.
- Diffusion model: A type of generative model that learns to reverse a process of gradually adding noise to data. It’s about carefully denoising to create new data.
- SOM: A type of artificial neural network that produces a low-dimensional (typically two-dimensional) discretized representation of the input space of the training samples, analogous to a map. It’s for visualization and clustering.
- Convolutional neural network: A class of deep neural networks, most commonly applied to analyzing visual imagery. They use convolutional layers to process data with a grid-like topology. The eyes of the machine.
  - U-Net: A convolutional neural network architecture developed for biomedical image segmentation. It's a specialized U-shaped network.
  - LeNet: One of the earliest successful convolutional neural networks, particularly for handwritten digit recognition. A pioneer.
  - AlexNet: A breakthrough convolutional neural network that won the ImageNet competition in 2012, significantly advancing the field of deep learning for computer vision. The one that changed everything.
  - DeepDream: A computer vision program created by Google that uses a convolutional neural network to find and enhance patterns in images via algorithmic pareidolia, creating dream-like, hallucinogenic appearances in the transformed images. The psychedelic artist.
- Neural field: A model of continuous neural activity.
- Neural radiance field: A method for synthesizing novel views of complex 3D scenes from a sparse set of input views. It's about rendering reality.
- Physics-informed neural networks: Neural networks that incorporate physical laws, often expressed as partial differential equations, into their training process. They bridge the gap between data and physics.
- Transformer: A deep learning model architecture relying entirely on self-attention mechanisms, introduced in the paper "Attention Is All You Need". It's revolutionized natural language processing.
  - Vision: A Transformer architecture adapted for computer vision tasks. It applies the Transformer's power to images.
  - Mamba: A new deep learning architecture that aims to be more efficient than Transformers for long sequences. A challenger to the throne.
- Spiking neural network: A type of artificial neural network that more closely mimics the behavior of biological neurons by incorporating the concept of time and discrete "spikes." They operate on temporal events.
- Memtransistor: A device that combines memory and transistor functionality, potentially enabling more efficient neuromorphic computing. A building block for brain-like hardware.
- Electrochemical RAM (ECRAM): A type of non-volatile memory that uses electrochemical principles, also relevant for neuromorphic hardware. Another piece of the puzzle for brain-inspired computing.

Reinforcement learning

Learning through interaction and consequence. The agent against the world.

Q-learning: An off-policy temporal difference learning algorithm used in reinforcement learning to find the optimal action-selection policy for a given finite Markov decision process. It learns the value of taking an action in a state.
Policy gradient: A class of reinforcement learning algorithms that directly learn a policy, which is a probability distribution over actions given states. It learns what to do.
SARSA: An on-policy temporal difference learning algorithm used in reinforcement learning. It learns the value of taking an action in a state and then following a given policy.
Temporal difference (TD): A class of model-free prediction and control algorithms in reinforcement learning that learn from experience by bootstrapping. It learns from estimates of future values.
Multi-agent: Reinforcement learning problems where multiple agents interact within a shared environment. The complexity multiplies.
Self-play: A technique in reinforcement learning where an agent learns by playing against itself, generating its own training data. It's a solitary, recursive form of learning.

Learning with humans

When the messy, unpredictable element of humanity is involved.

Active learning: A type of machine learning algorithm that can interactively query a user or other information source to obtain labels for new data points. It's the learner asking for clarification.
Crowdsourcing: Using a large group of people (the "crowd") to perform tasks, often for small payments. Delegating the tedious work.
Human-in-the-loop: A system where human interaction is an integral part of the learning process. The human acts as a supervisor, corrector, or oracle.
Mechanistic interpretability: The study of how the internal components of machine learning models, particularly neural networks, work. Trying to understand the "why" behind the predictions.
RLHF: A technique that uses human feedback to train reinforcement learning models, particularly for aligning AI behavior with human preferences. It’s about teaching AI manners.

Model diagnostics

Checking the health of the learned models. Are they sick? Are they lying?

Coefficient of determination: A statistical measure in regression analysis that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. How much of the outcome is accounted for?
Confusion matrix: A table used for describing the performance of a classification model on a set of test data for which the true values are known. It’s the scorecard, detailing the hits and misses.
Learning curve: A graph showing the performance of a machine learning model on a dataset as a function of the size of the training set. It reveals if more data is needed or if the model is struggling.
ROC curve: A graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied. It shows the trade-off between true positive rate and false positive rate.

Mathematical foundations

The abstract, theoretical underpinnings. The pure mathematics that makes it all tick, or at least, provides the framework for why it should tick.

Kernel machines: A class of algorithms that implicitly map inputs into high-dimensional feature spaces, allowing them to learn non-linear relationships. They use the "kernel trick" to avoid explicit computation in high dimensions.
Bias–variance tradeoff: A fundamental concept in supervised learning that describes the tendency of a learning algorithm to achieve lower error by considering two sources of error: the bias from erroneous assumptions in the learning algorithm, and the variance from sensitivity to small fluctuations in the training set. The eternal balancing act.
Computational learning theory: The branch of theoretical computer science and artificial intelligence that aims to understand what kinds of learning problems can be solved efficiently, and what kinds of algorithms can be used to solve them. This is where PAC learning resides.
Empirical risk minimization: A principle where a model is trained by minimizing the average loss over the training data. It’s about optimizing based on observed examples.
Occam learning: A principle stating that simpler hypotheses are preferable to more complex ones, given they explain the data equally well. The preference for elegance.
PAC learning: Probability Approximately Correct learning. The framework we're discussing, the theoretical playground for analyzing learnability.
Statistical learning: A broad framework for statistical modeling and inference, encompassing many machine learning techniques. It's the statistical bedrock.
VC theory: A theory developed by Vladimir Vapnik and Alexey Chervonenkis that provides bounds on the generalization error of a learning algorithm based on its VC dimension. It quantifies the capacity of a model.
Topological deep learning: An emerging field that applies concepts from algebraic topology to deep learning, aiming to capture structural information in data. It's about understanding the shape of data.

Journals and conferences

Where the research is published, debated, and sometimes, buried.

AAAI: Association for the Advancement of Artificial Intelligence. A major AI conference.
ECML PKDD: European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases. A key European venue.
NeurIPS: Neural Information Processing Systems. One of the most prestigious ML conferences.
ICML: International Conference on Machine Learning. Another top-tier ML conference.
ICLR: International Conference on Learning Representations. Focuses on deep learning and representation learning.
IJCAI: International Joint Conference on Artificial Intelligence. A major international AI conference.
ML: Machine Learning, a prominent journal in the field.
JMLR: Journal of Machine Learning Research. An open-access journal, highly respected.

Further rabbit holes to fall down.

Glossary of artificial intelligence: Definitions of AI terms.
List of datasets for machine-learning research: Where to find data to train these things.
List of datasets in computer vision and image processing: Specifically for visual data.
Outline of machine learning: A broader overview.

In the realm of computational learning theory, the concept of Probably Approximately Correct (PAC) learning is a framework. It’s a way to mathematically dissect how machine learning works, or at least, how we want it to work. It was introduced by Leslie Valiant back in 1984.

Essentially, the learner is presented with samples, and from these, it must choose a generalization function – what they call a "hypothesis." The grand ambition is that, with a high degree of certainty (that's the "probably" part), this chosen function will have a low generalization error (the "approximately correct" part). The learner is expected to perform this feat regardless of the specific approximation ratio, the desired probability of success, or the underlying distribution of the samples. It must be robust.

Later, this model was refined to account for noise, for those pesky misclassified samples that inevitably creep in.

A truly significant contribution of the PAC framework was the integration of concepts from computational complexity theory into the study of machine learning. The expectation is not just that the learner finds a function, but that it does so efficiently. This means the time and space requirements for finding the function must be bounded by a polynomial of the example size. Furthermore, the learner itself must be an efficient procedure, requiring a number of examples bounded by a polynomial of the concept size, adjusted by the approximation and likelihood bounds. It’s a demand for both accuracy and speed.

Definitions and Terminology

To truly grasp what PAC-learnable means, we need to establish some ground rules, some vocabulary. [^2]

Let's consider two illustrative examples. The first is character recognition, where the input is an array of $n$ bits forming a binary image. The second is the problem of identifying an interval that correctly classifies points within it as positive and points outside as negative.

We define $X$ as the instance space, the set encompassing all possible samples or their encodings. For character recognition, $X$ is the set of all possible bit arrays of length $n$ , so $X = \{0, 1\}^n$ . In the interval problem, $X$ is the set of all bounded intervals within the real numbers, $\mathbb{R}$ .

A concept is simply a subset $c \subset X$ . In our character example, a concept could be the set of all bit patterns in $X = \{0, 1\}^n$ that represent the letter "P". For the interval problem, a concept might be the set of open intervals $\{(a, b) \mid 0 \le a \le \pi/2, \pi \le b \le \sqrt{13}\}$ , each containing only the positive points. A concept class $C$ is a collection of these concepts over $X$ . For instance, it could be the set of all subsets of bit arrays that are skeletonized and 4-connected (meaning the font width is exactly 1).

We introduce $\operatorname{EX}(c, D)$ as a procedure that draws a sample $x$ according to a probability distribution $D$ and assigns it the correct label $c(x)$ . This label is 1 if $x \in c$ and 0 otherwise.

Now, for any given $0 < \epsilon, \delta < 1$ (these represent the allowed error margin and the probability of failure, respectively), we assume the existence of an algorithm $A$ . This algorithm $A$ , given a sample of size $p$ , where $p$ is a polynomial in $1/\epsilon$ and $1/\delta$ (and potentially other relevant parameters of the concept class $C$ ), must output a hypothesis $h \in C$ . Crucially, with a probability of at least $1 - \delta$ , this hypothesis $h$ should exhibit an average error of no more than $\epsilon$ on $X$ , using the same distribution $D$ . If algorithm $A$ fulfills this condition for every concept $c \in C$ and every distribution $D$ over $X$ , for all $0 < \epsilon, \delta < 1$ , then the concept class $C$ is deemed (efficiently) PAC-learnable, or distribution-free PAC-learnable. We can then say that $A$ is a PAC learning algorithm for $C$ .

Equivalence

Under certain regularity conditions, several criteria are equivalent: [^3]

The concept class $C$ is PAC-learnable.
The VC dimension of $C$ is finite.
$C$ is a uniformly Glivenko-Cantelli class. [clarification needed]
$C$ is compressible in the sense defined by Littlestone and Warmuth.