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For a list of modes of convergence, see Modes of convergence (annotated index) Each of the following objects is a special case of the types preceding it: sets , topological spaces , uniform spaces , topological abelian group , normed spaces , Euclidean spaces , and the real/complex numbers.
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick ...
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
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The PECEC mode has one fewer function evaluation than PECECE mode. More generally, if the corrector is run k times, the method is in P(EC) k or P(EC) k E mode. If the corrector method is iterated until it converges, this could be called PE(CE) ∞. [1]
This can perform significantly better than "true" stochastic gradient descent described, because the code can make use of vectorization libraries rather than computing each step separately as was first shown in [6] where it was called "the bunch-mode back-propagation algorithm". It may also result in smoother convergence, as the gradient ...
if d + 1 is even and C > 0 then convergence to a will be from values greater than a; if d + 1 is even and C < 0 then convergence to a will be from values less than a; if d + 1 is odd and C > 0 then convergence to a will be from the side where it starts; and; if d + 1 is odd and C < 0 then convergence to a will alternate sides.