In statistics and machine learning, double descent is the phenomenon where a statistical model with a small number of parameters and a model with an extremely large number of parameters have a small test 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 large error. [2] This phenomenon has been considered surprising, as it contradicts assumptions about overfitting in classical machine learning. [1]
Early observations of what would later be called double descent in specific models date back to 1989. [3] [4] The term "double descent" was coined in 2019, [1] when the phenomenon as a broader concept shared by many models gained popularity. [5] [6] [7] 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 bias-variance tradeoff), and the empirical observations in the 2010s that some modern machine learning models tend to perform better with larger models. [6] [8]
Double descent occurs in linear regression with isotropic Gaussian covariates and isotropic Gaussian noise. [9]
A model of double descent at the thermodynamic limit has been analyzed by the replica method, and the result has been confirmed numerically. [10]
The scaling behavior of double descent has been found to follow a broken neural scaling law [11] functional form.
Supervised learning (SL) is a paradigm in machine learning where input objects and a desired output value train a model. The training data is processed, building a function that maps new data to expected output values. An optimal scenario will allow for the algorithm to correctly determine output values for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way. This statistical quality of an algorithm is measured through the so-called generalization error.
In machine learning, a neural network is a model inspired by the structure and function of biological neural networks in animal brains.
In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be justified by the data. In a mathematical sense, these parameters represent the degree of a polynomial. The essence of overfitting is to have unknowingly extracted some of the residual variation as if that variation represented underlying model structure.
In machine learning, early stopping is a form of regularization used to avoid overfitting when training a learner with an iterative method, such as gradient descent. Such methods update the learner so as to make it better fit the training data with each iteration. Up to a point, this improves the learner's performance on data outside of the training set. Past that point, however, improving the learner's fit to the training data comes at the expense of increased generalization error. Early stopping rules provide guidance as to how many iterations can be run before the learner begins to over-fit. Early stopping rules have been employed in many different machine learning methods, with varying amounts of theoretical foundation.
Stochastic gradient descent is an iterative method for optimizing an objective function with suitable smoothness properties. It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient by an estimate thereof. Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate.
In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from input data. These input data used to build the model are usually divided into multiple data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and test sets.
Error-driven learning is a type of reinforcement learning method. This method tweaks a model’s parameters based on the difference between the proposed and actual results. These models stand out as they depend on environmental feedback instead of explicit labels or categories. They are based on the idea that language acquisition involves the minimization of the prediction error (MPSE). By leveraging these prediction errors, the models consistently refine expectations and decrease computational complexity. Typically, these algorithms are operated by the GeneRec algorithm.
There are many types of artificial neural networks (ANN).
In machine learning, a hyperparameter is a parameter, such as the learning rate or choice of optimizer, which specifies details of the learning process, hence the name hyperparameter. This is in contrast to parameters which determine the model itself. An additional contrast is that hyperparameters typically cannot be inferred while fitting the machine to the training set because the objective function is typically non-differentiable with respect to them. As a result, gradient based optimization methods cannot be applied directly.
Deep learning is the subset of machine learning methods based on neural networks with representation learning. The adjective "deep" refers to the use of multiple layers in the network. Methods used can be either supervised, semi-supervised or unsupervised.
In machine learning, feature learning or representation learning is a set of techniques that allows a system to automatically discover the representations needed for feature detection or classification from raw data. This replaces manual feature engineering and allows a machine to both learn the features and use them to perform a specific task.
A convolutional neural network (CNN) is a regularized type of feed-forward neural network that learns features by itself via filter optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer, 10,000 weights would be required for processing an image sized 100 × 100 pixels. However, applying cascaded convolution kernels, only 25 neurons are required to process 5x5-sized tiles. Higher-layer features are extracted from wider context windows, compared to lower-layer features.
In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions, and how well it can make predictions on previously unseen data that were not used to train the model. In general, as we increase the number of tunable parameters in a model, it becomes more flexible, and can better fit a training data set. It is said to have lower error, or bias. However, for more flexible models, there will tend to be greater variance to the model fit each time we take a set of samples to create a new training data set. It is said that there is greater variance in the model's estimated parameters.
Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity.
In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm. A hyperparameter is a parameter whose value is used to control the learning process.
In machine learning, a learning curve plots the optimal value of a model's loss function for a training set against this loss function evaluated on a validation data set with same parameters as produced the optimal function. Synonyms include error curve, experience curve, improvement curve and generalization curve.
Artificial neural networks (ANNs) are models created using machine learning to perform a number of tasks. Their creation was inspired by neural circuitry. While some of the computational implementations ANNs relate to earlier discoveries in mathematics, the first implementation of ANNs was by psychologist Frank Rosenblatt, who developed the perceptron. Little research was conducted on ANNs in the 1970s and 1980s, with the AAAI calling that period an "AI winter".
In the study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks during their training by gradient descent. It allows ANNs to be studied using theoretical tools from kernel methods.
Artificial neural networks are a class of models used in machine learning, and inspired by biological neural networks. They are the core component of modern deep learning algorithms. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width. Theoretical analysis of artificial neural networks sometimes considers the limiting case that layer width becomes large or infinite. This limit enables simple analytic statements to be made about neural network predictions, training dynamics, generalization, and loss surfaces. This wide layer limit is also of practical interest, since finite width neural networks often perform strictly better as layer width is increased.
In machine learning, grokking, or delayed generalization, is a transition to generalization that occurs many training iterations after the interpolation threshold, after many iterations of seemingly little progress, as opposed to the usual process where generalization occurs slowly and progressively once the interpolation threshold has been reached.
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