Showing papers on "Bounded function published in 2019"

OCGAN: One-Class Novelty Detection Using GANs With Constrained Latent Representations

Abstract: We present a novel model called OCGAN for the classical problem of one-class novelty detection, where, given a set of examples from a particular class, the goal is to determine if a query example is from the same class. Our solution is based on learning latent representations of in-class examples using a de-noising auto-encoder network. The key contribution of our work is our proposal to explicitly constrain the latent space to exclusively represent the given class. In order to accomplish this goal, firstly, we force the latent space to have bounded support by introducing a tanh activation in the encoder's output layer. Secondly, using a discriminator in the latent space that is trained adversarially, we ensure that encoded representations of in-class examples resemble uniform random samples drawn from the same bounded space. Thirdly, using a second adversarial discriminator in the input space, we ensure all randomly drawn latent samples generate examples that look real. Finally, we introduce a gradient-descent based sampling technique that explores points in the latent space that generate potential out-of-class examples, which are fed back to the network to further train it to generate in-class examples from those points. The effectiveness of the proposed method is measured across four publicly available datasets using two one-class novelty detection protocols where we achieve state-of-the-art results.

Neural Networks-Based Adaptive Control for Nonlinear State Constrained Systems With Input Delay

Abstract: This paper addresses the problem of adaptive tracking control for a class of strict-feedback nonlinear state constrained systems with input delay. To alleviate the major challenges caused by the appearances of full state constraints and input delay, an appropriate barrier Lyapunov function and an opportune backstepping design are used to avoid the constraint violation, and the Pade approximation and an intermediate variable are employed to eliminate the effect of the input delay. Neural networks are employed to estimate unknown functions in the design procedure. It is proven that the closed-loop signals are semiglobal uniformly ultimately bounded, and the tracking error converges to a compact set of the origin, as well as the states remain within a bounded interval. The simulation studies are given to illustrate the effectiveness of the proposed control strategy in this paper.

Finite-Time Adaptive Fuzzy Control for Nonlinear Systems With Full State Constraints

Abstract: In this paper, an adaptive fuzzy controller is constructed to address the finite-time tracking control problem for a class of strict-feedback nonlinear systems, where the full state constraints are strictly required in the systems. Backstepping design with a tan-type barrier Lyapunov function is proposed. Meanwhile, fuzzy logic systems are used to approximate the unknown nonlinear functions. The addressed control scheme guarantees that the output is followed the reference signals within a bounded error, and all the signals in the closed-loop system are bounded. The simulation results demonstrate the validity of the proposed method.

On deep learning as a remedy for the curse of dimensionality in nonparametric regression

Abstract: Assuming that a smoothness condition and a suitable restriction on the structure of the regression function hold, it is shown that least squares estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric regression. The proof is based on new approximation results concerning multilayer feedforward neural networks with bounded weights and a bounded number of hidden neurons. The estimates are compared with various other approaches by using simulated data.

Fuzzy-Based Multierror Constraint Control for Switched Nonlinear Systems and Its Applications

Abstract: In this paper, a framework of adaptive control for a switched nonlinear system with multiple prescribed performance bounds is established using an improved dwell time technique. Since the prescribed performance bounds for subsystems are different from each other, the different coordinate transformations have to be tackled when the system is transformed, which have not been encountered in some switched systems. We deal with the different coordinate transformations by finding a specific relationship between any two different coordinate transformations. To obtain a much less conservative result, in contrast to the common adaptive law, different adaptive laws are established for both active and inactive time-interval of each subsystem. The proposed controllers and switching signals guarantee that all signals appearing in the closed-loop system are bounded. Furthermore, both transient-state and steady-state performances of the switched system are obtained. Finally, the effectiveness of the developed method is verified by the application to a continuous stirred tank reactor system.

The string swampland constraints require multi-field inflation

Abstract: An important unsolved problem that affects practically all attempts to connect string theory to cosmology and phenomenology is how to distinguish effective field theories belonging to the string landscape from those that are not consistent with a quantum theory of gravity at high energies (the "string swampland"). It was recently proposed that potentials of the string landscape must satisfy at least two conditions, the "swampland criteria", that severely restrict the types of cosmological dynamics they can sustain. The first criterion states that the (multi-field) effective field theory description is only valid over a field displacement Δ ≤ Δ ~ (1) (in units where the Planck mass is 1), measured as a distance in the target space geometry. A second, more recent, criterion asserts that, whenever the potential V is positive, its slope must be bounded from below, and suggests |∇ V| / V ≥ c ~ (1). A recent analysis concluded that these two conditions taken together practically rule out slow-roll models of inflation. In this note we show that the two conditions rule out inflationary backgrounds that follow geodesic trajectories in field space, but not those following curved, non-geodesic, trajectories (which are parametrized by a non-vanishing bending rate Ω of the multi-field trajectory). We derive a universal lower bound on Ω (relative to the Hubble parameter H) as a function of Δ, c and the number of efolds Ne, assumed to be at least of order 60. If later studies confirm c and Δ to be strictly (1), the bound implies strong turns with Ω / H ≥ 3 Ne ~ 180. Slow-roll inflation in the landscape is not ruled out, but it is strongly multi-field.

Lower Bounds for Non-Convex Stochastic Optimization.

Abstract: We lower bound the complexity of finding $\epsilon$-stationary points (with gradient norm at most $\epsilon$) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least $\epsilon^{-4}$ queries to find an $\epsilon$ stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of $\epsilon^{-3}$ queries, establishing the optimality of recently proposed variance reduction techniques.

Anti-pluricanonical systems on Fano varieties

Abstract: In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon>0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

Abstract: We extend the De Giorgi–Nash–Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the Holder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations " of type II " , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law γ ∈ [−d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Holder regularity of these solutions.

Bounded Neural Network Control for Target Tracking of Underactuated Autonomous Surface Vehicles in the Presence of Uncertain Target Dynamics

Abstract: This paper is concerned with the target tracking of underactuated autonomous surface vehicles with unknown dynamics and limited control torques. The velocity of the target is unknown, and only the measurements of line-of-sight range and angle are obtained. First, a kinematic control law is designed based on an extended state observer, which is utilized to estimate the uncertain target dynamics due to the unknown velocities. Next, an estimation model based on a single-hidden-layer neural network is developed to approximate the unknown follower dynamics induced by uncertain model parameters, unmodeled dynamics, and environmental disturbances. A bounded control law is designed based on the neural estimation model and a saturated function. The salient feature of the proposed controller is twofold. First, only the measured line-of-sight range and angle are used, and the velocity information of the target is not required. Second, the control torques are bounded with the bounds known as a priori . The input-to-state stability of the closed-loop system is analyzed via cascade theory. Simulations illustrate the effectiveness of the proposed bounded controller for tracking a moving target.

Lineability: The Search For Linearity In Mathematics

Abstract: Preliminary Notions and Tools Cardinal numbers Cardinal arithmetic Basic concepts and results of abstract and linear algebra Residual subsets Lineability, spaceability, algebrability, and their variants Real Analysis What one needs to know Weierstrass' monsters Differentiable nowhere monotone functions Nowhere analytic functions and annulling functions Surjections, Darboux functions, and related properties Other properties related to the lack of continuity Continuous functions that attain their maximum at only one point Peano maps and space-filling curves Complex Analysis What one needs to know Nonextendable holomorphic functions: genericity Vector spaces of nonextendable functions Nonextendability in the unit disc Tamed entire functions Wild behavior near the boundary Nowhere Gevrey differentiability Sequence Spaces, Measure Theory, and Integration What one needs to know Lineability and spaceability in sequence spaces Non-contractive maps and spaceability in sequence spaces Lineability and spaceability in Lp[0, 1] Spaceability in Lebesgue spaces Lineability in sets of norm attaining operators in sequence spaces Riemann and Lebesgue integrable functions and spaceability Universality, Hypercyclicity, and Chaos What one needs to know Universal elements and hypercyclic vectors Lineability and dense-lineability of families of hypercyclic vectors Wild behavior near the boundary, universal series, and lineability Hypercyclicity and spaceability Algebras of hypercyclic vectors Supercyclicity and lineability Frequent hypercyclicity and lineability Distributional chaos and lineability Zeros of Polynomials in Banach Spaces What one needs to know Zeros of polynomials: the results Miscellaneous Series in classical Banach spaces Dirichlet series Non-convergent Fourier series Norm-attaining functionals Annulling functions and sequences with finitely many zeros Sierpinski-Zygmund functions Non-Lipschitz functions with bounded gradient The Denjoy-Clarkson property General Techniques What one needs to know The negative side When lineability implies dense-lineability General results about spaceability An algebrability criterion Additivity and cardinal invariants: a brief account Bibliography Index Exercises, Notes, and Remarks appear at the end of each chapter.

Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming

Abstract: Certifying the safety or robustness of neural networks against input uncertainties and adversarial attacks is an emerging challenge in the area of safe machine learning and control. To provide such a guarantee, one must be able to bound the output of neural networks when their input changes within a bounded set. In this paper, we propose a semidefinite programming (SDP) framework to address this problem for feed-forward neural networks with general activation functions and input uncertainty sets. Our main idea is to abstract various properties of activation functions (e.g., monotonicity, bounded slope, bounded values, and repetition across layers) with the formalism of quadratic constraints. We then analyze the safety properties of the abstracted network via the S-procedure and semidefinite programming. Our framework spans the trade-off between conservatism and computational efficiency and applies to problems beyond safety verification. We evaluate the performance of our approach via numerical problem instances of various sizes.

The reachability problem for Petri nets is not elementary

Abstract: Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. At the heart of our proof is a novel gadget so called the factorial amplifier that, assuming availability of counters that are zero testable and bounded by k, guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k. We also develop a novel construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R. Repeatedly composing the factorial amplifier with itself by means of the construction then enables us to compute in linear time Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we in fact establish hardness for h-exponential space already for Petri nets with h + 13 counters.

Universal approximation with deep narrow networks

Abstract: The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $\rho$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n + m + 2$, and activation function $\rho$, is dense in $C(K; \mathbb{R}^m)$ for $K \subseteq \mathbb{R}^n$ with $K$ compact. This covers every activation function possible to use in practice, and also includes polynomial activation functions, which is unlike the classical version of the theorem, and provides a qualitative difference between deep narrow networks and shallow wide networks. We then consider several extensions of this result. In particular we consider nowhere differentiable activation functions, density in noncompact domains with respect to the $L^p$-norm, and how the width may be reduced to just $n + m + 1$ for `most' activation functions.

Input-to-State Safety with Control Barrier Functions

Abstract: This letter presents a new notion of input-to-state safe control barrier functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under input disturbances. Similar to how safety conditions are specified in terms of forward invariance of a set, input-to-state safety (ISSf) conditions are specified in terms of forward invariance of a slightly larger set. In this context, invariance of the larger set implies that the states stay either inside or very close to the smaller safe set; and this closeness is bounded by the magnitude of the disturbances. The main contribution of the letter is the methodology used for obtaining a valid ISSf-CBF, given a control barrier function (CBF). The associated universal control law will also be provided. Towards the end, we will study unified quadratic programs (QPs) that combine control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single control law that ensures both safety and stability in systems with input disturbances.

Event-Triggered Partial-Nodes-Based State Estimation for Delayed Complex Networks With Bounded Distributed Delays

Abstract: In this paper, the state estimation problem is investigated for a class of continuous-time complex networks with bounded distributed delay. For the network under consideration, only the outputs from a fraction of nodes are available and this put forward the new challenge, that is, the so-called partial-nodes-based state estimation problem. In order to reduce the usage of the communication resources, a general event-triggering rule is considered in the design of the estimator. A novel estimator is constructed and, by constructing novel Lyapunov-Krasovskii functionals, some easy-to-check conditions are derived such that the error dynamics is exponentially ultimately bounded. Furthermore, it is shown that the Zeno behavior can be excluded from the event-triggering rules. Numerical simulations are presented to further illustrate the effectiveness of the theoretical results.

A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm

Abstract: In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.

Bounded Collection of Feynman Integral Calabi-Yau Geometries.

Abstract: We define the rigidity of a Feynman integral to be the smallest dimension over which it is nonpolylogarithmic. We prove that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops provided they are in the class that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless φ^{4} theory that saturate our predicted bound in rigidity at all loop orders.

Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms

Abstract: Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frust...

Four Lectures on Scalar Curvature

Abstract: We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures

Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing

Abstract: This paper deals with a Keller–Segel type parabolic–elliptic system involving nonlinear diffusion and chemotaxis in a smoothly bounded domain , , under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function , . The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when , or , . In addition, we find the conditions on parameters, and , that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.

Openness of K-semistability for Fano varieties

Abstract: In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.

The weak harnack inequality for the boltzmann equation without cut-off

Abstract: We obtain the weak Harnack inequality and Holder estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cutoff can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local Holder estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cutoff .

On the Bohr inequality with a fixed zero coefficient

Abstract: In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions.

Introduction to the theory of Gibbs point processes

Abstract: The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated with the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in \(\mathbb {R}^d\). In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space \(\mathbb {R}^d\). Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.

Consistent Discretization of Finite-time and Fixed-time Stable Systems

Abstract: Algorithms of implicit discretization for generalized homogeneous systems having discontinuity only at the origin are developed They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization schemes preserving the stability properties of the continuous-time system Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree), and practically fixed-time stable (in the case of positive homogeneity degree) The theoretical results are supported with numerical examples 1 Introduction Discretization issues are important for a digital implementation of estimation and control algorithms Construction of a consistent stable discretization is complex for essentially non-linear ordinary differential equations (ODEs), which do not satisfy some classical regularity assumptions For example, the sliding mode algorithms are known to be difficult in practical realization [1], [2], [3] due to discontinuous (set-valued) nature, which may invoke chattering caused by the discretization The mentioned papers have discovered that the implicit discretization technique is useful for practical implementation of non-smooth and discontinuous control and estimation algorithms In particular, chattering suppression in both input and output, as well as a good closed-loop performance has been confirmed experimentally in [1], [4], [5] Finite-time stability is a desirable property for many control and estimation algorithms [6], [7], [8], [9], [10], [11] It means that system trajectories reach a stable equilibrium (or a set) in a finite time, in contrast to asymptotic stability allowing this only for the time tending to infinity If the settling (reaching) time is globally bounded for all initial conditions then the origin is fixed-time stable (see, eg [12]) The corresponding ODE models do not satisfy Lipschitz condition (at least at the origin) In the general case, an application of the conventional implicit or explicit discretization schemes does not guarantee that finite-time or fixed-time stability properties will be preserved (see, eg [13], [14], [15]) The latter means that the discrete-time model may be inconsistent with the continuous-time one However, the discretized systems may remain globally finite-time stable in some cases (see [1], [2], [16], [17]) The aim of this paper is to study systematically the problem of consistent discretization of the so-called generalized homogeneous non-linear systems The discretized model is consistent if it preserves the stability property (eg exponential, finite-time or fixed-time stability) of the original continuous-time system Homogeneity is a certain form of symmetry studied in systems and control theory [9], [18], [19], [20],[21], [22], [23] The standard homogeneity (introduced originally by L Euler in 17th century) is the symmetry of a mathematical object f (eg function, vector field, operator, etc) with respect to the uniform dilation of the argument x → λx, namely, f (λx) = λ 1+ν f (x), λ > 0

Deep ReLU network approximation of functions on a manifold.

Abstract: Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a Holder function with smoothness index $\beta$ up to error $\epsilon$ using of the order of $\epsilon^{-d^*/\beta}\log(1/\epsilon)$ many non-zero network parameters. As an application, we derive statistical convergence rates for the estimator minimizing the empirical risk over all possible choices of bounded network parameters.

Tight Certificates of Adversarial Robustness for Randomly Smoothed Classifiers

Abstract: Strong theoretical guarantees of robustness can be given for ensembles of classifiers generated by input randomization. Specifically, an $\ell_2$ bounded adversary cannot alter the ensemble prediction generated by an additive isotropic Gaussian noise, where the radius for the adversary depends on both the variance of the distribution as well as the ensemble margin at the point of interest. We build on and considerably expand this work across broad classes of distributions. In particular, we offer adversarial robustness guarantees and associated algorithms for the discrete case where the adversary is $\ell_0$ bounded. Moreover, we exemplify how the guarantees can be tightened with specific assumptions about the function class of the classifier such as a decision tree. We empirically illustrate these results with and without functional restrictions across image and molecule datasets.