Many machine learning models depend on solving a large scale optimization problem. Recently, sub-sampled Newton methods have emerged to attract much attention for optimization due to their efficiency at each iteration, rectified a weakness in the ordinary Newton method of suffering a high cost at each iteration while commanding a high convergence rate. In this work we propose two new efficient ...

This paper explores Wallis’s role as editor of Newton’s mathematical work. My objective is to understand how two mathematicians who held different views concerning mathematical method could nonetheless cooperate with one another quite effectively. Most notably, Wallis and Newton pursued different policies as far as the printing of algebra is concerned. In the 1690s Newton held the view that alg...

Newton law arising due to the gravity localized on the general singular brane embedded in AdS5 bulk is examined in the absence or presence of the 4d induced Einstein term. For the RS brane, apart from the subleading correction, Newton potential obeys 4d-type and 5d-type gravitational law at longand short-ranges if it were not for the induced Einstein term. The 4d induced Einstein term generates...

This paper explores Wallis’s role as editor of Newton’s mathematical work. My objective is to understand how two mathematicians who held different views concerning mathematical method could nonetheless cooperate with one another quite effectively. Most notably, Wallis and Newton pursued different policies as far as the printing of algebra is concerned. In the 1690s Newton held the view that alg...

Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to N...

Interval Newton methods are widely used to find reliable enclosures for the roots of the polynomial systems. But the computation process needs the evaluation of an enclosure for the derivative of the polynomial and evaluation of the polynomial at a particular point. Again, the subdivisions (if any are needed) will require repeated evaluation of the function values. We propose an alternative app...

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners destroys in principle the...

In the Newton/log-barriermethod,Newton steps are taken for the log-barrier function for a xed value of the barrier parameter until a certain convergence criterion is satis ed. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton's method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier fun...

Newton iteration is known (under some precise conditions) to converge quadratically to zeros of non-degenerate systems of polynomials. This and other properties may be used to obtain theorems on the global complexity of solving systems of polynomial equations (See Shub and Smale in 6]), using a model of computability over the reals. However, it is not practical (and not desirable) to actually c...