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Double Descent
Double descent in statistics and machine learning is the phenomenon where a model with a small number of parameters and a model with an extremely large number of parameters both have a small training error, but a model whose number of parameters is about the same as the number of data points used to train the model will have a much greater test error than one with a much larger number of parameters. This phenomenon has been considered surprising, as it contradicts assumptions about overfitting in classical machine learning. History Early observations of what would later be called double descent in specific models date back to 1989. The term "double descent" was coined by Belkin et. al. in 2019, when the phenomenon gained popularity as a broader concept exhibited by many models. The latter development was prompted by a perceived contradiction between the conventional wisdom that too many parameters in the model result in a significant overfitting error (an extrapolation of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Descent In A Two-layer Neural Network (Figure 3a From Rocks Et Al
Double, The Double or Dubble may refer to: Mathematics and computing * Multiplication by 2 * Double precision, a floating-point representation of numbers that is typically 64 bits in length * A double number of the form x+yj, where j^2=+1 * A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted (a,b) * Double (manifold), in topology Food and drink * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubling cube, in backgammon ** Double, doubling a blackjack bet in a favorable situation ** Double, a bet offered by UK bookmakers which combines two selections * Double, villain in the video game ''Mega Man X4'' * A kart racing game '' Mario Kart: Double Dash'' * An arcade action game ''Double Dragon'' Sports * Double (association football), the act of a winning a division and primary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Selection
Model selection is the task of selecting a model from among various candidates on the basis of performance criterion to choose the best one. In the context of machine learning and more generally statistical analysis, this may be the selection of a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is well-suited to the problem of model selection. Given candidate models of similar predictive or explanatory power, the simplest model is most likely to be the best choice (Occam's razor). state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, has said, "How hetranslation from subject-matter problem to statistical model is done is often the most critical part of an analysis". Model selection may also refer to the problem of selecting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications On Pure And Applied Mathematics
''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or mathematical physics. The journal was established in 1948 as the ''Communications on Applied Mathematics'', obtaining its current title the next year. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 3.219. References External links * Mathematics journals Monthly journals Wiley (publisher) academic journals Academic journals established in 1948 English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Mathematics Of Data Science
Thailand, officially the Kingdom of Thailand and historically known as Siam (the official name until 1939), is a country in Southeast Asia on the Indochinese Peninsula. With a population of almost 66 million, it spans . Thailand is bordered to the northwest by Myanmar, to the northeast and east by Laos, to the southeast by Cambodia, to the south by the Gulf of Thailand and Malaysia, and to the southwest by the Andaman Sea; it also shares maritime borders with Vietnam to the southeast and Indonesia and India to the southwest. Bangkok is the state capital and largest city. Thai peoples migrated from southwestern China to mainland Southeast Asia from the 6th to 11th centuries. Indianised kingdoms such as the Mon, Khmer Empire, and Malay states ruled the region, competing with Thai states such as the Kingdoms of Ngoenyang, Sukhothai, Lan Na, and Ayutthaya, which also rivalled each other. European contact began in 1511 with a Portuguese diplomatic mission to Ayutthaya, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grokking (machine Learning)
In machine learning, grokking, or delayed generalization, is a phenomenon 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 of seemingly little progress. This contrasts with typical learning, where generalization occurs gradually alongside improved performance on training data. History Grokking was introduced in January 2022 by OpenAI researchers investigating how neural networks perform calculations. It is derived from the word ''grok'' coined by Robert Heinlein in his novel ''Stranger in a Strange Land''. 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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2210
In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 ( MMI) and will end on 31 December 3000 ( MMM), spanning the 21st to 30th centuries. Ongoing futures studies seek to understand what will likely continue and what could plausibly change in this period and beyond. Predictions and forecasts not included on this timeline * Climate change * Extinction * List of dates predicted for apocalyptic events * List of future astronomical events ** List of lunar eclipses in the 21st century ** List of solar eclipses in the 21st century * List of time capsules * Near future centennial (bi, tri, etc.) events. * Near future in fiction * Predictions and claims for the Second Coming * Projections of population growth ** Representative Concentration Pathway ** Shared Socioeconomic Pathways 21st century 2000s * See: 2000 * 2001 * 2002 * 2003 * 2004 * 2005 * 2006 * 2007 * 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neural Scaling Law
In machine learning, a neural scaling law is an empirical scaling law that describes how Neural network (machine learning), neural network performance changes as key factors are scaled up or down. These factors typically include the number of parameters, Training, validation, and test data sets, training dataset size, and training cost. Introduction In general, a deep learning model can be characterized by four parameters: model size, training dataset size, training cost, and the post-training error rate (e.g., the test set error rate). Each of these variables can be defined as a real number, usually written as N, D, C, L (respectively: parameter count, dataset size, computing cost, and Loss function, loss). A neural scaling law is a theoretical or empirical Empirical statistical laws, statistical law between these parameters. There are also other parameters with other scaling laws. Size of the model In most cases, the model's size is simply the number of parameters. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Replica Trick
In the statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula: \ln Z=\lim_ or: \ln Z = \lim_ \frac where Z is most commonly the partition function, or a similar thermodynamic function. It is typically used to simplify the calculation of \overline, the expected value of \ln Z, reducing the problem to calculating the disorder average \overline where n is assumed to be an integer. This is physically equivalent to averaging over n copies or ''replicas'' of the system, hence the name. The crux of the replica trick is that while the disorder averaging is done assuming n to be an integer, to recover the disorder-averaged logarithm one must send n continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundell and K.M. Blundell, "Concepts in Thermal Physics", Oxford University Press (2009) The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed: : N \to \infty,\, V \to \infty,\, \frac N V =\text In this limit, macroscopic thermodynamics is valid. There, thermal fluctuations in global quantities are negligible, and all List of thermodynamic properties, thermodynamic quantities, such as pressure and energy, are simply functions of the thermodynamic variables, such as temperature and density. For example, for a large volume of gas, the fluctuations of the total internal energy are negligible and can be ignored, and the average internal energy can be predicted fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropy
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
