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A mapping T of a metric space X,d into a metric space Y,ρ is called restrictive Lipschitz if there exist: a positive decreasing to zero sequence tn:n∈N and a nonnegative sequence Ln:n∈N, with L:=lim infn→∞Ln<∞, such that for all x,y∈X,n∈Ndx,y=tn⟹ρTx,Ty≤Lntn.Using a basis property of the sequence tn:n∈N (Lemma 1), we prove that if T is a continuous and restrictive Lipschitz mapping of a complete metrically convex space X,d into a metric space Y,ρ, then T is Lipschitz continuous with the constant L, that is ρTx,Ty≤Ldx,y,x,y∈X,and, in the case when the set n∈N:Ln0. This result leads to the following fixed-point principle: Every continuous selfmapping T of a nonempty metrically convex complete metric space X,d that is restrictive Lipschitz with a sequence Ln:n∈N, such that0≤Ln<1(n∈N) and lim infn→∞Ln≤1, has a unique fixed point, and either it is a Banach contraction, or there is an increasing concave function α:0,∞→0,∞, such that αt0 and dTx,Ty≤αdx,y,x,y∈X.Some applications of these results to the theory of iterative functional equations are proposed.

We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant—for multiple p-norms—of a feed forward neural network composed of commonly used layer types. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that the performance of the resulting models exceeds that of models trained with other common regularisers. We also provide evidence that the hyperparameters are intuitive to tune, demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model, and show that the performance gains provided by our method are particularly noticeable when only a small amount of training data is available.

A theorem of Stepanoff claims that approximate differentiability almost everywhere of a function u is equivalent to existence almost everywhere of approximate partial derivatives of the function, while Whitney proved that approximate differentiability almost everywhere of u is equivalent to the following Lusin-type property: (*) Given ε > 0, there is a C1 function ν on ℝn such that |{x ∈ D : u(x) ≠ ν(x)}| < ε. Federer then established that (*) is equivalent to having u be approximately locally Lipschitz almost everywhere in the sense that $\\underset{\\mathrm{y}\\to \\mathrm{x}}{\\mathrm{a}\\mathrm{p} \\ \\mathrm{lim} \\ \\mathrm{sup}}\\frac{\\left|\\mathrm{u}\\left(\\mathrm{y}\\right)-\\mathrm{u}\\left(\\mathrm{x}\\right)\\right|}{|\\mathrm{y}-\\mathrm{x}|}<\\mathrm{\\infty }$ holds almost everywhere. This paper extends these results to the case of approximate differentiability of general order ɤ which is not necessarily an integer.

This paper is about the exploitation of Lipschitz continuity properties for Markov Decision Processes to safely speed up policy-gradient algorithms. Starting from assumptions about the Lipschitz continuity of the state-transition model, the reward function, and the policies considered in the learning process, we show that both the expected return of a policy and its gradient are Lipschitz continuous w.r.t. policy parameters. By leveraging such properties, we define policy-parameter updates that guarantee a performance improvement at each iteration. The proposed methods are empirically evaluated and compared to other related approaches using different configurations of three popular control scenarios: the linear quadratic regulator, the mass-spring-damper system and the ship-steering control.

In this paper we investigate the solutions of the so-called α¯ -Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q) -harmonic functions.

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided.

In this article, we consider the Hölder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the Hölder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We also consider Hölder and Lipschitz continuity of harmonic mappings and in particular of harmonic mappings in Orlicz-Sobolev classes. In addition in planar case, we show in some situations that the map is bi-Lipschitzian if Beltrami coefficient is Hölder continuous.

Despite the recent success of reinforcement learning in various domains, these approaches remain, for the most part, deterringly sensitive to hyper-parameters and are often riddled with essential engineering feats allowing their success. We consider the case of off-policy generative adversarial imitation learning, and perform an in-depth review, qualitative and quantitative, of the method. We show that forcing the learned reward function to be local Lipschitz-continuous is a sine qua non condition for the method to perform well. We then study the effects of this necessary condition and provide several theoretical results involving the local Lipschitzness of the state-value function. We complement these guarantees with empirical evidence attesting to the strong positive effect that the consistent satisfaction of the Lipschitzness constraint on the reward has on imitation performance. Finally, we tackle a generic pessimistic reward preconditioning add-on spawning a large class of reward shaping methods, which makes the base method it is plugged into provably more robust, as shown in several additional theoretical guarantees. We then discuss these through a fine-grained lens and share our insights. Crucially, the guarantees derived and reported in this work are valid for any reward satisfying the Lipschitzness condition, nothing is specific to imitation. As such, these may be of independent interest.

This article is devoted to the Mizar formalization of various properties of differentiability of Lipschitzian bilinear operators in real normed spaces. Main results include the Lipschitz continuity of partial derivatives, the representation of the total derivative in terms of partial derivatives, and the continuous differentiability of Lipschitzian bilinear operators on open subsets of the product space.

In this paper, we introduce a new algorithm of inertial form for solving monotone variational inequalities (VI) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and Lipschitz continuity of the VI associated mapping, we establish the weak convergence of the scheme. Several numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature.