Neural field

In machine learning, a neural field (also known as implicit neural representation, neural implicit, or coordinate-based neural network), is a mathematical field that is fully or partially parametrized by a neural network. Initially developed to tackle visual computing tasks, such as rendering or reconstruction (e.g., neural radiance fields), neural fields emerged as a promising strategy to deal with a wider range of problems, including surrogate modelling of partial differential equations, such as in physics-informed neural networks. Differently from traditional machine learning algorithms, such as feed-forward neural networks, convolutional neural networks, or transformers, neural fields do not work with discrete data (e.g. sequences, images, tokens), but map continuous inputs (e.g., spatial coordinates, time) to continuous outputs (i.e., scalars, vectors, etc.). This makes neural fields not only discretization independent, but also easily differentiable. Moreover, dealing with continuous data allows for a significant reduction in space complexity, which translates to a much more lightweight network. == Formulation and training == According to the universal approximation theorem, provided adequate learning, sufficient number of hidden units, and the presence of a deterministic relationship between the input and the output, a neural network can approximate any function to any degree of accuracy. Hence, in mathematical terms, given a field y = Φ ( x ) {\textstyle {\boldsymbol {y}}=\Phi ({\boldsymbol {x}})} , with x ∈ R n {\displaystyle {\boldsymbol {x}}\in \mathbb {R} ^{n}} and y ∈ R m {\displaystyle {\boldsymbol {y}}\in \mathbb {R} ^{m}} , a neural field Ψ θ {\displaystyle \Psi _{\theta }} , with parameters θ {\displaystyle {\boldsymbol {\theta }}} , is such that: Ψ θ ( x ) = y ^ ≈ y {\displaystyle \Psi _{\theta }({\boldsymbol {x}})={\hat {\boldsymbol {y}}}\approx {\boldsymbol {y}}} === Training === For supervised tasks, given N {\displaystyle N} examples in the training dataset (i.e., ( x i , y i ) ∈ D t r a i n , i = 1 , … , N {\displaystyle ({\boldsymbol {x_{i}}},{\boldsymbol {y_{i}}})\in {\mathcal {D_{train}}},i=1,\dots ,N} ), the neural field parameters can be learned by minimizing a loss function L {\displaystyle {\mathcal {L}}} (e.g., mean squared error). The parameters θ ~ {\displaystyle {\tilde {\theta }}} that satisfy the optimization problem are found as: θ ~ = argmin θ 1 N ∑ ( x i , y i ) ∈ D t r a i n L ( Ψ θ ( x i ) , y i ) {\displaystyle {\tilde {\boldsymbol {\theta }}}={\underset {\boldsymbol {\theta }}{\text{argmin}}}\;{\frac {1}{N}}\sum _{({\boldsymbol {x_{i}}},{\boldsymbol {y_{i}}})\in {\mathcal {D_{train}}}}{\mathcal {L}}(\Psi _{\theta }({\boldsymbol {x}}_{i}),{\boldsymbol {y}}_{i})} Notably, it is not necessary to know the analytical expression of Φ {\displaystyle \Phi } , for the previously reported training procedure only requires input-output pairs. Indeed, a neural field is able to offer a continuous and differentiable surrogate of the true field, even from purely experimental data. Moreover, neural fields can be used in unsupervised settings, with training objectives that depend on the specific task. For example, physics-informed neural networks may be trained on just the residual. === Spectral bias === As for any artificial neural network, neural fields may be characterized by a spectral bias (i.e., the tendency to preferably learn the low frequency content of a field), possibly leading to a poor representation of the ground truth. In order to overcome this limitation, several strategies have been developed. For example, SIREN uses sinusoidal activations, while the Fourier-features approach embeds the input through sines and cosines. == Conditional neural fields == In many real-world cases, however, learning a single field is not enough. For example, when reconstructing 3D vehicle shapes from Lidar data, it is desirable to have a machine learning model that can work with arbitrary shapes (e.g., a car, a bicycle, a truck, etc.). The solution is to include additional parameters, the latent variables (or latent code) z ∈ R d {\displaystyle {\boldsymbol {z}}\in \mathbb {R} ^{d}} , to vary the field and adapt it to diverse tasks. === Latent code production === When dealing with conditional neural fields, the first design choice is represented by the way in which the latent code is produced. Specifically, two main strategies can be identified: Encoder: the latent code is the output of a second neural network, acting as an encoder. During training, the loss function is the objective used to learn the parameters of both the neural field and the encoder. Auto-decoding: each training example has its own latent code, jointly trained with the neural field parameters. When the model has to process new examples (i.e., not originally present in the training dataset), a small optimization problem is solved, keeping the network parameters fixed and only learning the new latent variables. Since the latter strategy requires additional optimization steps at inference time, it sacrifices speed, but keeps the overall model smaller. Moreover, despite being simpler to implement, an encoder may harm the generalization capabilities of the model. For example, when dealing with a physical scalar field f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} } (e.g., the pressure of a 2D fluid), an auto-decoder-based conditional neural field can map a single point to the corresponding value of the field, following a learned latent code z {\displaystyle {\boldsymbol {z}}} . However, if the latent variables were produced by an encoder, it would require access to the entire set of points and corresponding values (e.g. as a regular grid or a mesh graph), leading to a less robust model. === Global and local conditioning === In a neural field with global conditioning, the latent code does not depend on the input and, hence, it offers a global representation (e.g., the overall shape of a vehicle). However, depending on the task, it may be more useful to divide the domain of x {\displaystyle {\boldsymbol {x}}} in several subdomains, and learn different latent codes for each of them (e.g., splitting a large and complex scene in sub-scenes for a more efficient rendering). This is called local conditioning. === Conditioning strategies === There are several strategies to include the conditioning information in the neural field. In the general mathematical framework, conditioning the neural field with the latent variables is equivalent to mapping them to a subset θ ∗ {\displaystyle {\boldsymbol {\theta }}^{}} of the neural field parameters: θ ∗ = Γ ( z ) {\displaystyle {\boldsymbol {\theta }}^{}=\Gamma ({\boldsymbol {z}})} In practice, notable strategies are: Concatenation: the neural field receives, as input, the concatenation of the original input x {\displaystyle {\boldsymbol {x}}} with the latent codes z {\displaystyle {\boldsymbol {z}}} . For feed-forward neural networks, this is equivalent to setting θ ∗ {\displaystyle {\boldsymbol {\theta }}^{}} as the bias of the first layer and Γ ( z ) {\displaystyle \Gamma ({\boldsymbol {z}})} as an affine transformation. Hypernetworks: a hypernetwork is a neural network that outputs the parameters of another neural network. Specifically, it consists of approximating Γ ( z ) {\displaystyle \Gamma ({\boldsymbol {z}})} with a neural network Γ ^ γ ( z ) {\displaystyle {\hat {\Gamma }}_{\gamma }({\boldsymbol {z}})} , where γ {\displaystyle {\boldsymbol {\gamma }}} are the trainable parameters of the hypernetwork. This approach is the most general, as it allows to learn the optimal mapping from latent codes to neural field parameters. However, hypernetworks are associated to larger computational and memory complexity, due to the large number of trainable parameters. Hence, leaner approaches have been developed. For example, in the Feature-wise Linear Modulation (FiLM), the hypernetwork only produces scale and bias coefficients for the neural field layers. === Meta-learning === Instead of relying on the latent code to adapt the neural field to a specific task, it is also possible to exploit gradient-based meta-learning. In this case, the neural field is seen as the specialization of an underlying meta-neural-field, whose parameters are modified to fit the specific task, through a few steps of gradient descent. An extension of this meta-learning framework is the CAVIA algorithm, that splits the trainable parameters in context-specific and shared groups, improving parallelization and interpretability, while reducing meta-overfitting. This strategy is similar to the auto-decoding conditional neural field, but the training procedure is substantially different. == Applications == Thanks to the possibility of efficiently modelling diverse mathematical fields with neural networks, neural fields have been applied to a wide range of problems: 3D scene reconstruction: neural fields can be used to model t

LaMDA

LaMDA (Language Model for Dialogue Applications) is a family of conversational large language models developed by Google. Originally developed and introduced as Meena in 2020, the first-generation LaMDA was announced during the 2021 Google I/O keynote, while the second generation was announced the following year. In June 2022, LaMDA gained widespread attention when Google engineer Blake Lemoine made claims that the chatbot had become sentient. The scientific community has largely rejected Lemoine's claims, though it has led to conversations about the efficacy of the Turing test, which measures whether a computer can pass for a human. In February 2023, Google announced Gemini (then Bard), a conversational artificial intelligence chatbot powered by LaMDA, to counter the rise of OpenAI's ChatGPT. == History == === Background === On January 28, 2020, Google unveiled Meena, a neural network-powered chatbot with 2.6 billion parameters, which Google claimed to be superior to all other existing chatbots. The company previously hired computer scientist Ray Kurzweil in 2012 to develop multiple chatbots for the company, including one named Danielle. The Google Brain research team, who developed Meena, hoped to release the chatbot to the public in a limited capacity, but corporate executives refused on the grounds that Meena violated Google's "AI principles around safety and fairness". Meena was later renamed LaMDA as its data and computing power increased, and the Google Brain team again sought to deploy the software to the Google Assistant, the company's virtual assistant software, in addition to opening it up to a public demo. Both requests were once again denied by company leadership. LaMDA's two lead researchers, Daniel de Freitas and Noam Shazeer, eventually left the company in frustration. === First generation === Google announced the LaMDA conversational large language model during the Google I/O keynote on May 18, 2021, powered by artificial intelligence. The acronym stands for "Language Model for Dialogue Applications". Built on the seq2seq architecture, transformer-based neural networks developed by Google Research in 2017, LaMDA was trained on human dialogue and stories, allowing it to engage in open-ended conversations. Google states that responses generated by LaMDA have been ensured to be "sensible, interesting, and specific to the context". LaMDA has access to multiple symbolic text processing systems, including a database, a real-time clock and calendar, a mathematical calculator, and a natural language translation system, giving it superior accuracy in tasks supported by those systems, and making it among the first dual process chatbots. LaMDA is also not stateless because its "sensibleness" metric is fine-tuned by "pre-conditioning" each dialog turn by prepending many of the most recent dialog interactions, on a user-by-user basis. LaMDA is tuned on nine unique performance metrics: sensibleness, specificity, interestingness, safety, groundedness, informativeness, citation accuracy, helpfulness, and role consistency. Tests by Google indicated that LaMDA surpassed human responses in the area of interestingness. The pre-training dataset consists of 2.97B documents, 1.12B dialogs, and 13.39B utterances, for a total of 1.56T words. The largest LaMDA model has 137B non-embedding parameters. === Second generation === On May 11, 2022, Google unveiled LaMDA 2, the successor to LaMDA, during the 2022 Google I/O keynote. The new incarnation of the model draws examples of text from numerous sources, using it to formulate unique "natural conversations" on topics that it may not have been trained to respond to. === Sentience claims === On June 11, 2022, The Washington Post reported that Google engineer Blake Lemoine had been placed on paid administrative leave after Lemoine told company executives Blaise Agüera y Arcas and Jen Gennai that LaMDA had become sentient. Lemoine came to this conclusion after the chatbot made questionable responses to questions regarding self-identity, moral values, religion, and Isaac Asimov's Three Laws of Robotics. Google refuted these claims, insisting that there was substantial evidence to indicate that LaMDA was not sentient. In an interview with Wired, Lemoine reiterated his claims that LaMDA was "a person" as dictated by the Thirteenth Amendment to the U.S. Constitution, comparing it to an "alien intelligence of terrestrial origin". He further revealed that he had been dismissed by Google after he hired an attorney on LaMDA's behalf after the chatbot requested that Lemoine do so. On July 22, Google fired Lemoine, asserting that Blake had violated their policies "to safeguard product information" and rejected his claims as "wholly unfounded". Internal controversy instigated by the incident prompted Google executives to decide against releasing LaMDA to the public, which it had previously been considering. Lemoine's claims were widely pushed back by the scientific community. Many experts rejected the idea that LaMDA was sentient, including former New York University psychology professor Gary Marcus, David Pfau of Google sister company DeepMind, Erik Brynjolfsson of the Institute for Human-Centered Artificial Intelligence at Stanford University, and University of Surrey professor Adrian Hilton. Yann LeCun, who leads Meta Platforms' AI research team, stated that neural networks such as LaMDA were "not powerful enough to attain true intelligence". University of California, Santa Cruz professor Max Kreminski noted that LaMDA's architecture did not "support some key capabilities of human-like consciousness" and that its neural network weights were "frozen", assuming it was a typical large language model. Philosopher Nick Bostrom noted, however, that the lack of precise and consensual criteria for determining whether a system is conscious warrants some uncertainty. IBM Watson lead developer David Ferrucci compared how LaMDA appeared to be human in the same way Watson did when it was first introduced. Former Google AI ethicist Timnit Gebru called Lemoine a victim of a "hype cycle" initiated by researchers and the media. Lemoine's claims have also generated discussion on whether the Turing test remained useful to determine researchers' progress toward achieving artificial general intelligence, with Will Omerus of the Post opining that the test actually measured whether machine intelligence systems were capable of deceiving humans, while Brian Christian of The Atlantic said that the controversy was an instance of the ELIZA effect. == Products == === AI Test Kitchen === With the unveiling of LaMDA 2 in May 2022, Google also launched the AI Test Kitchen, a mobile application for the Android operating system powered by LaMDA capable of providing lists of suggestions on-demand based on a complex goal. Originally open only to Google employees, the app was set to be made available to "select academics, researchers, and policymakers" by invitation sometime in the year. In August, the company began allowing users in the U.S. to sign up for early access. In November, Google released a "season 2" update to the app, integrating a limited form of Google Brain's Imagen text-to-image model. A third iteration of the AI Test Kitchen was in development by January 2023, expected to launch at I/O later that year. Following the 2023 I/O keynote in May, Google added MusicLM, an AI-powered music generator first previewed in January, to the AI Test Kitchen app. In August, the app was delisted from Google Play and the Apple App Store, instead moving completely online. === Bard === On February 6, 2023, Google announced Bard, a conversational AI chatbot powered by LaMDA, in response to the unexpected popularity of OpenAI's ChatGPT chatbot. Google positions the chatbot as a "collaborative AI service" rather than a search engine. Bard became available for early access on March 21. === Other products === In addition to Bard, Pichai also unveiled the company's Generative Language API, an application programming interface also based on LaMDA, which he announced would be opened up to third-party developers in March 2023. == Architecture == LaMDA is a decoder-only Transformer language model. It is pre-trained on a text corpus that includes both documents and dialogs consisting of 1.56 trillion words, and is then trained with fine-tuning data generated by manually annotated responses for "sensibleness, interestingness, and safety". LaMDA was retrieval-augmented to improve the accuracy of facts provided to the user. Three different models were tested, with the largest having 137 billion non-embedding parameters:

Lukas Biewald

Lukas Biewald (born 1981) is an American entrepreneur and a prominent figure in artificial intelligence. He is recognized for his contributions to machine learning and as the CEO and co-founder of Weights & Biases, a company that builds developer tools for AI, that sold to CoreWeave in 2025 for $1.7B. He previously founded and was CEO of Figure Eight, a human-in-the-loop machine learning platform. He has co-authored 26 AI research papers from 2004 through 2018. == Early life and education == Biewald was born in Boston, Massachusetts in 1981. He attended Cambridge Rindge and Latin School and later earned both a Bachelor's and Master's degree in Computer science from Stanford University. == Early Career and Founding Figure Eight == After graduation, Biewald joined Yahoo! as an engineer, working on machine translations to improve search results, and eventually led the Search Relevance Team for Yahoo! Japan. He later joined Powerset, a natural language search technology company, as their Senior Scientist, which was acquired by Microsoft in 2008 for an estimated $100M. In 2007, Biewald co-founded Figure Eight (formerly CrowdFlower), a data labeling and crowdsourcing company that created datasets for training machine learning models. Figure Eight was acquired by Appen in 2019 for $300 million. == Weights and Biases == In 2017, Biewald co-founded Weights & Biases with Chris Van Pelt and Shawn Lewis. The company provides tools for tracking machine learning experiments, model management, and collaborative AI and LLM app development. The platform has been adopted by organizations such as OpenAI, Salesforce, and Microsoft. In March 2025 Coreweave acquired Weights and Biases at $1.7 billion, with the transaction closing on May 5, 2025. == Gradient Dissent == Biewald hosts the bi-weekly podcast Gradient Dissent. Guest have included: Anthony Goldbloom – Co-founder & CEO of Kaggle. “How to Win Kaggle Competitions” (podcast, Sep. 9, 2020). Shared tips on data-science competitions from the founder of the largest ML community. Richard Socher – Founder & CEO of You.com; former Chief Scientist at Salesforce. “The Challenges of Making ML Work in the Real World” (podcast, September 28, 2020). A leading NLP researcher, he spoke on multimodal search engines powered by large language models. Jensen Huang – Founder & CEO of NVIDIA. “NVIDIA’s CEO on the Next Generation of AI and MLOps” (podcast, March 3, 2022). Huang’s GPUs power modern ML research and production. Emad Mostaque – Co-founder & CEO of Stability AI. “Stable Diffusion, Stability AI, and What’s Next” (podcast, Nov. 15, 2022). Leads the company behind Stable Diffusion, which helped spark the generative-AI imaging boom. Drago Anguelov – Head of Research at Waymo. “Robustness, Safety, and Scalability at Waymo” (podcast, July 14, 2022). Covered Waymo’s self-driving AI advances and deployment challenges. Jeremy Howard – Co-founder of fast.ai. “The Simple but Profound Insight Behind Diffusion” (podcast, Jan. 5, 2023). Known for democratizing deep-learning education; discussed diffusion models and accessible AI tooling. Aidan Gomez – Co-founder & CEO of Cohere. “Scaling LLMs and Accelerating Adoption” (podcast, April 20, 2023). Co-author of “Attention Is All You Need,” he shared how Cohere delivers large-scale NLP models as a service. Chelsea Finn – Stanford Assistant Professor (AI & Robotics). “Shaping the World of Robotics with Chelsea Finn” (podcast, February 15, 2024). A pioneer in meta-learning and robotics, she detailed robots learning complex tasks like cooking. Andrew Feldman – Co-founder & CEO of Cerebras Systems. "Launching the Fastest AI Inference Solution" (podcast, August 27, 2024). Described wafer-scale AI chips achieving new training performance records. Thomas Dohmke – CEO of GitHub. “GitHub CEO on Copilot and the Future of Software Development” (podcast, June 10, 2025). Discussed building Copilot and the future of AI-assisted coding. Martin Shkreli – Founder of Godel Terminal. “From Pharma to AGI Hype, and Developing AI in Finance: Martin Shkreli’s Journey” (podcast, May 20, 2025). Shkreli reflects on his pharma controversies, prison experience, and his new AI-driven trading platform. Jarek Kutylowski – Founder & CEO of DeepL. “How DeepL Built a Translation Powerhouse with AI” (podcast, July 8, 2025). Shared how DeepL’s neural-MT rivals Google Translate through model and infrastructure innovation. == Awards and recognition == In 2010, Lukas Biewald won the Netexplorateur Award for creating the GiveWork iPhone app, which allows users to perform small tasks that assist refugees and people in developing countries. In 2010, Inc Magazine included Biewald and Van Pelt on its list of the Top 30 Entrepreneurs Under 30. == Publications == Ensuring quality in crowdsourced search relevance evaluation: The effects of training question distribution by John Le, Andy Edmonds, Vaughn Hester, Lukas Biewald. SIGIR 2010 Workshop on Crowdsourcing for Search Evaluation, July 2010. Superficial Data Analysis: Exploring Millions of Social Stereotypes by Lukas Biewald, Brendan O’Connor. O’Reilly July 2009 Biewald has co-authored 26 AI research papers from 2004 through 2018.

Reification (knowledge representation)

Reification in knowledge representation is the process of turning a predicate or statement into an addressable object. Reification allows the representation of assertions so that they can be referred to or qualified by other assertions, i.e., meta-knowledge. The message "John is six feet tall" is an assertion involving truth that commits the speaker to its factuality, whereas the reified statement "Mary reports that John is six feet tall" defers such commitment to Mary. In this way, the statements can be incompatible without creating contradictions in reasoning. For example, the statements "John is six feet tall" and "John is five feet tall" are mutually exclusive (and thus incompatible), but the statements "Mary reports that John is six feet tall" and "Paul reports that John is five feet tall" are not incompatible, as they are both governed by a conclusive rationale that either Mary or Paul is (or both are), in fact, incorrect. In linguistics, reporting, telling, and saying are recognized as verbal processes that project a wording (or locution). If a person says that "Paul told x" and "Mary told y", this person stated only that the telling took place. In this case, the person who made these two statements did not represent a person inconsistently. In addition, if two people are talking to each other, let's say Paul and Mary, and Paul tells Mary "John is five feet tall" and Mary rejects Paul's statement by saying "No, he is actually six feet tall", the socially constructed model of John does not become inconsistent. The reason for that is that statements are to be understood as an attempt to convince the addressee of something (Austin's How to do things with words), alternatively as a request to add some attribute to the model of Paul. The response to a statement can be an acknowledgement, in which case the model is changed, or it can be a statement rejection, in which case the model does not get changed. Finally, the example above for which John is said to be "five feet tall" or "six feet tall" is only incompatible because John can only be a single number of feet tall. If the attribute were a possession as in "he has a dog" or "he also has a cat", a model inconsistency would not happen. In other words, the issue of model inconsistency has to do with our model of the domain element (John) and not with the ascription of different range elements (measurements such as "five feet tall" or "six feet tall").

ChessMachine

The ChessMachine was a chess computer sold between 1991 and 1995 by TASC (The Advanced Software Company). It was unique at the time for incorporating both an ARM2 coprocessor for the chess engine on an ISA card which plugged into an IBM PC and a software interface running on the PC to display a chess board and control the engine. The ISA card was sold with a CPU running at either 16 MHz or 32 MHz, and 128 KB, 512 KB, or 1 MB of onboard memory for transposition tables. This made economic sense at the time of introduction because mainstream PCs were only running from 10 MHz to 25 MHz. Two engines were sold with the card: The King by Johann de Koning and Gideon by Ed Schröder. Gideon was famed for winning two World Computer Chess Championships on this hardware. The King later became the engine used in the popular Chessmaster series of chess programs. TASC later incorporated the technology into a dedicated unit, sold from 1993 to 1997. There were two models, the R30 and R40, running at 30 MHz and 40 MHz respectively, and having 512 KB and 1 MB of transposition tables, respectively. The SmartBoard, a wooden sensory board, was connected to the units, which were in tiny boxes approximately the size of chess clocks. They were only sold with The King chess engine. This was the end of the era of strong dedicated chess computers, and these two models are acknowledged as the strongest dedicated chess computers that were ever sold. At the height of its strength, the R30 attained a rating over 2350 on computer rating lists, higher than any other dedicated unit. According to the SSDF rating list, the R30 held its own against its contemporary programs running a Pentium-90 MHz and won against other dedicated units.

Keyword extraction

Keyword extraction is tasked with the automatic identification of terms that best describe the subject of a document. Key phrases, key terms, key segments or just keywords are the terminology which is used for defining the terms that represent the most relevant information contained in the document. Although the terminology is different, function is the same: characterization of the topic discussed in a document. The task of keyword extraction is an important problem in text mining, information extraction, information retrieval and natural language processing (NLP). == Keyword assignment vs. extraction == Keyword assignment methods can be roughly divided into: keyword assignment (keywords are chosen from controlled vocabulary or taxonomy) and keyword extraction (keywords are chosen from words that are explicitly mentioned in original text). Methods for automatic keyword extraction can be supervised, semi-supervised, or unsupervised. Unsupervised methods can be further divided into simple statistics, linguistics or graph-based, or ensemble methods that combine some or most of these methods.

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