Abstract: We study the problem of transferring a sample in one domain to an analog sample in another domain. Given two related domains, S and T, we would like to learn a generative function G that maps an input sample from S to the domain T, such that the output of a given function f, which accepts inputs in either domains, would remain unchanged. Other than the function f, the training data is unsupervised and consist of a set of samples from each domain. The Domain Transfer Network (DTN) we present employs a compound loss function that includes a multiclass GAN loss, an f-constancy component, and a regularizing component that encourages G to map samples from T to themselves. We apply our method to visual domains including digits and face images and demonstrate its ability to generate convincing novel images of previously unseen entities, while preserving their identity.

Abstract: Here F(x) and G(x) are less-than cumulative probability distributionis where x is a continuous or discrete random variable representing the outcome of a prospect. The closed interval [a, b] is the sample space of both prospects. The integral shown in Rule 2 and those shown throughout the paper are Stieltjes integrals. Recall that the Stieltjes integral fb f(x)dg(x) exists if one of the functions f and g is continuous and the other has finite variation in [a, b]. Let D1, D2, and D3 be three sets of utility functions ?(x). D1 is the set containing all utility functions with 4(x) and +1(x) continuous, and 41(x) >0 for all xE[a, b]. D2 is the set with ?(x), ?1(x), ?2(x) continuous, and q$j(x)>0, 02(x)?O for all xC[a, b]. D3 is the set with ?(x), ?1(x), ?2(X), ?3(X) continuous, and +1(x) > 04 2(x) O O for all xC[a, b]. Here +1(x) denotes the ith derivative of +(x). Hadar and Russell proved that Rule 1 is valid for all ,CD1 and Rutle 2 is valid for all ED2. The authors point out that the set of probability distributions that can be ordered by means of second-degree stochastic dominance is, in general, larger than that which can be ordered by means of first-degree stochastic dominance. Note that in Rule 2, they assume that +(x) is not only an increasing function of x but also exhibits weak global risk aversion, a condition guaranteed by requiring the second derivative of ?(x) to be nonpositive. In this paper, a condition which will be called third-degree stochastic dominance is considered. It is based on the following assumption about the form of the utility function ?(x). From a normative point of view, one expects the risk premium associated with an uncertain prospect to become smaller the greater is the individual's wealth. The plausibility and implications of this assumption h'ave been explored by John Pratt, as well as others. The risk premium of an uncertain prospect is that amount by which the certainty equivalent of the prospect differs from its expected value. In mathematical terms, given the prospect F(x) with expected value A, the corresponding risk premium -t is obtained by solving the following equation. rb

Abstract: We study the problem of transferring a sample in one domain to an analog sample in another domain. Given two related domains, S and T, we would like to learn a generative function G that maps an input sample from S to the domain T, such that the output of a given function f, which accepts inputs in either domains, would remain unchanged. Other than the function f, the training data is unsupervised and consist of a set of samples from each domain. The Domain Transfer Network (DTN) we present employs a compound loss function that includes a multiclass GAN loss, an f-constancy component, and a regularizing component that encourages G to map samples from T to themselves. We apply our method to visual domains including digits and face images and demonstrate its ability to generate convincing novel images of previously unseen entities, while preserving their identity.

Abstract: In this paper, a value iteration adaptive dynamic programming (ADP) algorithm is developed to solve infinite horizon undiscounted optimal control problems for discrete-time nonlinear systems. The present value iteration ADP algorithm permits an arbitrary positive semi-definite function to initialize the algorithm. A novel convergence analysis is developed to guarantee that the iterative value function converges to the optimal performance index function. Initialized by different initial functions, it is proven that the iterative value function will be monotonically nonincreasing, monotonically nondecreasing, or nonmonotonic and will converge to the optimum. In this paper, for the first time, the admissibility properties of the iterative control laws are developed for value iteration algorithms. It is emphasized that new termination criteria are established to guarantee the effectiveness of the iterative control laws. Neural networks are used to approximate the iterative value function and compute the iterative control law, respectively, for facilitating the implementation of the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the present method.

Abstract: There has been a growing effort in studying the distributed optimization problem over a network. The objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. Literature has developed consensus-based distributed (sub)gradient descent (DGD) methods and has shown that they have the same convergence rate O(log t/√t) as the centralized (sub)gradient methods (CGD) when the function is convex but possibly nonsmooth. However, when the function is convex and smooth, under the framework of DGD, it is unclear how to harness the smoothness to obtain a faster convergence rate comparable to CGD's convergence rate. In this paper, we propose a distributed algorithm that, despite using the same amount of communication per iteration as DGD, can effectively harnesses the function smoothness and converge to the optimum with a rate of O(1/t). If the objective function is further strongly convex, our algorithm has a linear convergence rate. Both rates match the convergence rate of CGD. The key step in our algorithm is a novel gradient estimation scheme that uses history information to achieve fast and accurate estimation of the average gradient. To motivate the necessity of history information, we also show that it is impossible for a class of distributed algorithms like DGD to achieve a linear convergence rate without using history information even if the objective function is strongly convex and smooth.

Abstract: In this work, we present a unified performance analysis of a free-space optical (FSO) link that accounts for pointing errors and both types of detection techniques [i.e., intensity modulation/direct detection (IM/DD) and heterodyne detection]. More specifically, we present unified exact closed-form expressions for the cumulative distribution function, the probability density function, the moment generating function, and the moments of the end-to-end signal-to-noise ratio (SNR) of a single link FSO transmission system, all in terms of the Meijer’s G function except for the moments that is in terms of simple elementary functions. We then capitalize on these unified results to offer unified exact closed-form expressions for various performance metrics of FSO link transmission systems, such as the outage probability, the scintillation index (SI), the average error rate for binary and $M$ -ary modulation schemes, and the ergodic capacity (except for IM/DD technique, where we present closed-form lower bound results), all in terms of Meijer’s G functions except for the SI that is in terms of simple elementary functions. Additionally, we derive the asymptotic results for all the expressions derived earlier in terms of Meijer’s G function in the high SNR regime in terms of simple elementary functions via an asymptotic expansion of the Meijer’s G function. We also derive new asymptotic expressions for the ergodic capacity in the low as well as high SNR regimes in terms of simple elementary functions via utilizing moments. All the presented results are verified via computer-based Monte-Carlo simulations.

Abstract: Policy gradient methods are an appealing approach in reinforcement learning because they directly optimize the cumulative reward and can straightforwardly be used with nonlinear function approximators such as neural networks. The two main challenges are the large number of samples typically required, and the difficulty of obtaining stable and steady improvement despite the nonstationarity of the incoming data. We address the first challenge by using value functions to substantially reduce the variance of policy gradient estimates at the cost of some bias, with an exponentially-weighted estimator of the advantage function that is analogous to TD(lambda). We address the second challenge by using trust region optimization procedure for both the policy and the value function, which are represented by neural networks. Our approach yields strong empirical results on highly challenging 3D locomotion tasks, learning running gaits for bipedal and quadrupedal simulated robots, and learning a policy for getting the biped to stand up from starting out lying on the ground. In contrast to a body of prior work that uses hand-crafted policy representations, our neural network policies map directly from raw kinematics to joint torques. Our algorithm is fully model-free, and the amount of simulated experience required for the learning tasks on 3D bipeds corresponds to 1-2 weeks of real time.

Abstract: We propose mS2GD: a method incorporating a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent (S2GD). We consider the problem of minimizing a strongly convex function represented as the sum of an average of a large number of smooth convex functions, and a simple nonsmooth convex regularizer. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate becoming the new starting point. The novelty of our method is in introduction of mini-batching into the computation of stochastic steps. In each step, instead of choosing a single function, we sample $b$ functions, compute their gradients, and compute the direction based on this. We analyze the complexity of the method and show that it benefits from two speedup effects. First, we prove that as long as $b$ is below a certain threshold, we can reach any predefined accuracy with less overall work than without mini-batching. Second, our mini-batching scheme admits a simple parallel implementation, and hence is suitable for further acceleration by parallelization.

Abstract: Function Secret Sharing (FSS), introduced by Boyle et al. (Eurocrypt 2015), provides a way for additively secret-sharing a function from a given function family F. More concretely, an m-party FSS scheme splits a function f : {0, 1}n -> G, for some abelian group G, into functions f1,...,fm, described by keys k1,...,km, such that f = f1 + ... + fm and every strict subset of the keys hides f. A Distributed Point Function (DPF) is a special case where F is the family of point functions, namely functions f_{a,b} that evaluate to b on the input a and to 0 on all other inputs. FSS schemes are useful for applications that involve privately reading from or writing to distributed databases while minimizing the amount of communication. These include different flavors of private information retrieval (PIR), as well as a recent application of DPF for large-scale anonymous messaging. We improve and extend previous results in several ways: * Simplified FSS constructions. We introduce a tensoring operation for FSS which is used to obtain a conceptually simpler derivation of previous constructions and present our new constructions. * Improved 2-party DPF. We reduce the key size of the PRG-based DPF scheme of Boyle et al. roughly by a factor of 4 and optimize its computational cost. The optimized DPF significantly improves the concrete costs of 2-server PIR and related primitives. * FSS for new function families. We present an efficient PRG-based 2-party FSS scheme for the family of decision trees, leaking only the topology of the tree and the internal node labels. We apply this towards FSS for multi-dimensional intervals. We also present a general technique for extending FSS schemes by increasing the number of parties. * Verifiable FSS. We present efficient protocols for verifying that keys (k*/1,...,k*/m ), obtained from a potentially malicious user, are consistent with some f in F. Such a verification may be critical for applications that involve private writing or voting by many users.

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Abstract: In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calculation on the active region of interest, and a low level calculation on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: it density functional embedding, Green's function embedding, and density matrix embedding, respectively use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used: functional equations for density functional embedding; diagrammatics for Green's function embedding; and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications, and possible future developments.

Abstract: In this paper, we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build preaggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators such as the maximum and the Choquet integral.

Abstract: This paper addresses the problem of distributed multi-agent optimization in which each agent i has a local cost function hi(x), and the goal is to optimize a global cost function that aggregates the local cost functions. Such optimization problems are of interest in many contexts, including distributed machine learning, distributed resource allocation, and distributed robotics.We consider the distributed optimization problem in the presence of faulty agents. We focus primarily on Byzantine failures, but also briey discuss some results for crash failures. For the Byzantine fault-tolerant optimization problem, the ideal goal is to optimize the average of local cost functions of the non-faulty agents. However, this goal also cannot be achieved. Therefore, we consider a relaxed version of the fault-tolerant optimization problem.The goal for the relaxed problem is to generate an output that is an optimum of a global cost function formed as a convex combination of local cost functions of the non-faulty agents. More precisely, there must exist weights αi for i∈N such that αi ≥ 0 and ∑i≥ Nαi=1, and the output is an optimum of the cost function ∑i≥ N αihi(x). Ideally, we would like αi=1/|N| for all i≥ N, however, this cannot be guaranteed due to the presence of faulty agents. In fact, the maximum number of nonzero weights (αi's) that can be guaranteed is |N|-f, where f is the maximum number of Byzantine faulty agents.We present an iterative distributed algorithm that achieves optimal fault-tolerance. Specifically, it ensures that at least |N|-f agents have weights that are bounded away from 0 (in particular, lower bounded by 1/2|N|-f}). The proposed distributed algorithm has a simple iterative structure, with each agent maintaining only a small amount of local state. We show that the iterative algorithm ensures two properties as time goes to ∞: consensus (i.e., output of non-faulty agents becomes identical in the time limit), and optimality (in the sense that the output is the optimum of a suitably defined global cost function).

Abstract: The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painleve transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are reduced to the study of a nonlinear special function in particular limits. The universal object that these functions share is a Riemann – Hilbert representation: the nonlinear special function can be recovered from the solution of a Riemann-Hilbert problem (RHP). A RHP consists of finding a piecewise-analytic function in the complex plane when the behavior of its discontinuities is specified. In this dissertation, the applied theory of Riemann-Hilbert problems, using both Holder and Lebesgue spaces, is reviewed. The numerical solution of RHPs is discussed. Furthermore, the uniform approximation theory for the numerical solution of RHPs is presented, proving that in certain cases the convergence of the numerical method is uniform with respect to a parameter. This theory shares close relation to the method of nonlinear steepest descent for RHPs. The inverse scattering transform for the Korteweg – de Vries and Nonlinear Schroedinger equation is made effective by solving the associated RHPs numerically. This technique is extended to solve the Painleve II equation numerically. Similar Riemann-Hilbert techniques are used to compute the so-called finite-genus solutions of the Korteweg-de Vries equation. This involves ideas from Riemann surface theory. Finally, the methodology is applied to compute orthogonal polynomials with exponential weights. This allows for the computation of statistical quantities stemming from random matrix ensembles.

Abstract: A standard model for Recommender Systems is the Matrix Completion setting: given partially known matrix of ratings given by users (rows) to items (columns), infer the unknown ratings. In the last decades, few attempts where done to handle that objective with Neural Networks, but recently an architecture based on Autoencoders proved to be a promising approach. In current paper, we enhanced that architecture (i) by using a loss function adapted to input data with missing values, and (ii) by incorporating side information. The experiments demonstrate that while side information only slightly improve the test error averaged on all users/items, it has more impact on cold users/items.

Abstract: Neural networks are a powerful class of functions that can be trained with simple gradient descent to achieve state-of-the-art performance on a variety of applications. Despite their practical success, there is a paucity of results that provide theoretical guarantees on why they are so effective. Lying in the center of the problem is the difficulty of analyzing the non-convex loss function with potentially numerous local minima and saddle points. Can neural networks corresponding to the stationary points of the loss function learn the true target function? If yes, what are the key factors contributing to such nice optimization properties? In this paper, we answer these questions by analyzing one-hidden-layer neural networks with ReLU activation, and show that despite the non-convexity, neural networks with diverse units have no spurious local minima. We bypass the non-convexity issue by directly analyzing the first order optimality condition, and show that the loss can be made arbitrarily small if the minimum singular value of the "extended feature matrix" is large enough. We make novel use of techniques from kernel methods and geometric discrepancy, and identify a new relation linking the smallest singular value to the spectrum of a kernel function associated with the activation function and to the diversity of the units. Our results also suggest a novel regularization function to promote unit diversity for potentially better generalization.

Abstract: We study the minimization of a convex function $f(X)$ over the set of $n\times n$ positive semi-definite matrices, but when the problem is recast as $\min_U g(U) := f(UU^\top)$, with $U \in \mathbb{R}^{n \times r}$ and $r \leq n$. We study the performance of gradient descent on $g$---which we refer to as Factored Gradient Descent (FGD)---under standard assumptions on the original function $f$. We provide a rule for selecting the step size and, with this choice, show that the local convergence rate of FGD mirrors that of standard gradient descent on the original $f$: i.e., after $k$ steps, the error is $O(1/k)$ for smooth $f$, and exponentially small in $k$ when $f$ is (restricted) strongly convex. In addition, we provide a procedure to initialize FGD for (restricted) strongly convex objectives and when one only has access to $f$ via a first-order oracle; for several problem instances, such proper initialization leads to global convergence guarantees. FGD and similar procedures are widely used in practice for problems that can be posed as matrix factorization. To the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions.

Abstract: We consider parallel global optimization of derivative-free expensive-to-evaluate functions, and propose an efficient method based on stochastic approximation for implementing a conceptual Bayesian optimization algorithm proposed by Ginsbourger et al. (2007). At the heart of this algorithm is maximizing the information criterion called the "multi-points expected improvement'', or the q-EI. To accomplish this, we use infinitessimal perturbation analysis (IPA) to construct a stochastic gradient estimator and show that this estimator is unbiased. We also show that the stochastic gradient ascent algorithm using the constructed gradient estimator converges to a stationary point of the q-EI surface, and therefore, as the number of multiple starts of the gradient ascent algorithm and the number of steps for each start grow large, the one-step Bayes optimal set of points is recovered. We show in numerical experiments that our method for maximizing the q-EI is faster than methods based on closed-form evaluation using high-dimensional integration, when considering many parallel function evaluations, and is comparable in speed when considering few. We also show that the resulting one-step Bayes optimal algorithm for parallel global optimization finds high-quality solutions with fewer evaluations than a heuristic based on approximately maximizing the q-EI. A high-quality open source implementation of this algorithm is available in the open source Metrics Optimization Engine (MOE).

Abstract: Many physical systems can be modeled by a Wiener nonlinear model, which consists of a linear dynamic system followed by a nonlinear static function. This work is concerned with the identification of Wiener systems whose output nonlinear function is assumed to be continuous and invertible. A recursive least squares algorithm is presented based on the auxiliary model identification idea. To solve the difficulty of the information vector including the unmeasurable variables, the unknown terms in the information vector are replaced with their estimates, which are computed through the preceding parameter estimates. Finally, an example is given to support the proposed method.

Abstract: We explore the efficiency of a statistical learning technique based on Gaussian process (GP) regression as an efficient non-parametric method for constructing multi-dimensional potential energy surfaces (PESs) for polyatomic molecules. Using an example of the molecule N4, we show that a realistic GP model of the six-dimensional PES can be constructed with only 240 potential energy points. We construct a series of the GP models and illustrate the accuracy of the resulting surfaces as a function of the number of ab initio points. We show that the GP model based on ~1500 potential energy points achieves the same level of accuracy as the conventional regression fits based on 16 421 points. The GP model of the PES requires no fitting of ab initio data with analytical functions and can be readily extended to surfaces of higher dimensions.

Abstract: Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that functions of random variables are typically near their means. Of particular importance is the case where f(X) is a function of independent random variables X = (X 1, . . ., Xn ). Here the well-known bounded differences inequality (also called McDiarmid's inequality or the Hoeffding–Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X) − f(X′)| ⩽ ck whenever X, X′ differ only in Xk . While this is easy to check, the main disadvantage is that it considers worst-case changes ck , which often makes the resulting bounds too weak to be useful. In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f(X) where (i) the typical changes are small, although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations as to why concentration should hold. Indeed, given an event Γ that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where Γ occurs. The point is that the resulting typical changes ck are often much smaller than the worst case ones. To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobas and Erdős.

Abstract: We integrated recent improvements within the floating catchment area (FCA) method family into an integrated ‘iFCA`method. Within this method we focused on the distance decay function and its parameter. So far only distance decay functions with constant parameters have been applied. Therefore, we developed a variable distance decay function to be used within the FCA method. We were able to replace the impedance coefficient β by readily available distribution parameter (i.e. median and standard deviation (SD)) within a logistic based distance decay function. Hence, the function is shaped individually for every single population location by the median and SD of all population-to-provider distances within a global catchment size. Theoretical application of the variable distance decay function showed conceptually sound results. Furthermore, the existence of effective variable catchment sizes defined by the asymptotic approach to zero of the distance decay function was revealed, satisfying the need for variable catchment sizes. The application of the iFCA method within an urban case study in Berlin (Germany) confirmed the theoretical fit of the suggested method. In summary, we introduced for the first time, a variable distance decay function within an integrated FCA method. This function accounts for individual travel behaviors determined by the distribution of providers. Additionally, the function inherits effective variable catchment sizes and therefore obviates the need for determining variable catchment sizes separately.

Abstract: We present measurements of the galaxy luminosity and stellar mass function in a 3.71 deg$^2$ (0.3 Mpc$^2$) area in the core of the Virgo cluster, based on $ugriz$ data from the Next Generation Virgo Cluster Survey (NGVS). The galaxy sample consists of 352 objects brighter than $M_g=-9.13$ mag, the 50% completeness limit of the survey. Using a Bayesian analysis, we find a best-fit faint end slope of $\\alpha=-1.33 \\pm 0.02$ for the g-band luminosity function; consistent results are found for the stellar mass function as well as the luminosity function in the other four NGVS bandpasses. We discuss the implications for the faint-end slope of adding 92 ultra compact dwarfs galaxies (UCDs) -- previously compiled by the NGVS in this region -- to the galaxy sample, assuming that UCDs are the stripped remnants of nucleated dwarf galaxies. Under this assumption, the slope of the luminosity function (down to the UCD faint magnitude limit, $M_g = -9.6$ mag) increases dramatically, up to $\\alpha = -1.60 \\pm 0.06$ when correcting for the expected number of disrupted non-nucleated galaxies. We also calculate the total number of UCDs and globular clusters that may have been deposited in the core of Virgo due to the disruption of satellites, both nucleated and non-nucleated. We estimate that ~150 objects with $M_g\\lesssim-9.6$ mag and that are currently classified as globular clusters, might, in fact, be the nuclei of disrupted galaxies. We further estimate that as many as 40% of the (mostly blue) globular clusters in the core of Virgo might once have belonged to such satellites; these same disrupted satellites might have contributed ~40% of the total luminosity in galaxies observed in the core region today. Finally, we use an updated Local Group galaxy catalog to provide a new measurement of the luminosity function of Local Group satellites, $\\alpha=-1.21\\pm0.05$.

Abstract: We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has k hidden layers and that the l1-norm of the incoming weights of any neuron is bounded by L. We present a kernel-based method, such that with probability at least 1 - δ, it learns a predictor whose generalization error is at most e worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in (1/e, log(1/δ), F(k, L)), where F(k, L) is a function depending on (k, L) and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time.

Abstract: We employ the exponential parametrization of the metric and a “physical” gauge fixing procedure to write a functional flow equation for the gravitational effective average action in an f(R) truncation. The background metric is a four-sphere and the coarse-graining procedure contains three free parameters. We look for scaling solutions, i.e. non-Gaussian fixed points for the function f. For a discrete set of values of the parameters, we find simple global solutions of quadratic polynomial form. For other values, global solutions can be found numerically. Such solutions can be extended in certain regions of parameter space and have two relevant directions. We discuss the merits and the shortcomings of this procedure.

Abstract: In this paper, an effective adaptive control approach is constructed to stabilize a class of nonlinear discrete-time systems, which contain unknown functions, unknown dead-zone input, and unknown control direction. Different from linear dead zone, the dead zone, in this paper, is a kind of nonlinear dead zone. To overcome the noncausal problem, which leads to the control scheme infeasible, the systems can be transformed into a $m$ -step-ahead predictor. Due to nonlinear dead-zone appearance, the transformed predictor still contains the nonaffine function. In addition, it is assumed that the gain function of dead-zone input and the control direction are unknown. These conditions bring about the difficulties and the complicacy in the controller design. Thus, the implicit function theorem is applied to deal with nonaffine dead-zone appearance, the problem caused by the unknown control direction can be resolved through applying the discrete Nussbaum gain, and the neural networks are used to approximate the unknown function. Based on the Lyapunov theory, all the signals of the resulting closed-loop system are proved to be semiglobal uniformly ultimately bounded. Moreover, the tracking error is proved to be regulated to a small neighborhood around zero. The feasibility of the proposed approach is demonstrated by a simulation example.