In machine learning, grokking, or delayed generalization, is a phenomenon observed in some settings where a model abruptly transitions from overfitting (performing well only on training data) to generalizing (performing well on both training and test data), after many training iterations with little or no improvement on the held-out data. This contrasts with what is typically observed in machine learning, where generalization occurs gradually alongside improved performance on training data. == Origin == Grokking was introduced by OpenAI researcher Alethea Power and colleagues in the January 2022 paper "Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets". It is derived from the word grok coined by Robert Heinlein in his novel Stranger in a Strange Land. In ML research, "grokking" is not used as a synonym for "generalization"; rather, it names a sometimes-observed delayed‑generalization training phenomenon in which training and held‑out performance do not improve in tandem, and in which held‑out performance rises abruptly later. Authors also analyze the "grokking time", the epoch or step at which this transition occurs in those scenarios. == Interpretations == Grokking can be understood as a phase transition during the training process. In particular, recent work has shown that grokking may be due to a complexity phase transition in the model during training. While grokking has been thought of as largely a phenomenon of relatively shallow models, grokking has been observed in deep neural networks and non-neural models and is the subject of active research. One potential explanation is that the weight decay (a component of the loss function that penalizes higher values of the neural network parameters, also called regularization) slightly favors the general solution that involves lower weight values, but that is also harder to find. According to Neel Nanda, the process of learning the general solution may be gradual, even though the transition to the general solution occurs more suddenly later. Recent theories have hypothesized that grokking occurs when neural networks transition from a "lazy training" regime where the weights do not deviate far from initialization, to a "rich" regime where weights abruptly begin to move in task-relevant directions. Follow-up empirical and theoretical work has accumulated evidence in support of this perspective, and it offers a unifying view of earlier work as the transition from lazy to rich training dynamics is known to arise from properties of adaptive optimizers, weight decay, initial parameter weight norm, and more. This perspective is complementary to a unifying "pattern learning speeds" framework that links grokking and double descent; within this view, delayed generalization can arise across training time ("epoch‑wise") or across model size ("model‑wise"), and the authors report "model‑wise grokking".
Tensor (machine learning)
In machine learning, the term tensor informally refers to two different concepts: (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space. Observations, such as images, movies, volumes, sounds, and relationships among words and concepts, stored in an M-way array ("data tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factors data tensors into smaller tensors. Operations on data tensors can be expressed in terms of matrix multiplication and the Kronecker product. The computation of gradients, a crucial aspect of backpropagation, can be performed using software libraries such as PyTorch and TensorFlow. Computations are often performed on graphics processing units (GPUs) using CUDA, and on dedicated hardware such as Google's Tensor Processing Unit or Nvidia's Tensor core. These developments have greatly accelerated neural network architectures, and increased the size and complexity of models that can be trained. == History == A tensor is by definition a multilinear map. In mathematics, this may express a multilinear relationship between sets of algebraic objects. In physics, tensor fields, considered as tensors at each point in space, are useful in expressing mechanics such as stress or elasticity. In machine learning, the exact use of tensors depends on the statistical approach being used. In 2001, the field of signal processing and statistics were making use of tensor methods. Pierre Comon surveys the early adoption of tensor methods in the fields of telecommunications, radio surveillance, chemometrics and sensor processing. Linear tensor rank methods (such as, Parafac/CANDECOMP) analyzed M-way arrays ("data tensors") composed of higher order statistics that were employed in blind source separation problems to compute a linear model of the data. He noted several early limitations in determining the tensor rank and efficient tensor rank decomposition. In the early 2000s, multilinear tensor methods crossed over into computer vision, computer graphics and machine learning with papers by Vasilescu or in collaboration with Terzopoulos, such as Human Motion Signatures, TensorFaces TensorTextures and Multilinear Projection. Multilinear algebra, the algebra of higher-order tensors, is a suitable and transparent framework for analyzing the multifactor structure of an ensemble of observations and for addressing the difficult problem of disentangling the causal factors based on second order or higher order statistics associated with each causal factor. Tensor (multilinear) factor analysis disentangles and reduces the influence of different causal factors with multilinear subspace learning. When treating an image or a video as a 2- or 3-way array, i.e., "data matrix/tensor", tensor methods reduce spatial or time redundancies as demonstrated by Wang and Ahuja. Yoshua Bengio, Geoff Hinton and their collaborators briefly discuss the relationship between deep neural networks and tensor factor analysis beyond the use of M-way arrays ("data tensors") as inputs. One of the early uses of tensors for neural networks appeared in natural language processing. A single word can be expressed as a vector via Word2vec. Thus a relationship between two words can be encoded in a matrix. However, for more complex relationships such as subject-object-verb, it is necessary to build higher-dimensional networks. In 2009, the work of Sutskever introduced Bayesian Clustered Tensor Factorization to model relational concepts while reducing the parameter space. From 2014 to 2015, tensor methods become more common in convolutional neural networks (CNNs). Tensor methods organize neural network weights in a "data tensor", analyze and reduce the number of neural network weights. Lebedev et al. accelerated CNN networks for character classification (the recognition of letters and digits in images) by using 4D kernel tensors. == Definition == Let F {\displaystyle \mathbb {F} } be a field (such as the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ). A tensor T ∈ F I 1 × I 2 × … × I C {\displaystyle {\mathcal {T}}\in {\mathbb {F} }^{I_{1}\times I_{2}\times \ldots \times I_{C}}} is a multilinear transformation from a set of domain vector spaces to a range vector space: T : { F I 1 × F I 2 × … F I C } ↦ F I 0 {\displaystyle {\mathcal {T}}:\{{\mathbb {F} }^{I_{1}}\times {\mathbb {F} }^{I_{2}}\times \ldots {\mathbb {F} }^{I_{C}}\}\mapsto {\mathbb {F} }^{I_{0}}} Here, C {\displaystyle C} and I 0 , I 1 , … , I C {\displaystyle I_{0},I_{1},\ldots ,I_{C}} are positive integers, and ( C + 1 ) {\displaystyle (C+1)} is the number of modes of a tensor (also known as the number of ways of a multi-way array). The dimensionality of mode c {\displaystyle c} is I c {\displaystyle I_{c}} , for 0 ≤ c ≤ C {\displaystyle 0\leq c\leq C} . In statistics and machine learning, an image is vectorized when viewed as a single observation, and a collection of vectorized images is organized as a "data tensor". For example, a set of facial images { d i p , i e , i l , i v ∈ R I X } {\displaystyle \{{\mathbb {d} }_{i_{p},i_{e},i_{l},i_{v}}\in {\mathbb {R} }^{I_{X}}\}} with I X {\displaystyle I_{X}} pixels that are the consequences of multiple causal factors, such as a facial geometry i p ( 1 ≤ i p ≤ I P ) {\displaystyle i_{p}(1\leq i_{p}\leq I_{P})} , an expression i e ( 1 ≤ i e ≤ I E ) {\displaystyle i_{e}(1\leq i_{e}\leq I_{E})} , an illumination condition i l ( 1 ≤ i l ≤ I L ) {\displaystyle i_{l}(1\leq i_{l}\leq I_{L})} , and a viewing condition i v ( 1 ≤ i v ≤ I V ) {\displaystyle i_{v}(1\leq i_{v}\leq I_{V})} may be organized into a data tensor (ie. multiway array) D ∈ R I X × I P × I E × I L × V {\displaystyle {\mathcal {D}}\in {\mathbb {R} }^{I_{X}\times I_{P}\times I_{E}\times I_{L}\times V}} where I P {\displaystyle I_{P}} are the total number of facial geometries, I E {\displaystyle I_{E}} are the total number of expressions, I L {\displaystyle I_{L}} are the total number of illumination conditions, and I V {\displaystyle I_{V}} are the total number of viewing conditions. Tensor factorizations methods such as TensorFaces and multilinear (tensor) independent component analysis factorizes the data tensor into a set of vector spaces that span the causal factor representations, where an image is the result of tensor transformation T {\displaystyle {\mathcal {T}}} that maps a set of causal factor representations to the pixel space. Another approach to using tensors in machine learning is to embed various data types directly. For example, a grayscale image, commonly represented as a discrete 2-way array D ∈ R I R X × I C X {\displaystyle {\mathbf {D} }\in {\mathbb {R} }^{I_{RX}\times I_{CX}}} with dimensionality I R X × I C X {\displaystyle I_{RX}\times I_{CX}} where I R X {\displaystyle I_{RX}} are the number of rows and I C X {\displaystyle I_{CX}} are the number of columns. When an image is treated as 2-way array or 2nd order tensor (i.e. as a collection of column/row observations), tensor factorization methods compute the image column space, the image row space and the normalized PCA coefficients or the ICA coefficients. Similarly, a color image with RGB channels, D ∈ R N × M × 3 . {\displaystyle {\mathcal {D}}\in \mathbb {R} ^{N\times M\times 3}.} may be viewed as a 3rd order data tensor or 3-way array.-------- In natural language processing, a word might be expressed as a vector v {\displaystyle v} via the Word2vec algorithm. Thus v {\displaystyle v} becomes a mode-1 tensor v ↦ A ∈ R N . {\displaystyle v\mapsto {\mathcal {A}}\in \mathbb {R} ^{N}.} The embedding of subject-object-verb semantics requires embedding relationships among three words. Because a word is itself a vector, subject-object-verb semantics could be expressed using mode-3 tensors v a × v b × v c ↦ A ∈ R N × N × N . {\displaystyle v_{a}\times v_{b}\times v_{c}\mapsto {\mathcal {A}}\in \mathbb {R} ^{N\times N\times N}.} In practice the neural network designer is primarily concerned with the specification of embeddings, the connection of tensor layers, and the operations performed on them in a network. Modern machine learning frameworks manage the optimization, tensor factorization and backpropagation automatically. === As unit values === Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered through the activation function f {\displaystyle f} : y m = f ( ∑ n x n u m , n ) , {\displaystyle y_{m}=f\left(\sum _{n}x_{n}u_{m,n}\right),} where y m ∈ R .
Education by algorithm
Education by algorithm refers to automated solutions that algorithmic agents or social bots offer to education, to assist with mundane educational tasks. These are often instrumentalist “educational reforms” or “curriculum transformations”, which have been implemented by policy makers and are supported by proprietary education technologies. New educational policies, mandated by transnational governance forums (like the OECD), have manufactured a connection between economies and education. Governments, schools and universities are expected to introduce or prepare students for an “unknown future”, to “future proof” them against an identified issue or to mitigate a national crisis. Technologies are seen as a catalyst to effect these changes. However, these policies mask a deeper problem, which include the assetization of education and the use of technologies as a means for surveillance and behavior modification. The traces that students and leave, through cookies, logins learning activities, assignments and tests, are collected, facetted, and shared with commercial organizations by these agents, to both predict future behavior and shape it. Techno solutionist thinking has led to managers adopting educational policies and reforms, and looking towards technologies to act as disrupters, liberators or agents to improve efficiency. During the COVID-19 pandemic, many more students had to modify their learning and working circumstances to protect themselves. Academics shifted their assessment practices from the dominant assessment of learning paradigm to an orientation that saw value in "assessment for learning". Big tech assisted, and teaching infrastructure became further privatized, and unbundling of education provision went a step further. Following the return to class, this assessment paradigm became rationalised in education. Leaving the space for algorithmic agents to step in. Academics work was increasingly driven by learning experience platforms and student understanding was extended through interleaving, behavior modification nudges and rewards and scheduled high stakes assessments. This data collection may also be construed as surveillance., or perceived as evidence of a Fourth Industrial Revolution
Australian Geoscience Data Cube
The Australian Geoscience Data Cube (AGDC) is an approach to storing, processing and analyzing large collections of Earth observation data. The technology is designed to meet challenges of national interest by being agile and flexible with vast amounts of layered grid data. The AGDC reduces processing time of traditional image analysis by calibrating, pre-computing known extents, pixel alignment and storing metadata in a cell lattice structure. The temporal-pixel aligned data can often be analysed faster across space and time dimensions than previous scene based techniques. This allows the AGDC to be flexible in tackling future challenges and improve analysis times on every-increasing data repositories of earth observation. The AGDC has also been used internationally to allow countries to maintain ecologically sustainable programs and reduce the difficulty curve of utilizing Remote Sensing data. == Background == The AGDC was originally conceived by Geoscience Australia but is now maintained in a partnership between Geoscience Australia, Commonwealth Scientific and Industrial Research Organisation (CSIRO) and National Computational Infrastructure National Facility (Australia) (NCI). This is made possible by the funding from the partnership and a number of organisations such as National Collaborative Research Infrastructure Strategy (NCRIS). == Analysis ready data, ingestion and indexing == The data processed in the cube is made analysis ready before being ingested and indexed into the AGDC. Analysis ready data is pre-processed data that has applied corrections for instrument calibration (gains and offsets), geolocation (spatial alignment) and radiometry (solar illumination, incidence angle, topography, atmospheric interference). The ingestion process manages the translation of datasets into the storage units while maintaining a database index. The data within the storage and index can be accessed via API calls often compiled within code such as Python (programming language). Example: s2a_l1c = dc.load(product='s2a_level1c_granule',x=(147.36, 147.41), y=(-35.1, -35.15), measurements=['04','03','02'], output_crs='EPSG:4326', resolution=(-0.00025,0.00025)) === Datasets currently stored === Geoscience Australia Landsat Surface Reflectance (1987 to present) Landsat Pixel Quality Landsat Fractional Cover Landsat NDVI === Datasets that have been piloted === USGS Landsat Surface Reflectance SRTM DEM Himawari 8 MODIS Sentinel-2 L1C / S2A Australian Gridded Climate Data == Open source == The AGDC code base is situated in GitHub as an open repository. The core code base moved to the Open Data Cube in early 2017 as part of an international collaboration. Whilst the code base is the Open Data Cube, individual cubes exist as their own right such as the AGDC on the National Computational Infrastructure National Facility (Australia) (NCI) using the High-Performance Computing Cluster HPCC. The core code can be installed on personal computers or public computers (using git) and has many unit tests. Documentation for the code base exists on Read the Docs. == Challenges of the AGDC == The AGDC is designed to meet nationally significant challenges such as the following. Sustainability Environment Water resource management Disaster assist Policy development Community planning Forest preservation Carbon measurement == International awards == The AGDC won the 2016 Content Platform of the Year award from Geospatial World Forum.
Data management plan
A data management plan or DMP is a formal document that outlines how data are to be handled both during a research project, and after the project is completed. The goal of a data management plan is to consider the many aspects of data management, metadata generation, data preservation, and analysis before the project begins; this may lead to data being well-managed in the present, and prepared for preservation in the future. DMPs were originally used in 1966 to manage aeronautical and engineering projects' data collection and analysis, and expanded across engineering and scientific disciplines in the 1970s and 1980s. Up until the early 2000s, DMPs were used "for projects of great technical complexity, and for limited mid-study data collection and processing purposes". In the 2000s and later, E-research and economic policies drove the development and uptake of DMPs. == Importance == Preparing a data management plan before data are collected is claimed to ensure that data are in the correct format, organized well, and better annotated. This could arguably save time in the long term because there is no need to re-organize, re-format, or try to remember details about data. It is also claimed to increase research efficiency since both the data collector and other researchers might be able to understand and use well-annotated data in the future. One component of a data management plan is data archiving and preservation. By deciding on an archive ahead of time, the data collector can format data during collection to make its future submission to a database easier. If data are preserved, they are more relevant since they can be re-used by other researchers. It also allows the data collector to direct requests for data to the database, rather than address requests individually. A frequent argument in favor of preservation is that data that are preserved have the potential to lead to new, unanticipated discoveries, and they prevent duplication of scientific studies that have already been conducted. Data archiving also provides insurance against loss by the data collector. In the 2010s, funding agencies increasingly required data management plans as part of the proposal and evaluation process, despite little or no evidence of their efficacy. == Major components == "There is no general and definitive list of topics that should be covered in a DMP for a research project", and researchers are often left to their own devices as to how to fill out a DMP. === Information about data and data format === A description of data to be produced by the project. This might include (but is not limited to) data that are: Experimental Observational Raw or derived Physical collections Models Simulations Curriculum materials Software Images How will the data be acquired? When and where will they be acquired? After collection, how will the data be processed? Include information about Software used Algorithms Scientific workflows File formats that will be used, justify those formats, and describe the naming conventions used. Quality assurance & quality control measures that will be taken during sample collection, analysis, and processing. If existing data are used, what are their origins? How will the data collected be combined with existing data? What is the relationship between the data collected and existing data? How will the data be managed in the short-term? Consider the following: Version control for files Backing up data and data products Security & protection of data and data products Who will be responsible for management === Metadata content and format === Metadata are the contextual details, including any information important for using data. This may include descriptions of temporal and spatial details, instruments, parameters, units, files, etc. Metadata is commonly referred to as "data about data". Issues to be considered include: How detailed has the metadata to be in order to make the data meaningful? How will the metadata be created and/or captured? Examples include lab notebooks, GPS hand-held units, Auto-saved files on instruments, etc. What format will be used for the metadata? What are the metadata standards commonly used in the respective scientific discipline? There should be justification for the format chosen. === Policies for access, sharing, and re-use === Describe any obligations that exist for sharing data collected. These may include obligations from funding agencies, institutions, other professional organizations, and legal requirements. Include information about how data will be shared, including when the data will be accessible, how long the data will be available, how access can be gained, and any rights that the data collector reserves for using data. Address any ethical or privacy issues with data sharing Address intellectual property & copyright issues. Who owns the copyright? What are the institutional, publisher, and/or funding agency policies associated with intellectual property? Are there embargoes for political, commercial, or patent reasons? Describe the intended future uses/users for the data Indicate how the data should be cited by others. How will the issue of persistent citation be addressed? For example, if the data will be deposited in a public archive, will the dataset have a persistent identifier (e.g., ARK, DOI, Handle, PURL, URN) assigned to it? === Long-term storage and data management === Researchers should identify an appropriate archive for the long-term preservation of their data. By identifying the archive early in the project, the data can be formatted, transformed, and documented appropriately to meet the requirements of the archive. Researchers should consult colleagues and professional societies in their discipline to determine the most appropriate database, and include a backup archive in their data management plan in case their first choice goes out of existence. Early in the project, the primary researcher should identify what data will be preserved in an archive. Usually, preserving the data in its most raw form is desirable, although data derivatives and products can also be preserved. An individual should be identified as the primary contact person for archived data, and ensure contact information is always kept up-to-date in case there are requests for data or information about data. === Budget === Data management and preservation costs may be considerable, depending on the nature of the project. By anticipating costs ahead of time, researchers ensure that the data will be properly managed and archived. Potential expenses that should be considered are Human resources and staff as they handle data preparation, management, documentation, and preservation Hardware and/or software needed for data management, backing up, security, documentation, and preservation Costs associated with submitting the data to an archive The data management plan should include how these costs will be paid. == NSF Data Management Plan == All grant proposals submitted to National Science Foundation (NSF) must include a Data Management Plan that is no more than two pages. This is a supplement (not part of the 15-page proposal) and should describe how the proposal will conform to the Award and Administration Guide policy (see below). It may include the following: The types of data The standards to be used for data and metadata format and content Policies for access and sharing Policies and provisions for re-use Plans for archiving data Policy summarized from the NSF Award and Administration Guide, Section 4 (Dissemination and Sharing of Research Results): Promptly publish with appropriate authorship Share data, samples, physical collections, and supporting materials with others, within a reasonable time frame Share software and inventions Investigators can keep their legal rights over their intellectual property, but they still have to make their results, data, and collections available to others Policies will be implemented via Proposal review Award negotiations and conditions Support/incentives == ESRC Data Management Plan == Since 1995, the UK's Economic and Social Research Council (ESRC) have had a research data policy in place. The current ESRC Research Data Policy states that research data created as a result of ESRC-funded research should be openly available to the scientific community to the maximum extent possible, through long-term preservation and high-quality data management. ESRC requires a data management plan for all research award applications where new data are being created. Such plans are designed to promote a structured approach to data management throughout the data lifecycle, resulting in better quality data that is ready to archive for sharing and re-use. The UK Data Service, the ESRC's flagship data service, provides practical guidance on research data management planning suitable for social science researchers in the UK and around the world. ESRC has a longstanding arrangement with the UK Data A
Order-independent transparency
Order-independent transparency (OIT) is a class of techniques in rasterisational computer graphics for rendering transparency in a 3D scene, which do not require rendering geometry in sorted order for alpha compositing. == Description == Commonly, 3D geometry with transparency is rendered by blending (using alpha compositing) all surfaces into a single buffer (think of this as a canvas). Each surface occludes existing color and adds some of its own color depending on its alpha value, a ratio of light transmittance. The order in which surfaces are blended affects the total occlusion or visibility of each surface. For a correct result, surfaces must be blended from farthest to nearest or nearest to farthest, depending on the alpha compositing operation, over or under. Ordering may be achieved by rendering the geometry in sorted order, for example sorting triangles by depth, but can take a significant amount of time, not always produce a solution (in the case of intersecting or circularly overlapping geometry) and the implementation is complex. Instead, order-independent transparency sorts geometry per-pixel, after rasterisation. For exact results this requires storing all fragments before sorting and compositing. == History == The A-buffer is a computer graphics technique introduced in 1984 which stores per-pixel lists of fragment data (including micro-polygon information) in a software rasteriser, REYES, originally designed for anti-aliasing but also supporting transparency. More recently, depth peeling in 2001 described a hardware accelerated OIT technique. With limitations in graphics hardware the scene's geometry had to be rendered many times. A number of techniques have followed, to improve on the performance of depth peeling, still with the many-pass rendering limitation. For example, Dual Depth Peeling (2008). In 2009, two significant features were introduced in GPU hardware/drivers/Graphics APIs that allowed capturing and storing fragment data in a single rendering pass of the scene, something not previously possible. These are, the ability to write to arbitrary GPU memory from shaders and atomic operations. With these features a new class of OIT techniques became possible that do not require many rendering passes of the scene's geometry. The first was storing the fragment data in a 3D array, where fragments are stored along the z dimension for each pixel x/y. In practice, most of the 3D array is unused or overflows, as a scene's depth complexity is typically uneven. To avoid overflow the 3D array requires large amounts of memory, which in many cases is impractical. Two approaches to reducing this memory overhead exist. Packing the 3D array with a prefix sum scan, or linearizing, removed the unused memory issue but requires an additional depth complexity computation rendering pass of the geometry. The "Sparsity-aware" S-Buffer, Dynamic Fragment Buffer, "deque" D-Buffer, Linearized Layered Fragment Buffer all pack fragment data with a prefix sum scan and are demonstrated with OIT. Storing fragments in per-pixel linked lists provides tight packing of this data and in late 2011, driver improvements reduced the atomic operation contention overhead making the technique very competitive. == Exact OIT == Exact, as opposed to approximate, OIT accurately computes the final color, for which all fragments must be sorted. For high depth complexity scenes, sorting becomes the bottleneck. One issue with the sorting stage is local memory limited occupancy, in this case a SIMT attribute relating to the throughput and operation latency hiding of GPUs. Backwards memory allocation (BMA) groups pixels by their depth complexity and sorts them in batches to improve the occupancy and hence performance of low depth complexity pixels in the context of a potentially high depth complexity scene. Up to a 3× overall OIT performance increase is reported. Sorting is typically performed in a local array, however performance can be improved further by making use of the GPU's memory hierarchy and sorting in registers, similarly to an external merge sort, especially in conjunction with BMA. == Approximate OIT == Approximate OIT techniques relax the constraint of exact rendering to provide faster results. Higher performance can be gained from not having to store all fragments or only partially sorting the geometry. A number of techniques also compress, or reduce, the fragment data. These include: Stochastic Transparency: draw in a higher resolution in full opacity but discard some fragments. Downsampling will then yield transparency. Adaptive Transparency, a two-pass technique where the first constructs a visibility function which compresses on the fly (this compression avoids having to fully sort the fragments) and the second uses this data to composite unordered fragments. Intel's pixel synchronization avoids the need to store all fragments, removing the unbounded memory requirement of many other OIT techniques. Weighted Blended Order-Independent Transparency replaced the over operator with a commutative approximation. Feeding depth information into the weight produces visually-acceptable occlusion. == OIT in Hardware == The Sega Dreamcast games console included hardware support for automatic OIT.
Unrestricted algorithm
An unrestricted algorithm is an algorithm for the computation of a mathematical function that puts no restrictions on the range of the argument or on the precision that may be demanded in the result. The idea of such an algorithm was put forward by C. W. Clenshaw and F. W. J. Olver in a paper published in 1980. In the problem of developing algorithms for computing, as regards the values of a real-valued function of a real variable (e.g., g[x] in "restricted" algorithms), the error that can be tolerated in the result is specified in advance. An interval on the real line would also be specified for values when the values of a function are to be evaluated. Different algorithms may have to be applied for evaluating functions outside the interval. An unrestricted algorithm envisages a situation in which a user may stipulate the value of x and also the precision required in g(x) quite arbitrarily. The algorithm should then produce an acceptable result without failure.