boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

square root penalty adaptation to the margin in classification and in edge estimation we consider the problem of adaptation to the margin in binary classification we suggest a penalized empirical risk minimization classifier that adaptively attains up to a logarithmic factor fast optimal rates of convergence for the excess risk that is rates that can be faster than n 1 2 where n is the sample size we show that our method also gives adaptive estimators for the problem of edge estimation

algebraic reflexivity of the isometry group of some spaces of lipschitz functions abstract we show that the isometry groups of lip x d and lip x d with 0 1 for a compact metric space x d are algebraically reflexive we also prove that the sets of isometric reflections and generalized bi circular projections on such spaces are algebraically reflexive in order to achieve this we characterize generalized bi circular projections on these spaces

data driven calibration of penalties for least squares regression penalization procedures often suffer from their dependence on multiplying factors whose optimal values are either unknown or hard to estimate from data we propose a completely data driven calibration algorithm for these parameters in the least squares regression framework without assuming a particular shape for the penalty our algorithm relies on the concept of minimal penalty recently introduced by birge and massart 2007 in the context of penalized least squares for gaussian homoscedastic regression on the positive side the minimal penalty can be evaluated from the data themselves leading to a data driven estimation of an optimal penalty which can be used in practice on the negative side their approach heavily relies on the homoscedastic gaussian nature of their stochastic framework the purpose of this paper is twofold stating a more general heuristics for designing a data driven penalty the slope heuristics and proving that it works for penalized least squares regression with a random design even for heteroscedastic non gaussian data for technical reasons some exact mathematical results will be proved only for regressogram bin width selection this is at least a first step towards further results since the approach and the method that we use are indeed general

risk bounds for statistical learning we propose a general theorem providing upper bounds for the risk of an empirical risk minimizer erm we essentially focus on the binary classification framework we extend tsybakov s analysis of the risk of an erm under margin type conditions by using concentration inequalities for conveniently weighted empirical processes this allows us to deal with ways of measuring the size of a class of classifiers other than entropy with bracketing as in tsybakov s work in particular we derive new risk bounds for the erm when the classification rules belong to some vc class under margin conditions and discuss the optimality of these bounds in a minimax sense

on the stabilizability of a structural acoustic model which incorporates shear effects in the structural component abstract in this paper we consider a linear three dimensional structural acoustic model which takes account of displacement rotational inertia and shear effects in the flat flexible structural component of the model thus the deflections of the structural component of the structure are governed by the reissner mindlin plate equations we show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions it turns out that damping is needed in the interior of the plate to which end kelvin voigt damping is introduced in the plate equations as our main tool we use a resolvent criterion for strong stability due to tomilov

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

heat wave interaction in 2 3 dimensions optimal rational decay rate abstract in this paper we consider a simplified version of a fluid structure pde model which has been of longstanding interest within the mathematical and biological sciences in it a n dimensional heat equation replaces the original stokes system so as to ultimately have a vector valued heat equation and vector valued wave equation compose the coupled pde system under study the coupling between the two disparate pde components occurs across a boundary interface as such the entire pde dynamics manifests features of both hyperbolicity and parabolicity for this heat structure system our main result of uniform stability is as follows given smooth initial data i e data in the domain of the underlying semigroup generator of the coupled pde system the corresponding solutions decay at the rate of o t 1 this establishes the long conjectured optimal rate the problem of obtaining sharp rational decay rates for the heat wave pde under present consideration has been a much considered problem with the modus operandi of earlier efforts taking place within the time domain by contrast we adopt here a frequency domain approach which is based upon a recent resolvent criterion and which was initiated in our prior effort wherein we obtained the rate of decay o 1 t the present optimal improvement o t 1 is achieved by employing an additional tool in our analysis a microlocal analysis argument to estimate a critical term involving two problematic boundary traces

rejoinder local rademacher complexities and oracle inequalities in risk minimization of the true risk function f 3 f pf the first quantity of interest is the l2 diameter of this set d f 8 and the second one is the function fa f 8 that is equal to the expected supremum of the empirical process indexed by the differences g f g e f 8 these two functions are then combined in the expression n 8 t that has its roots in talagrand s concentration inequalities for empirical processes the j transform of n t which is just a way to write solu tions of fixed point type equations is then used to define the localized complexities that provide upper bounds on the excess risk under further assumptions such as mean variance relationships discussed in detail by shen and wang bartlett and mendelson also discuss this and call the function classes satisfying these relation ships bernstein classes these complexities can be redefined in terms of the local l2 continuity modulus of empirical processes since the rademacher process can be used as a data dependent bootstrap type estimate of the empirical process this approach also leads to data dependent local rademacher complexities the use of the whole minimal set is not the only possibility one can also look at its slices f 8 82 f 82 f 8 and define the excess risk bounds in terms of the accuracy of empirical approximation on the slices one can even make the slices really thin and look at g f pf inf pg 8 this was the approach taken by peter bartlett and shahar mendelson under an additional and relatively innocent assumption that the class f is star shaped they established excess risk bounds and also ratio type bounds in terms of complexities of such thin slices

termination and continuity of greedy growing for tree structured vector quantizers tree structured vector quantizers tsvq provide a computationally efficient variable rate method of compressing vector valued data in applications the problem of designing a tsvq from empirical training data is critical greedy growing algorithms are a common and effective approach to the design problem they are recursive procedures that produce a tsvq one node at a time by optimizing a simple splitting criterion at each step while unsupervised greedy growing algorithms are well understood from an experimental point of view there has been little theory to support their use or to examine their behavior on large training sets the authors present a rigorous analysis of a greedy growing algorithm proposed by riskin 1990 riskin and gray 1991 and balakrishnan 1991 the first part of the paper is a description of the algorithm and an examination of its asymptotic behavior as it applies to a fixed absolutely continuous distribution the second part of the paper establishes the structural consistency of the algorithm with respect to a convergent sequence of distributions as an application the authors obtain results concerning the large sample empirical behavior of the algorithm when it is applied to stationary ergodic training vectors

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

banach stone theorems for vector valued functions on completely regular spaces abstract we obtain several banach stone type theorems for vector valued functions in this paper let x y be realcompact or metric spaces e f locally convex spaces and a bijective linear map from c x e onto c y f if preserves zero set containments i e z f z g z f z g f g c x e then x is homeomorphic to y and is a weighted composition operator the above conclusion also holds if we assume a seemingly weaker condition that preserves nonvanishing functions i e z f 0 z f 0 f c x e these two results are special cases of the theorems in a very general setting in this paper covering bounded continuous vector valued functions on general completely regular spaces and uniformly continuous vector valued functions on metric spaces our results extend and generalize many recent ones

interface feedback control stabilization of a nonlinear fluid structure interaction we consider a model of fluid structure interaction in a bounded domain rn n 2 where is comprised of two open adjacent sub domains occupied respectively by the solid and the fluid this leads to a study of the navier stokes equation coupled on the boundary with the dynamic system of elasticity we shall consider models where the elastic body exhibits small but rapid oscillations these are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid questions related to the stability of finite energy solutions are of paramount interest it was shown in lasiecka and lu 2011 14 that all data of finite energy produce solutions whose energy converges strongly to zero the cited result holds under partial flatness geometric condition whose role is to control the effects of the pressure in the ns equation related conditions has been used in avalos and triggiani 2008 23 for the analysis of the linear model the goal of the present work is to study uniform stability of all finite energy solutions corresponding to nonlinear interaction this particular question of interest in its own rights is also a necessary preliminary step for the analysis of optimal control strategies arising in infinite horizon control problems associated with the structure it is shown in this paper that a stress type feedback control applied on the interface of the structure produces solutions whose energy is exponentially stable

on the bayes risk consistency of regularized boosting methods the probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated the main result of the paper is that certain regularized boosting algorithms provide bayes risk consistent classifiers under the sole assumption that the bayes classifier may be approximated by a convex combination of the base classifiers nonasymptotic distribution free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

cart and best ortho basis a connection we study what we call dyadic cart a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares where the optimization is over all recursive partitions arising from midpoint splits we show that the method is adaptive to unknown degrees of anisotropic smoothness specifically consider the anisotropic smoothness classes of nikol skii consisting of bivariate functions f xl x2 whose finite difference of distance h in direction i is bounded in lp norm by ch8i i 1 2 we show that dyadic cart with an appropriate complexity penalty parameter a u2 const log n is within logarithmic terms of minimax over every anisotropic smoothness class 0 c x 0 81 82 1 the proof shows that dyadic cart is identical to a certain adaptive best ortho basis algorithm based on the library of all anisotropic haar bases then it applies empirical basis selection ideas of donoho and johnstone the basis empirically selected by dyadic cart is shown to be nearly as good as a basis ideally adapted to the underlying f the risk of estimation in an ideally adapted anisotropic haar basis is shown to be comparable to the minimax risk over anisotropic smoothness classes underlying the success of this argument is harmonic analysis of anisotropic smoothness classes we show that for each anisotropic smoothness class there is an anisotropic haar basis which is a best orthogonal basis for representing that smoothness class the basis is optimal not just within the library of anisotropic haar bases but among all orthogonal bases of l2 0 1 2

optimal rates of decay for operator semigroups on hilbert spaces abstract we investigate rates of decay for c 0 semigroups on hilbert spaces under assumptions on the resolvent growth of the semigroup generator our main results show that one obtains the best possible estimate on the rate of decay that is to say an upper bound which is also known to be a lower bound under a comparatively mild assumption on the growth behaviour this extends several statements obtained by batty et al 2016 6 in fact for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly we illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary

2 local automorphisms of operator algebras abstract a not necessarily continuous linear or multiplicative function from an algebra a into itself is called a 2 local automorphism if agrees with an automorphism of a at each pair of points in a in this paper we study when a 2 local automorphism of a c algebra or a standard operator algebra on a locally convex space is an automorphism

analysis of a complexity based pruning scheme for classification trees a complexity based pruning procedure for classification trees is described and bounds on its finite sample performance are established the procedure selects a subtree of a possibly random initial tree in order to minimize a complexity penalized measure of empirical risk the complexity assigned to a subtree is proportional to the square root of its size two cases are considered in the first the growing and pruning data sets are identical and in the second they are independent using the performance bound the bayes risk consistency of pruned trees obtained via the procedure is established when the sequence of initial trees satisfies suitable geometric and structural constraints the pruning method and its analysis are motivated by work on adaptive model selection using complexity regularization

wellposedness spectral analysis and asymptotic stability of a multilayered heat wave wave system abstract in this work we consider a multilayered heat wave system where a 3 d heat equation is coupled with a 3 d wave equation via a 2 d interface whose dynamics is described by a 2 d wave equation this system can be viewed as a simplification of a certain fluid structure interaction fsi pde model where the structure is of composite type namely it consists of a thin layer and a thick layer we associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay our first result is semigroup well posedness for the fsi pde dynamics utilizing here a lumer phillips approach we show that the fluid structure system generates a c 0 semigroup on a chosen finite energy space of data as our second result we prove that the solution to the fsi dynamics generated by the c 0 semigroup tends asymptotically to the zero state for all initial data that is the semigroup of the fsi system is strongly stable for this stability work we analyze the spectrum of the generator a and show that the spectrum of a does not intersect the imaginary axis

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

unital and multiplicatively spectrum preserving surjections between semi simple commutative banach algebras are linear and multiplicative abstract let t be a surjective map from a unital semi simple commutative banach algebra a onto a unital commutative banach algebra b suppose that t preserves the unit element and the spectrum f g of the product of any two elements f and g in a coincides with the spectrum t f t g then b is semi simple and t is an isomorphism the condition that t is surjective is essential an example of a non linear and non multiplicative unital map from a commutative c algebra into itself such that t f t g f g holds for every f g are given we also show an example of a surjective unital map from a commutative c algebra onto itself which is neither linear nor multiplicative such that t f t g f g holds for every f g

arcing classifier with discussion and a rejoinder by the author recent work has shown that combining multiple versions of unstable classifiers such as trees or neural nets results in reduced test set error one of the more effective is bagging here modified training sets are formed by resampling from the original training set classifiers constructed using these training sets and then combined by voting freund and schapire propose an algorithm the basis of which is to adaptively resample and combine hence the acronym arcing so that the weights in the resampling are increased for those cases most often misclassified and the combining is done by weighted voting arcing is more successful than bagging in test set error reduction we explore two arcing algorithms compare them to each other and to bagging and try to understand how arcing works we introduce the definitions of bias and variance for a classifier as components of the test set error unstable classifiers can have low bias on a large range of data sets their problem is high variance combining multiple versions either through bagging or arcing reduces variance significantly

tree pruning with subadditive penalties in this paper we study the problem of pruning a binary tree by minimizing over all pruned subtrees of the given tree an objective function that combines an additive cost term with a penalty term that depends only on tree size we present algorithms for general size based penalties although our focus is on subadditive penalties roughly penalties that grow more slowly than linear penalties with increasing tree size such penalties are motivated by recent results in statistical learning theory for decision trees but may have wider application as well we show that the family of pruned subtrees induced by a subadditive penalty is a subset of the family induced by an additive penalty this implies by known results about additive penalties that the family induced by a subadditive penalty 1 is nested 2 is unique and 3 can be computed efficiently it also implies that when a single tree is to be selected by cross validation from the family of prunings subadditive penalties will never present a richer set of options than an additive penalty

on the rate of convergence of local averaging plug in classification rules under a margin condition the rates of convergence of plug in kernel partitioning and nearest neighbors classification rules are analyzed a margin condition which measures how quickly the a posteriori probabilities cross the decision boundary smoothness conditions on the a posteriori probabilities and boundedness of the feature vector are imposed the rates of convergence of the plug in classifiers shown in this paper are faster than previously known

long time behavior of a coupled heat wave system arising in fluid structure interaction this paper is devoted to the analysis of the long time behavior of a coupled wave heat system in which a wave and a heat equation evolve in two bounded domains with natural transmission conditions at a common interface these conditions couple in particular the heat unknown with the velocity of the wave solution this model may be viewed as a simplified version of linearized models which arise in fluid structure interaction first we show the strong asymptotic stability of solutions to this system then based on the construction of ray like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface we show the lack of uniform decay in general domains further we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain finally in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface we also analyze the difficulty there is to extend this result to the more natural transmission conditions

minimax optimal classification with dyadic decision trees decision trees are among the most popular types of classifiers with interpretability and ease of implementation being among their chief attributes despite the widespread use of decision trees theoretical analysis of their performance has only begun to emerge in recent years in this paper it is shown that a new family of decision trees dyadic decision trees ddts attain nearly optimal in a minimax sense rates of convergence for a broad range of classification problems furthermore ddts are surprisingly adaptive in three important respects they automatically 1 adapt to favorable conditions near the bayes decision boundary 2 focus on data distributed on lower dimensional manifolds and 3 reject irrelevant features ddts are constructed by penalized empirical risk minimization using a new data dependent penalty and may be computed exactly with computational complexity that is nearly linear in the training sample size ddts comprise the first classifiers known to achieve nearly optimal rates for the diverse class of distributions studied here while also being practical and implementable this is also the first study of which we are aware to consider rates for adaptation to intrinsic data dimension and relevant features

risk bounds for cart classifiers under a margin condition non asymptotic risk bounds for classification and regression trees cart classifiers are obtained in the binary supervised classification framework under a margin assumption on the joint distribution of the covariates and the labels these risk bounds are derived conditionally on the construction of the maximal binary tree and allow to prove that the linear penalty used in the cart pruning algorithm is valid under the margin condition it is also shown that conditionally on the construction of the maximal tree the final selection by test sample does not alter dramatically the estimation accuracy of the bayes classifier

strong solutions to a nonlinear fluid structure interaction system abstract in this paper we prove the existence of local in time smooth solutions to the nonlinear fluid structure interaction model first introduced in j l lions quelques methodes de resolution des problemes aux limites non lineaires dunod 1969 and considered in v barbu z grujic i lasiecka a tuffaha existence of the energy level weak solutions for a nonlinear fluid structure interaction model in fluids and waves in contemp math vol 440 amer math soc providence ri 2007 pp 55 82 v barbu z grujic i lasiecka a tuffaha smoothness of weak solutions to a nonlinear fluid structure interaction model indiana univ math j 57 3 2008 1173 1207 in particular the strong solutions here are obtained given initial datum for the navier stokes equation in the space h 1 and initial data for the wave equation w 0 and w 1 in the spaces h 2 e and h 1 e respectively

asymptotic stability of semigroups associated with linear weak dissipative systems with memory we study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations we show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution in the case of the lack of exponential decay we find the polynomial rate of decay of the solution some examples are given

regularity and asymptotic behavior for a damped plate membrane transmission problem abstract we consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions we show that exponential stability holds in the damped damped situation and polynomial stability but no exponential stability holds in the damped undamped case additionally we show that the solutions first defined by the weak formulation in fact have higher sobolev space regularity

model selection for cart regression trees the performance of the classification and regression trees cart pruning algorithm and the final discrete selection by test sample as a functional estimation procedure are considered the validation of the pruning procedure applied to gaussian and bounded regression is of primary interest on the one hand the paper shows that the complexity penalty used in the pruning algorithm is valid in both cases and on the other hand that conditionally to the construction of the maximal tree the final selection does not alter dramatically the estimation accuracy of the regression function in both cases the risk bounds that are proved obtained by using the penalized model selection validate the cart algorithm which is used in many applications such as meteorology biology medicine pollution monitoring or image coding

an iterative growing and pruning algorithm for classification tree design a critical issue in classification tree design obtaining right sized trees i e trees which neither underfit nor overfit the data is addressed instead of stopping rules to halt partitioning the approach of growing a large tree with pure terminal nodes and selectively pruning it back is used a new efficient iterative method is proposed to grow and prune classification trees this method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset successively interchanging the roles of the two subsets the convergence and other properties of the algorithm are established theoretical and practical considerations suggest that the iterative free growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms numerical results on a waveform recognition problem are presented to support this view

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

norm linear and norm additive operators between uniform algebras abstract let a c x and b c y be uniform algebras with choquet boundaries a and b a map t a b is called norm linear if t f t g f g norm additive if t f t g f g and norm additive in modulus if t f t g f g for each c and all algebra elements f and g we show that for any norm linear surjection t a b there exists a homeomorphism a b such that t f y f y for every f a and y b sufficient conditions for norm additive and norm linear surjections not assumed a priori to be linear or continuous to be unital isometric algebra isomorphisms are given we prove that any unital norm linear surjection t for which t i i or which preserves the peripheral spectra of c peaking functions of a is a unital isometric algebra isomorphism in particular we show that if a linear operator between two uniform algebras which is surjective and norm preserving is unital or preserves the peripheral spectra of c peaking functions then it is automatically multiplicative and in fact an algebra isomorphism

optimal polynomial decay of functions and operator semigroups we characterize the polynomial decay of orbits of hilbert space c 0 semigroups in resolvent terms we also show that results of the same type for general banach space semigroups and functions obtained recently in batty and duyckaerts j evol eq 8 765 780 2008 are sharp this settles a conjecture posed in batty and duyckaerts 2008

recursive partitioning to reduce distortion adaptive partitioning of a multidimensional feature space plays a fundamental role in the design of data compression schemes most partition based design methods operate in an iterative fashion seeking to reduce distortion at each stage of their operation by implementing a linear split of a selected cell the operation and eventual outcome of such methods is easily described in terms of binary tree structured vector quantizers this paper considers a class of simple growing procedures for tree structured vector quantizers of primary interest is the asymptotic distortion of quantizers produced by the unsupervised implementation of the procedures it is shown that application of the procedures to a convergent sequence of distributions with a suitable limit yields quantizers whose distortion tends to zero analogous results are established for tree structured vector quantizers produced from stationary ergodic training data the analysis is applicable to procedures employing both axis parallel and oblique splitting and a variety of distortion measures the results of the paper apply directly to unsupervised procedures that may be efficiently implemented on a digital computer

on well posedness for a free boundary fluid structure model we address a fluid structure interaction model describing the motion of an elastic body immersed in an incompressible fluid we establish a priori estimates for the local existence of solutions for a class of initial data which also guarantees uniqueness

simultaneous adaptation to the margin and to complexity in classification we consider the problem of adaptation to the margin and to complexity in binary classification we suggest a learning method with a numerically easy aggregation step adaptivity both to the margin and complexity in classification usually involves empirical risk minimization or rademacher complexities which lead to numerical difficulties on the other hand there exist classifiers that are easy to compute and that converge with fast rates but are not adaptive combining these classifiers by our aggregation procedure we get numerically realizable adaptive classifiers that converge with fast rates