Enhanced Differentiable Architecture Search Based on Asymptotic Regularization

In differentiable search architecture search methods,a more efficient search space design can significantly improve the performance of the searched architecture,thus requiring people to carefully define the search space with different complexity according to various operations.Meanwhile rationalizing the search strategies to explore the well-defined search space will further improve the speed and efficiency of architecture search.With this in mind,we propose a faster and more efficient differentiable architecture search method,AllegroNAS.Firstly,we introduce a more efficient search space enriched by the introduction of two redefined convolution modules.Secondly,we utilize a more efficient architectural parameter regularization method,mitigating the overfitting problem during the search process and reducing the error brought about by gradient approximation.Meanwhile,we introduce a natural exponential cosine annealing method to make the learning rate of the neural network training process more suitable for the search procedure.Moreover,group convolution and data augmentation are employed to reduce the computational cost.Finally,through extensive experiments on several public datasets,we demonstrate that our method can more swiftly search for better-performing neural network architectures in a more efficient search space,thus validating the effectiveness of our approach.

FIXED POINTS OF A PAIR OF ASYMPTOTICALLY REGULAR MAPPINGS

In this paper some theorems on fixed points of pair of asymptotically regularmappings in p-uniformly convex Banach space are proved For these mappings somefixed point theorems in a Hilbert space.in Lp spaces in Hardy spaces Hp and in Sobolev spaces Hpk for 1<P<+∞ and K>0 are also established.Thus resultsof Gornicki Kruppel and others are extended.

Generalized Kannan-type contraction and fixed point theorems

In this paper,the generalized Kannan-type contraction in cone metric spaces over Banach algebras is introduced.The fixed point theorems satisfying generalized contractive conditions are obtained,without appealing to completeness of X or normality of the cone.The continuity of the mapping is relaxed.Furthermore,we prove that the completeness in cone metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X.These results greatly generalize several well-known comparable results in the literature.

ON THE EXISTENCE OF COMMON FIXED POINTS FOR A PAIR OF LIPSCHITZIAN MAPPINGS IN BANACH SPACES

The existence of common fixed points for a pair of Lipschitzian mappings in Banach spaces is proved. By using this result, some common fixed point theorems are also established for these mappings in Hilbert spaces, in L p spaces, in Hardy spaces H p, and in Sobolev spaces H r,p , for 1<p<+∞ and r≥0.

COMMON FIXED POINTS WITH APPLICATIONS TO BEST SIMULTANEOUS APPROXIMATIONS

For a subset K of a metric space （X,d） and x ∈ X,Px（x）={y ∈ K ： d（x,y） = d（x,K）≡ inf{d（x,k） ： k ∈ K}}is called the set of best K-approximant to x. An element go E K is said to be a best simulta- neous approximation of the pair y1,y2 E ∈ if max{d（y1,go）,d（y2,go）}=inf g∈K max {d（y1,g）,d（y2,g）}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T- and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas^[1], S. Chandok and T.D. Narang^[2], T.D. Narang and S. Chandok^[11], S.A. Sahab, M.S. Khan and S. Sessa^[14], P. Vijayaraju^[20] and P. Vijayaraju and M. Marudai^[21].

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈? S} of selfmappings on C satisfying