Misplaced Pages

#940059

43-473: NTM may refer to: Medicine and Science [ edit ] Neural Turing machine , a recurrent neural network model Nondeterministic Turing machine , a theoretical model of computation Nontuberculous mycobacteria , a class of bacteria NTM (gene) , which encodes the protein neurotrimin Organisations [ edit ] National Taiwan Museum ,

86-494: A {\displaystyle \mathbf {a} } in the direction of the negative gradient of F {\displaystyle F} at a , − ∇ F ( a ) {\displaystyle \mathbf {a} ,-\nabla F(\mathbf {a} )} . It follows that, if for a small enough step size or learning rate γ ∈ R + {\displaystyle \gamma \in \mathbb {R} _{+}} , then F (

129-413: A n ) ≥ F ( a n + 1 ) {\displaystyle F(\mathbf {a_{n}} )\geq F(\mathbf {a_{n+1}} )} . In other words, the term γ ∇ F ( a ) {\displaystyle \gamma \nabla F(\mathbf {a} )} is subtracted from a {\displaystyle \mathbf {a} } because we want to move against

172-407: A monotonic sequence so the sequence ( x n ) {\displaystyle (\mathbf {x} _{n})} converges to the desired local minimum. Note that the value of the step size γ {\displaystyle \gamma } is allowed to change at every iteration. It is possible to guarantee the convergence to a local minimum under certain assumptions on

215-677: A museum in Taipei, Taiwan National Technical Museum (Prague) , an institution in the Czech Republic National Theatre Movement , in Victoria, Australia National Transformation Movement , two political parties in Trinidad and Tobago National Translation Mission , an Indian initiative to make texts accessible Neil Thomas Ministries , a Christian organization Network Television Marketing ,

258-423: A neural network controller coupled to external memory resources, which it interacts with through attentional mechanisms. The memory interactions are differentiable end-to-end, making it possible to optimize them using gradient descent . An NTM with a long short-term memory (LSTM) network controller can infer simple algorithms such as copying, sorting, and associative recall from examples alone. The authors of

301-453: A simple algorithm can be as follows, To avoid multiplying by A {\displaystyle A} twice per iteration, we note that x := x + γ r {\displaystyle \mathbf {x} :=\mathbf {x} +\gamma \mathbf {r} } implies r := r − γ A r {\displaystyle \mathbf {r} :=\mathbf {r} -\gamma \mathbf {Ar} } , which gives

344-422: A sophisticated instrument to measure, which the persons happen to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track. In this analogy,

387-461: A system of nonlinear equations . Below is an example that shows how to use the gradient descent to solve for three unknown variables, x 1 , x 2 , and x 3 . This example shows one iteration of the gradient descent. Consider the nonlinear system of equations Let us introduce the associated function where One might now define the objective function which we will attempt to minimize. As an initial guess, let us use We know that where

430-554: A television station in Pakistan New Tribes Mission , now Ethnos360, a Christian mission in Sanford, Florida, USA Other [ edit ] Miracema do Tocantins Airport ( IATA: NTM ), Brazil Nateni language (by ISO 639 code) Non-tariff measures , a type of barrier to trade Notice to mariners , a navigational safety publication Suprême NTM , a French hip-hop duo Topics referred to by

473-579: Is an important practical problem. Philip Wolfe also advocated using "clever choices of the [descent] direction" in practice. Whilst using a direction that deviates from the steepest descent direction may seem counter-intuitive, the idea is that the smaller slope may be compensated for by being sustained over a much longer distance. To reason about this mathematically, consider a direction p n {\displaystyle \mathbf {p} _{n}} and step size γ n {\displaystyle \gamma _{n}} and consider

SECTION 10

#1732787328941

516-436: Is different from Wikidata All article disambiguation pages All disambiguation pages Neural Turing machine A neural Turing machine ( NTM ) is a recurrent neural network model of a Turing machine . The approach was published by Alex Graves et al. in 2014. NTMs combine the fuzzy pattern matching capabilities of neural networks with the algorithmic power of programmable computers . An NTM has

559-437: Is generally attributed to Augustin-Louis Cauchy , who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. A simple extension of gradient descent, stochastic gradient descent , serves as

602-418: Is illustrated in the adjacent picture. Here, F {\displaystyle F} is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines , that is, the regions on which the value of F {\displaystyle F} is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that

645-441: Is maximized when they are colinear . In the case of gradient descent, that would be when the vector of independent variable adjustments is proportional to the gradient vector of partial derivatives. The gradient descent can take many iterations to compute a local minimum with a required accuracy , if the curvature in different directions is very different for the given function. For such functions, preconditioning , which changes

688-474: Is real symmetric and positive-definite , an objective function is defined as the quadratic function, with minimization of so that For a general real matrix A {\displaystyle A} , linear least squares define In traditional linear least squares for real A {\displaystyle A} and b {\displaystyle \mathbf {b} } the Euclidean norm

731-422: Is that evaluating the second term in square brackets requires evaluating ∇ F ( a n − t γ n p n ) {\displaystyle \nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})} , and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are: Usually by following one of

774-435: Is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a trajectory that maximizes that function; the procedure is then known as gradient ascent . It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization . Gradient descent

817-420: Is the step size. If they step off a cliff in the fog they will have optimised their descent path. Since using a step size γ {\displaystyle \gamma } that is too small would slow convergence, and a γ {\displaystyle \gamma } too large would lead to overshoot and divergence, finding a good setting of γ {\displaystyle \gamma }

860-462: Is used, in which case The line search minimization, finding the locally optimal step size γ {\displaystyle \gamma } on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal γ {\displaystyle \gamma } are known. For example, for real symmetric and positive-definite matrix A {\displaystyle A} ,

903-522: The Barzilai-Borwein method , or a sequence γ n {\displaystyle \gamma _{n}} satisfying the Wolfe conditions (which can be found by using line search ). When the function F {\displaystyle F} is convex , all local minima are also global minima, so in this case gradient descent can converge to the global solution. This process

SECTION 20

#1732787328941

946-510: The Hessian using conjugate gradient techniques can be better alternatives. Generally, such methods converge in fewer iterations, but the cost of each iteration is higher. An example is the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find

989-525: The Jacobian matrix J G {\displaystyle J_{G}} is given by We calculate: Thus and Now, a suitable γ 0 {\displaystyle \gamma _{0}} must be found such that This can be done with any of a variety of line search algorithms. One might also simply guess γ 0 = 0.001 , {\displaystyle \gamma _{0}=0.001,} which gives Evaluating

1032-452: The (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function F {\displaystyle F} is minimal. The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. Persons are stuck in the mountains and are trying to get down (i.e., trying to find

1075-415: The angle between − ∇ F ( a n ) {\displaystyle -\nabla F(\mathbf {a_{n}} )} and p n {\displaystyle \mathbf {p} _{n}} , this requires that cos ⁡ θ n > 0. {\displaystyle \cos \theta _{n}>0.} To say more, we need more information about

1118-428: The angle between the descent direction and the negative gradient. The second term measures how quickly the gradient changes along the descent direction. In principle inequality ( 1 ) could be optimized over p n {\displaystyle \mathbf {p} _{n}} and γ n {\displaystyle \gamma _{n}} to choose an optimal step size and direction. The problem

1161-1093: The function F {\displaystyle F} (for example, F {\displaystyle F} convex and ∇ F {\displaystyle \nabla F} Lipschitz ) and particular choices of γ {\displaystyle \gamma } . Those include the sequence γ n = | ( x n − x n − 1 ) T [ ∇ F ( x n ) − ∇ F ( x n − 1 ) ] | ‖ ∇ F ( x n ) − ∇ F ( x n − 1 ) ‖ 2 {\displaystyle \gamma _{n}={\frac {\left|\left(\mathbf {x} _{n}-\mathbf {x} _{n-1}\right)^{T}\left[\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})\right]\right|}{\left\|\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})\right\|^{2}}}} as in

1204-515: The geometry of the space to shape the function level sets like concentric circles , cures the slow convergence. Constructing and applying preconditioning can be computationally expensive, however. The gradient descent can be modified via momentums ( Nesterov , Polyak, and Frank-Wolfe ) and heavy-ball parameters (exponential moving averages and positive-negative momentum ). The main examples of such optimizers are Adam, DiffGrad, Yogi, AdaBelief, etc. Methods based on Newton's method and inversion of

1247-417: The global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so they must use local information to find the minimum. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). If they were trying to find

1290-481: The gradient, toward the local minimum. With this observation in mind, one starts with a guess x 0 {\displaystyle \mathbf {x} _{0}} for a local minimum of F {\displaystyle F} , and considers the sequence x 0 , x 1 , x 2 , … {\displaystyle \mathbf {x} _{0},\mathbf {x} _{1},\mathbf {x} _{2},\ldots } such that We have

1333-524: The latter case, the search space is typically a function space , and one calculates the Fréchet derivative of the functional to be minimized to determine the descent direction. That gradient descent works in any number of dimensions (finite number at least) can be seen as a consequence of the Cauchy-Schwarz inequality , i.e. the magnitude of the inner (dot) product of two vectors of any dimension

NTM - Misplaced Pages Continue

1376-422: The maximum to minimum eigenvalues of A T A {\displaystyle A^{T}A} ) , while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from preconditioning , where gradient descent may require less assumptions on the preconditioner. Gradient descent can also be used to solve

1419-409: The more general update: Finding good settings of p n {\displaystyle \mathbf {p} _{n}} and γ n {\displaystyle \gamma _{n}} requires some thought. First of all, we would like the update direction to point downhill. Mathematically, letting θ n {\displaystyle \theta _{n}} denote

1462-480: The most basic algorithm used for training most deep networks today. Gradient descent is based on the observation that if the multi-variable function F ( x ) {\displaystyle F(\mathbf {x} )} is defined and differentiable in a neighborhood of a point a {\displaystyle \mathbf {a} } , then F ( x ) {\displaystyle F(\mathbf {x} )} decreases fastest if one goes from

1505-446: The objective function at this value, yields The decrease from F ( 0 ) = 58.456 {\displaystyle F(\mathbf {0} )=58.456} to the next step's value of is a sizable decrease in the objective function. Further steps would reduce its value further until an approximate solution to the system was found. Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In

1548-428: The objective function that we are optimising. Under the fairly weak assumption that F {\displaystyle F} is continuously differentiable, we may prove that: This inequality implies that the amount by which we can be sure the function F {\displaystyle F} is decreased depends on a trade off between the two terms in square brackets. The first term in square brackets measures

1591-558: The original NTM paper did not publish their source code . The first stable open-source implementation was published in 2018 at the 27th International Conference on Artificial Neural Networks, receiving a best-paper award. Other open source implementations of NTMs exist but as of 2018 they are not sufficiently stable for production use. The developers either report that the gradients of their implementation sometimes become NaN during training for unknown reasons and cause training to fail; report slow convergence; or do not report

1634-457: The persons represent the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the function at that point. The instrument used to measure steepness is differentiation . The direction they choose to travel in aligns with the gradient of the function at that point. The amount of time they travel before taking another measurement

1677-440: The recipes above, convergence to a local minimum can be guaranteed. When the function F {\displaystyle F} is convex , all local minima are also global minima, so in this case gradient descent can converge to the global solution. Gradient descent can be used to solve a system of linear equations reformulated as a quadratic minimization problem. If the system matrix A {\displaystyle A}

1720-403: The same term [REDACTED] This disambiguation page lists articles associated with the title NTM . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=NTM&oldid=1127483345 " Category : Disambiguation pages Hidden categories: Short description

1763-407: The speed of learning of their implementation. Differentiable neural computers are an outgrowth of Neural Turing machines, with attention mechanisms that control where the memory is active, and improve performance. Gradient descent The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this

NTM - Misplaced Pages Continue

1806-419: The top of the mountain (i.e., the maximum), then they would proceed in the direction of steepest ascent (i.e., uphill). Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point ), like a mountain lake. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires

1849-427: The traditional algorithm, The method is rarely used for solving linear equations, with the conjugate gradient method being one of the most popular alternatives. The number of gradient descent iterations is commonly proportional to the spectral condition number κ ( A ) {\displaystyle \kappa (A)} of the system matrix A {\displaystyle A} (the ratio of

#940059