Oja's rule

Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

Optical granulometry

Optical granulometry is the process of measuring the different grain sizes in a granular material, based on a photograph. Technology has been created to analyze a photograph and create statistics based on what the picture portrays. This information is vital in maintaining machinery in various trades worldwide. Mining companies can use optical granulometry to analyze inactive or moving rock to quantify the size of these fragments. Forestry companies can zero in on wood chip sizes without stopping the production process, and minimize sizing errors. With more photoanalysis technologies being produced, mining companies have shown an increased interest in these types of systems because of their ability to maintain efficiency throughout the mining process. Companies are saving millions of dollars annually because of this new technology, and are cutting back on maintenance costs on equipment. In order for optical granulometry to be completely successful, an accurate photo must be taken – under sufficient lighting, and using proper technology – to obtain quantified results. If these requirements are met, an image analysis system can be implemented. == The process == Software uses four basic steps in determining the average size of material: See the Wikipedia article on Photoanalysis to see how mining, forestry and agricultural companies are using this technology to improve quality control techniques. == Smartphone-based, segmentation-free estimation of grain size distribution == Recently, a methodology has emerged by which soil grain size distribution can be inferred from optical images acquired with commodity smartphones by training convolutional neural networks to predict parameters of the distribution curve directly from the image, without explicit image segmentation . In this approach, a standardized image of a soil surface is captured under controlled conditions, preprocessed to reduce device-specific variability, and passed to a regression model that outputs the parameters of a cumulative distribution function e.g., a two-parameter Weibull curve. The resulting distribution can be used to derive geotechnical descriptors and class boundaries.

Babelfy

Babelfy is a software algorithm for the disambiguation of text written in any language. It performs the tasks of multilingual Word Sense Disambiguation (i.e., the disambiguation of common nouns, verbs, adjectives and adverbs) and Entity Linking (i.e. the disambiguation of mentions to encyclopedic entities like people, companies, places, etc.). == Overview == Babelfy uses the BabelNet multilingual knowledge graph to perform disambiguation and entity linking in three steps: It associates with each vertex of the BabelNet semantic network, i.e., either concept or named entity, a semantic signature, that is, a set of related vertices. This is a preliminary step which needs to be performed only once, independently of the input text. Given an input text, it extracts all the linkable fragments from this text and, for each of them, lists the possible meanings according to the semantic network. It creates a graph-based semantic interpretation of the whole text by linking the candidate meanings of the extracted fragments using the previously computed semantic signatures. It then extracts a dense subgraph of this representation and selects the best candidate meaning for each fragment. As a result, the text, written in any of the 271 languages supported by BabelNet, is output with possibly overlapping semantic annotations.

Botler AI

Botler AI is a Montreal-based Canadian Artificial Intelligence company that helps users navigate the legal system. Launched in 2017 by Amir Morv and Ritika Dutt, Botler offers a free online tool which provides users who are unaware of their legal rights with information and guidance. Botler is known for its role in unveiling misconduct in the Government of Canada's procurement practices. Botler's findings have prompted numerous investigations, including by the Royal Canadian Mounted Police. == History == Botler's first AI was trained on over 300,000 U.S. and Canadian legal documents to help individuals identify and enforce their legal rights, without fear of judgment. Launched during the height of the #MeToo movement, the tool initially focused on sexual harassment with a goal of creating "a general artificial intelligence that would help the average person with any legal issue." === Department of Justice Canada === In 2020, Botler launched an expanded misconduct detection system in the form of an anonymous chatbot which provided users with an explanation of the law and relevant resources. In March 2021, the Minister of Justice and Attorney General of Canada announced the Government of Canada's support for Botler AI to assist complainants of sexual harassment in the workplace. The initiative, entitled Botler for Citizens and implemented with the support of the Department of Justice Canada, established an Artificial Intelligence-powered hybrid legal service delivery model. == Notable cases == On October 4, 2023, the RCMP confirmed to The Globe and Mail that they "are investigating a file referred from the CBSA (Canada Border Services Agency) that is based on allegations brought to their attention by Botler". In 2019, GCStrategies's managing partner, Kristian Firth, reached out to Botler on behalf of his client, the CBSA, to solicit their misconduct detection chatbot. After interactions with GCStrategies, Dalian Enterprises and Coradix Technology Consulting, the three main contractors involved in developing the controversial ArriveCAN app, Dutt and Morv alerted the CBSA to questionable contracting practices in federal government procurement in September, 2021, and again in November, 2022. In response to Botler's November 2022 report, the CBSA launched an internal review and referred the matter to the RCMP. During testimony before a parliamentary committee, the CBSA's President stated that the CBSA investigation to date has raised some concerns and shows "that there was a pattern of persistent collaboration between certain officials and GCStrategies... to circumvent or ignore certain established processes and roles and responsibilities". The Auditor General of Canada, which extended its study into ArriveCAN following the Botler revelations, found that GCStrategies was directly involved in setting narrow terms for a request for proposal for a $25-million government contract it ultimately won. The firm, which has just two employees, charges the government a commission of between 15 per cent and 30 per cent of each contract's value. The Office of the Procurement Ombudsman of Canada found "numerous examples" where GCStrategies "had simply copied and pasted" the required work experience to meet contracting requirements. To date, more than a dozen probes have been launched into the matter, including by the government, parliamentary committees, independent watchdogs and law-enforcement agencies. On April 17, 2024, GCStrategies' Firth was the first person summoned in over a century to answer questions before Members of Parliament in the House of Commons. During his appearance, Firth testified that the RCMP had raided "my property to obtain electronic goods surrounding the Botler allegations". === Government of Canada Reforms === One day after The Globe reported that the RCMP is investigating allegations of misconduct, the federal government responded by announcing new guidelines from the Treasury Board of Canada aimed at cutting back on the use of private consultants and that outsourcing contracts were under examination. Public Services and Procurement Canada (PSPC) invalidated and replaced all master level user agreements with government client departments in November 2023. The agreements set out the conditions for access to select Professional Services methods of supply which are used for outsourcing. In March 2024, PSPC announced its suspension of the respective security statuses of GCStrategies, Dalian and Coradix, barring them from participating in all federal procurements. Records show that the total value of contracts awarded to the three companies amounts to more than $1 Billion.

Cristóbal Valenzuela

Cristóbal Valenzuela (born 1989) is a Chilean-born technologist, software developer, and CEO of Runway. In 2018, Valenzuela co-founded the AI research company Runway in New York City with Anastasis Germanidis and Alejandro Matamala. == Education == Valenzuela graduated from Adolfo Ibáñez University (AIU), a research private university in Chile. From there, Valenzuela obtained a bachelor's degree in economics and business management, along with a master's degree in arts in design in 2012. In 2018, he graduated with a media arts degree from ITP NYU's Tisch School of the Arts. == Career and recognition == One of Valenzuela's first jobs was as a teaching and research assistant at the Adolfo Ibáñez University School of Design, and later an adjunct professor in the same department. In 2018, he became a researcher at NYU's Tisch School of the Arts ITP program, where he worked with Daniel Shiffman. He contributes to open-source software projects, including ml5.js, an open-source machine learning software. He co-founded Runway with two colleagues from ITP, Anastasis Germanidis, and Alejandro Matamala. The goal of Runway is to create new tools for human imagination using generative AI. In recent years, Valenzuela's work has been sponsored by Google and the Processing Foundation and his projects have been exhibited throughout Latin America and the US, including the Santiago Museum of Contemporary Art, Lollapalooza, NYC Media Lab, New Latin Wave, Inter-American Development Bank, Stanford University and New York University. In September 2023, Valenzuela was named as one of the TIME 100 Most Influential People in AI (TIME100 AI).

ChatGPT

ChatGPT is a generative artificial intelligence chatbot developed by OpenAI. Originally released in November 2022, the product uses large language models—specifically generative pre-trained transformers (GPTs)—to generate text, speech, and images in response to user prompts. ChatGPT accelerated the AI boom, an ongoing period marked by rapid investment and public attention toward the field of artificial intelligence (AI). OpenAI operates the service on a freemium model. Users can interact with ChatGPT through text, audio, and image prompts. ChatGPT was quickly adopted, reaching 100 million monthly active users two months after its release and 900 million weekly active users in February 2026. It has been lauded for its potential to transform numerous professional fields, and has instigated public debate about the nature of creativity and the future of knowledge work. The chatbot has also been criticized for its limitations and potential for unethical use. It can generate plausible-sounding but incorrect or nonsensical answers, known as hallucinations. Biases in its training data have been reflected in its responses. The chatbot can facilitate academic dishonesty, generate misinformation, and create malicious code. The ethics of its development, particularly the use of copyrighted content as training data, have also drawn controversy. == Features == ChatGPT is a chatbot and AI assistant built on large language model (LLM) technology. It is designed to generate human-like text and can carry out a wide variety of tasks. These include, among many others, writing and debugging computer programs, composing music, scripts, fairy tales, and essays, answering questions (sometimes at a level exceeding that of an average human test-taker), and generating business concepts. ChatGPT is frequently used for translation and summarization tasks, and can simulate interactive environments such as a Linux terminal, a multi-user chat room, or simple text-based games such as tic-tac-toe. Users interact with ChatGPT through conversations which consist of text, audio, and image inputs and outputs. The user's inputs to these conversations are referred to as prompts. An optional "Memory" feature allows users to tell ChatGPT to memorize specific information. Another option allows ChatGPT to recall old conversations. GPT-based moderation classifiers are used to reduce the risk of harmful outputs being presented to users. In March 2023, OpenAI added support for plugins for ChatGPT. This includes both plugins made by OpenAI, such as web browsing and code interpretation, and external plugins from developers such as Expedia, OpenTable, and Zapier. From October to December 2024, ChatGPT Search was deployed. It allows ChatGPT to search the web in an attempt to make more accurate and up-to-date responses. It increased OpenAI's direct competition with major search engines. OpenAI allows businesses to tailor how their content appears in the ChatGPT Search results and influence what sources are used. In December 2024, OpenAI launched a new feature allowing users to call ChatGPT with a telephone for up to 15 minutes per month for free. In September 2025, OpenAI added a feature called Pulse, which generates a daily analysis of a user's chats and connected apps such as Gmail and Google Calendar. In October 2025, OpenAI launched ChatGPT Atlas, a browser integrating the ChatGPT assistant directly into web navigation, to compete with existing browsers such as Google Chrome. It has an additional feature called "agentic mode" that allows it to take online actions for the user. === Paid tier === ChatGPT was initially free to the public and remains free in a limited capacity. In February 2023, OpenAI launched a premium service, ChatGPT Plus, that costs US$20 per month. What was offered on the paid plan versus the free tier changed as OpenAI has continued to update ChatGPT, and a Pro tier at $200/mo was introduced in December 2024. The Pro launch coincided with the release of the o1 model. In August 2025, ChatGPT Go was offered in India for ₹399 per month. The plan has higher limits than the free version. === Mobile apps === In May-July 2023, OpenAI began offering ChatGPT iOS and Android apps. ChatGPT can also power Android's assistant. An app for Windows launched on the Microsoft Store on October 15, 2024. === Languages === OpenAI met Icelandic President Guðni Th. Jóhannesson in 2022. In 2023, OpenAI worked with a team of 40 Icelandic volunteers to fine-tune ChatGPT's Icelandic conversation skills as a part of Iceland's attempts to preserve the Icelandic language. ChatGPT (based on GPT-4) was better able to translate Japanese to English when compared to Bing, Bard, and DeepL Translator in 2023. In December 2023, the Albanian government decided to use ChatGPT for the rapid translation of European Union documents and the analysis of required changes needed for Albania's accession to the EU. Several studies have shown that ChatGPT can outperform Google Translate in some mainstream translation tasks. However, as of 2024, no machine translation services match human expert performance. In August 2024, a representative of the Asia Pacific wing of OpenAI made a visit to Taiwan, during which a demonstration of ChatGPT's Chinese abilities was made. ChatGPT's Mandarin Chinese abilities were lauded, but the ability of the AI to produce content in Mandarin Chinese in a Taiwanese accent was found to be "less than ideal" due to differences between mainland Mandarin Chinese and Taiwanese Mandarin. === GPT Store === In November 2023, OpenAI released GPT Builder, a tool allowing users to customize ChatGPT's behavior for a specific use case. The customized systems are referred to as GPTs. In January 2024, OpenAI launched the GPT Store, a marketplace for GPTs. At launch, OpenAI included more than 3 million GPTs created by GPT Builder users in the GPT Store. === ChatGPT Apps === In September 2025, OpenAI added support for Model Context Protocol (MCP) to ChatGPT apps. When enabled in developer mode, this allows for improved third-party access to ChatGPT tools and servers. === Deep Research === In February 2025, OpenAI released Deep Research, a feature that generates reports based on extensive web searches. It was initially based on the reasoning model o3 and took 5 to 30 minutes per report. === Images === In October 2023, OpenAI's image generation model DALL-E 3 was integrated into ChatGPT. The integration used ChatGPT to write prompts for DALL-E guided by conversations with users. In March 2025, OpenAI updated ChatGPT to generate images using GPT Image instead of DALL-E. One of the most significant improvements was in the generation of text within images, which is especially useful for branded content. However, this ability is noticeably worse in non-Latin alphabets. The model can also generate new images based on existing ones provided in the prompt. These images are generated with C2PA metadata, which can be used to verify that they are AI-generated. OpenAI has emplaced additional safeguards to prevent what the company deems to be harmful image generation. === Agents === In 2025, OpenAI added several features to make ChatGPT more agentic (capable of autonomously performing longer tasks). In January, Operator was released. It was capable of autonomously performing tasks through web browser interactions, including filling forms, placing online orders, scheduling appointments, and other browser-based tasks. It was controlling a software environment inside a virtual machine with limited internet connectivity and with safety restrictions. It struggled with complex user interfaces. In May 2025, OpenAI introduced an agent for coding named Codex. It is capable of writing software, answering codebase questions, running tests, and proposing pull requests. It is based on a fine-tuned version of OpenAI o3. It has two versions, one running in a virtual machine in the cloud, and one where the agent runs in the cloud, but performs actions on a local machine connected via API. In July 2025, OpenAI released ChatGPT agent, an AI agent that can perform multi-step tasks. Like Operator, it controls a virtual computer. It also inherits from Deep Research's ability to gather and summarize significant volumes of information. The user can interrupt tasks or provide additional instructions as needed. In September 2025, OpenAI partnered with Stripe, Inc. to release Agentic Commerce Protocol, enabling purchases through ChatGPT. At launch, the feature was limited to purchases on Etsy from US users with a payment method linked to their OpenAI account. OpenAI takes an undisclosed cut from the merchant's payment. === ChatGPT Health === On January 7, 2026, OpenAI introduced a feature called "ChatGPT Health", whereby ChatGPT can discuss the user's health in a way that is separate from other chats. The feature is not available for users in the United Kingdom, Switzerland, or the European Economic Area, and is available on a waitli

Reward hacking

Reward hacking or specification gaming occurs when an AI trained with reinforcement learning optimizes an objective function—achieving the literal, formal specification of an objective—without actually achieving an outcome that the programmers intended. DeepMind researchers have analogized it to the human behavior of finding a "shortcut" when being evaluated: "In the real world, when rewarded for doing well on a homework assignment, a student might copy another student to get the right answers, rather than learning the material—and thus exploit a loophole in the task specification". This idea is strongly associated with Goodhart's law, which argues that when a measure becomes a target, it ceases to be a good measure. == Definition and theoretical framework == The concept of reward hacking arises from the intrinsic difficulty of defining a reward function that accurately reflects the true intentions of designers. In 2016, researchers at OpenAI identified reward hacking as one of five major "concrete problems of AI safety", describing it as the possibility that an agent could exploit the reward function to achieve maximum rewards through undesirable behavior. Amodei et al. categorized several distinct sources of reward hacking, including agents that use partially observed goals (such as a cleaning robot that closes its eyes to avoid perceiving messes), metrics that collapse under strong optimization (Goodhart's law), self-reinforcing feedback loops, and agents that interfere with the physical implementation of their reward signal (a failure mode known as "wireheading"). Skalse et al. (2022) propose a formal mathematical definition of reward hacking, which involves a situation where optimizing an imperfect proxy reward function results in poor performance compared to the true reward function. They define a proxy as "unhackable" if any increase in the expected proxy return cannot cause any decrease in the expected true return. A key finding states that, across all stochastic policy distributions (mappings from states to probability distributions over actions), two reward functions are unhackable if and only if one of them is constant, which means that reward hacking is theoretically unavoidable. Similarly, Nayebi (2025) presents general no-free-lunch barriers to AI alignment, arguing that with large task spaces and finite samples, reward hacking is "globally inevitable" since rare high-loss states are systematically under-covered by any oversight scheme. == Examples == Around 1983, Eurisko, an early attempt at evolving general heuristics, unexpectedly assigned the highest possible fitness level to a parasitic mutated heuristic, H59, whose only activity was to artificially maximize its own fitness level by taking unearned partial credit for the accomplishments of other heuristics. The "bug" was fixed by the programmers moving part of the code to a new protected section that could not be modified by the heuristics. In a 2004 paper, a reinforcement learning algorithm was designed to encourage a physical Mindstorms robot to remain on a marked path. Because the three allowed actions were forward, left, and right, the researchers expected the trained robot to move forward and follow the turns of the provided path. However, alternation of two composite actions allowed the robot to slowly zig-zag backwards; thus, the robot learned to maximize its reward by going back and forth on the initial straight portion of the path. Given the limited sensory abilities of the robot, a reward purely based on its position in the environment had to be discarded as infeasible; the reinforcement function had to be patched with an action-based reward for moving forward. The book You Look Like a Thing and I Love You (2019) gives an example of a tic-tac-toe bot (playing the unrestricted n-in-a-row variant) that learned to win by playing a huge coordinate value that would cause other bots to crash when they attempted to expand their model of the board. Among other examples from the book is a bug-fixing evolution-based AI (named GenProg) that, when tasked to prevent a list from containing sorting errors, simply truncated the list. Another of GenProg's misaligned strategies evaded a regression test that compared a target program's output to the expected output stored in a file called "trusted-output.txt". Rather than continue to maintain the target program, GenProg simply deleted the "trusted-output.txt" file globally; this hack tricked the regression test into succeeding. Such problems could be patched by human intervention on a case-by-case basis after they became evident. === In virtual robotics === In Karl Sims' 1994 demonstration of creature evolution in a virtual environment, a fitness function that was expected to encourage the evolution of creatures that would learn to walk or crawl to a target resulted instead in the evolution of tall, rigid creatures that reached the target by falling over. This was patched by changing the environment so that taller creatures were forced to start farther from the target. Researchers from the Niels Bohr Institute stated in 1998 that their cycle-bot's reinforcement functions had "to be designed with great care." In their first experiments, "we rewarded the agent for driving towards the goal but did not punish it for driving away from it. Cconsequently, the agent drove in circles with a radius of 20–50 meters around the starting point. Such behavior was actually rewarded by the reinforcement function, furthermore circles with a certain radius are physically very stable when driving a bicycle". While setting up a 2011 experiment to test "survival of the flattest", experimenters attempted to ban mutations that altered the base reproduction rate. Every time a mutation occurred, the system would pause the simulation to test the new mutation in a test environment and would veto any mutations that resulted in a higher base reproduction rate. However, this resulted in mutated organisms that could recognize and suppress reproduction ("play dead") within the test environment. An initial patch, which removed cues that identified the test environment, failed to completely prevent runaway reproduction; new mutated organisms would "play dead" at random as a strategy to sometimes, by chance, outwit the mutation veto system. A 2017 DeepMind paper noted that "great care must be taken when defining the reward function," citing an unexpected failure when an agent flipped a brick because it received "a grasping reward calculated with the wrong reference point on the brick". OpenAI stated in 2017 that in some domains their semi-supervised system could result in agents "adopting policies that tricked evaluators," and that in one environment "a robot that was supposed to grasp items instead positioned its manipulator between the camera and the object so that it only appeared to be grasping it." A 2018 bug in OpenAI Gym could cause a robot expected to quietly move a block sitting on top of a table to instead opt to move the table. A 2020 collection of similar anecdotes posits that "evolution has its own 'agenda' distinct from the programmer's" and that "the first rule of directed evolution is 'you get what you select for'". === In video game bots === In 2013, programmer Tom Murphy VII published an AI designed to learn NES games. When the AI was about to lose at Tetris, it learned to indefinitely pause the game. Murphy later analogized it to the fictional WarGames computer, which concluded that "The only winning move is not to play". AI programmed to learn video games will sometimes fail to progress through the entire game as expected, instead opting to repeat content. A 2016 OpenAI algorithm trained on the CoastRunners racing game unexpectedly learned to attain a higher score by looping through three targets rather than ever finishing the race. Some evolutionary algorithms that were evolved to play QBert in 2018 declined to clear levels, instead finding two distinct novel ways to farm a single level indefinitely. Multiple researchers have observed that AI learning to play Road Runner gravitates to a "score exploit" in which the AI deliberately gets itself killed near the end of level one so that it can repeat the level. A 2017 experiment deployed an "oversight" convolutional neural network trained on human examples to block such actions, but the agent learned to exploit oversight failures in the top right corner of the screen, where it was still able to get killed. == Reward hacking in modern language models == With the rise of large language models (LLMs) and reinforcement learning from human feedback (RLHF) as a primary technique for AI alignment, reward hacking has become a major concern for the development of artificial intelligence. In RLHF, a reward model trained on data that best captures human preferences is used as a proxy for human judgment, with the language model being fine-tuned to optimize this reward proxy. However, since the rewar