least squares temporal difference methods an analysis under general conditions we consider approximate policy evaluation for finite state and action markov de cision processes mdp with the least squares temporal difference lstd algorithm lstd in an exploration enhanced learning context where policy costs are computed from observations of a markov chain different from the one corresponding to the policy under evaluation we establish for the discounted cost criterion that lstd converges almost surely under mild minimal conditions we also analyze other properties of the iterates involved in the algorithm including convergence in mean and boundedness our analysis draws on theories of both finite space markov chains and weak feller markov chains on a topological space our results can be applied to other temporal difference algorithms and mdp models as examples we give a convergence analysis of a td algorithm and extensions to mdp with compact state and action spaces as well as a convergence proof of a new lstd algorithm with state dependent parameters

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

u p q u p 1 q is a generalized gelfand pair denote by g u p q the orthogonal group of the sesqui linear quadratic form x y x 1 overline y 1 cdots x p overline y p x p 1 overline y p 1 cdots x p q overline y p q on mathbb c p q and let h1 u p 1 q be the stabilizer of the first unit vector e1 let h0 u 1 and set h h0 h1 define the character l of h by chi l h chi l h 0h 1 h 0 l h 0 in h 0 h 1 in h 1 where l in mathbb z define the anti involution on g by sigma g overline g 1 in this note we show that any distribution t on g satisfying t h1gh2 l h1h2 t g g g h1 h2 h is invariant under the anti involution this result implies that g h1 is a generalized gelfand pair

on the langlands functoriality for automorphic forms this is the extended notes of my lcctures on the langlands functoriality of automorphic forms which are in particular on the topics closely related to my recent work

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

continuous dependence for backward parabolic operators with log lipschitz coefficients we prove continuous dependence on cauchy data for a backward parabolic operator whose coefficients are log lipschitz continuous in time

finite sample analysis of least squares policy iteration in this paper we report a performance bound for the widely used least squares policy iteration lspi algorithm we first consider the problem of policy evaluation in reinforcement learning that is learning the value function of a fixed policy using the least squares temporal difference lstd learning method and report finite sample analysis for this algorithm to do so we first derive a bound on the performance of the lstd solution evaluated at the states generated by the markov chain and used by the algorithm to learn an estimate of the value function this result is general in the sense that no assumption is made on the existence of a stationary distribution for the markov chain we then derive generalization bounds in the case when the markov chain possesses a stationary distribution and is b mixing finally we analyze how the error at each policy evaluation step is propagated through the iterations of a policy iteration method and derive a performance bound for the lspi algorithm

cubic unipotent arthur parameters and multiplicities of square integrable automorphic forms l g f g a ld g f g a lcont g f g a into the direct sum of its discrete spectrum and its continuous spectrum the theory of eisenstein series reduces the description of lcont g f g a to that of the discrete spectrum of certain reductive subgroups of g and thus the basic question is the understanding of the discrete spectrum ld g f g a the discrete spectrum has a further orthogonal decomposition

global jacquet langlands correspondence multiplicity one and classification of automorphic representations in this paper we generalize the local jacquet langlands correspondence to all unitary irreducible representations we prove the global jacquet langlands correspondence in characteristic zero as consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of gl n as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by m glin waldspurger and jacquet shalika for gl n

rankin selberg integrals for the group g2 introduction the problem of determining the analytic continuation and functional equation of general l functions associated to automorphic representations of reductive lie groups and representations of the l groups has been pursued by 2 different techniques the first is the interpretation of the l function as arising in the constant term of eisenstein series the langlands shahidi method the second is the generalized rankin selberg integral representation of the l function we refer the reader to gs for an excellent description of both methods and a comparison of the various techniques used in any event the langlands shahidi method determines a prescribed list of l functions this list does not include all the possible cases of l functions that arise the rankin selberg method has so far yielded only some of the l functions that come about by the first method one of the major questions thus is whether the rankin selberg method will give new examples of l functions not possible by langlands shahidi method we show that in fact this is the case we construct here via the rankin selberg method the l function associated to the tensor product of the standard 7 dimensional representation of g2 with the standard 2 dimensional representation of gl2 in fact as a bonus of our method we also represent the l function of the standard 7 dimensional representation of g2

automorphic forms with degenerate fourier coefficients the main theme of this paper is that singular automorphic forms on classical groups are given by theta series liftings we establish several inequalities relating the automorphic multiplicities of a given representation and that of its abstract theta lift our methods allow one to lift noncuspidal square integrable automorphic forms when the dual pair involved is in the stable range in this way we construct new families of singular automorphic forms many of which are clearly unipotent in fact starting from one dimensional representations and repeating the procedure of lifting in the stable range one may obtain all automorphic forms which are quadratic unipotent in the sense of moeglin

multiplicity theorems modulo p for mathbf gl _ 2 mathbf q _p let f be a non archimedean local field pi an admissible irreducible mathbf gl 2 f representation with complex coefficients for a quadratic extension l f and an l times character chi a classical result of tunnell and saito establish a precise connection between the dimension of the hom space hbox hom l times big pi vert l times chi big and the normalized local factor of the pair pi chi the study of analogous hom spaces for complex valued representations has recently been generalized to mathbf gl n in aizenbud et al ann math 172 1407 1434 2010 and their connections with local factors have been established by work of moeglin and waldspurger la conjecture locale de gross prasad pour les groupes speciaux orthogonaux le cas general preprint available at http www math jussieu fr moeglin gp pdf 2009 in this paper we approach the analogous problem in the context of the p modular langlands correspondence for mathbf gl 2 mathbf q p we describe the restriction to cartan subgroups of an irreducible p modular representation pi of mathbf gl 2 mathbf q p and deduce generalized multiplicity results on the dimension of the ext spaces mathrm ext fancyscript o l times i big pi vert fancyscript o l times chi big where fancyscript o l times is the ring of integers of a quadratic extension of mathbf q p and chi a smooth character of fancyscript o l times

multiplicity one theorems for fourier jacobi models for every genuine irreducible admissible smooth representation pi of the metaplectic group widetilde rm sp 2n over a p adic field and every smooth oscillator representation omega psi of widetilde rm sp 2n we prove that the tensor product pi otimes omega psi is multiplicity free as a smooth representation of the symplectic group rm sp 2n similar results are proved for general linear groups and unitary groups

nontempered a packets of g 2 liftings from sl 2 we give a construction of a family of nontempered local and global arthur packets of the exceptional group g 2 one particular local packet contains a reducible representation using a rankin selberg integral we show that the subspace of the discrete spectrum associated to each packet is a full near equivalence class and this implies the arthur multiplicity formula for these packets

finite multiplicity theorems for induction and restriction abstract we find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple lie group g occurring in the induced representations ind h g from irreducible representations of a closed subgroup h as corollaries we establish geometric criteria for finiteness of the dimension of hom g ind h g induction and of hom h h restriction by means of the real flag variety g p and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety

homological multiplicities in representation theory of p adic groups for a spherical variety x of a reductive group g over a non archimedean local field f and for a smooth representation pi of g we study homological multiplicities dim text ext g mathcal s x pi based on bernstein s decomposition of the category of smooth representations of g we introduce a sheaf that measures these multiplicities we show that these multiplicities are finite whenever the usual mutliplicities begin aligned dim text hom g mathcal s x pi end aligned are finite the latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties furthermore we show that the euler poincare characteristic is constant in families parabolically induced from finite length representations of a levi subgroup m g in the case when m g we compute these multiplicities more explicitly

the gross prasad conjecture and local theta correspondence we establish the fourier jacobi case of the local gross prasad conjecture for unitary groups by using local theta correspondence to relate the fourier jacobi case with the bessel case established by beuzart plessis to achieve this we prove two conjectures of prasad on the precise description of the local theta correspondence for almost equal rank unitary dual pairs in terms of the local langlands correspondence the proof uses arthur s multiplicity formula and thus is one of the first examples of a concrete application of this global reciprocity law

local periods for discrete series representations abstract let g h be a symmetric pair over a p adic field and a discrete series representation of g in this paper for some type of symmetric pairs g h we show that local periods in hom h c can be constructed by integrating the matrix coefficients of over h

error bounds for approximations from projected linear equations we consider linear fixed point equations and their approximations by projection on a low dimensional subspace we derive new bounds on the approximation error of the solution which are expressed in terms of low dimensional matrices and can be computed by simulation when the fixed point mapping is a contraction as is typically the case in markov decision processes mdp one of our bounds is always sharper than the standard contraction based bounds and another one is often sharper the former bound is also tight in a worst case sense our bounds also apply to the noncontraction case including policy evaluation in mdp with nonstandard projections that enhance exploration there are no error bounds currently available for this case to our knowledge

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

regularized policy iteration with nonparametric function spaces we study two regularization based approximate policy iteration algorithms namely reg lspi and reg brm to solve reinforcement learning and planning problems in discounted markov decision processes with large state and finite action spaces the core of these algorithms are the regularized extensions of the least squares temporal difference lstd learning and bellman residual minimization brm which are used in the algorithms policy evaluation steps regularization provides a convenient way to control the complexity of the function space to which the estimated value function belongs and as a result enables us to work with rich nonparametric function spaces we derive efficient implementations of our methods when the function space is a reproducing kernel hilbert space we analyze the statistical properties of reg lspi and provide an upper bound on the policy evaluation error and the performance loss of the policy returned by this method our bound shows the dependence of the loss on the number of samples the capacity of the function space and some intrinsic properties of the underlying markov decision process the dependence of the policy evaluation bound on the number of samples is minimax optimal this is the first work that provides such a strong guarantee for a nonparametric approximate policy iteration algorithm

o v oplus f o v is a gelfand pair for any quadratic space v over a local field f let v be a quadratic space with a form q over an arbitrary local field f of characteristic different from 2 let w v oplus fe with the form q extending q with q e 1 consider the standard embedding mathrm o v hookrightarrow mathrm o w and the two sided action of mathrm o v times mathrm o v on mathrm o w in this note we show that any mathrm o v times mathrm o v invariant distribution on mathrm o w is invariant with respect to transposition this result was earlier proven in a bit different form in van dijk math z 193 581 593 1986 for f mathbb r in aparicio and van dijk complex generalized gelfand pairs tambov university 2006 for f mathbb c and in bosman and van dijk geometriae dedicata 50 261 282 1994 for p adic fields here we give a different proof using results from aizenbud et al arxiv 0709 1273 math rt submitted we show that this result on invariant distributions implies that the pair o v o w is a gelfand pair in the archimedean setting this means that for any irreducible admissible smooth frechet representation e of mathrm o w we have dim hom mathrm o v e mathbb c leq 1 a stronger result for p adic fields is obtained in aizenbud et al arxiv 0709 4215 math rt submitted

conditional stability for backward parabolic equations with loglipt lipx coefficients abstract in this work we present an improvement of del santo and prizzi 2009 where the authors proved a result concerning continuous dependence for backward parabolic operators whose coefficients are log lipschitz in t and c 2 in x in that paper the c 2 regularity with respect to x had to be assumed for technical reasons here we remove this assumption replacing it with lipschitz continuity the main tools in the proof are littlewood paley theory and bony s paraproduct

constructing dynamic treatment regimes over indefinite time horizons summaryexisting methods for estimating optimal dynamic treatment regimes are limited to cases where a utility function is optimized over a fixed time period we develop an estimation procedure for the optimal dynamic treatment regime over an indefinite time period and derive associated large sample results the proposed method can be used to estimate the optimal dynamic treatment regime in chronic disease settings we illustrate this by simulating a dataset corresponding to a cohort of patients with diabetes that mimics the third wave of the national health and nutrition examination survey and examining the performance of the proposed method in controlling the level of haemoglobin a1c

analysis of classification based policy iteration algorithms we introduce a variant of the classification based approach to policy iteration which uses a cost sensitive loss function weighting each classification mistake by its actual regret that is the difference between the action value of the greedy action and of the action chosen by the classifier for this algorithm we provide a full finite sample analysis our results state a performance bound in terms of the number of policy improvement steps the number of rollouts used in each iteration the capacity of the considered policy space classifier and a capacity measure which indicates how well the policy space can approximate policies that are greedy with respect to any of its members the analysis reveals a tradeoff between the estimation and approximation errors in this classification based policy iteration setting furthermore it confirms the intuition that classification based policy iteration algorithms could be favorably compared to value based approaches when the policies can be approximated more easily than their corresponding value functions we also study the consistency of the algorithm when there exists a sequence of policy spaces with increasing capacity

brief paper average cost temporal difference learning we propose a variant of temporal difference learning that approximates average and differential costs of an irreducible aperiodic markov chain approximations are comprised of linear combinations of fixed basis functions whose weights are incrementally updated during a single endless trajectory of the markov chain we present a proof of convergence with probability 1 and a characterization of the limit of convergence we also provide a bound on the resulting approximation error that exhibits an interesting dependence on the mixing time of the markov chain the results parallel previous work by the authors involving approximations of discounted cost to go

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

weak convergence properties of constrained emphatic temporal difference learning with constant and slowly diminishing stepsize we consider the emphatic temporal difference td algorithm etd for learning the value functions of stationary policies in a discounted finite state and action markov decision process the etd algorithm was recently proposed by sutton mahmood and white 2016 to solve a long standing divergence problem of the standard td algorithm when it is applied to off policy training where data from an exploratory policy are used to evaluate other policies of interest the almost sure convergence of etd has been proved in our recent work under general off policy training conditions but for a narrow range of diminishing stepsize in this paper we present convergence results for constrained versions of etd with constant stepsize and with diminishing stepsize from a broad range our results characterize the asymptotic behavior of the trajectory of iterates produced by those algorithms and are derived by combining key properties of etd with powerful convergence theorems from the weak convergence methods in stochastic approximation theory for the case of constant stepsize in addition to analyzing the behavior of the algorithms in the limit as the stepsize parameter approaches zero we also analyze their behavior for a fixed stepsize and bound the deviations of their averaged iterates from the desired solution these results are obtained by exploiting the weak feller property of the markov chains associated with the algorithms and by using ergodic theorems for weak feller markov chains in conjunction with the convergence results we get from the weak convergence methods besides etd our analysis also applies to the off policy td algorithm when the divergence issue is avoided by setting sufficiently large it yields for that case new results on the asymptotic convergence properties of constrained off policy td with constant or slowly diminishing stepsize

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

generalized polynomial approximations in markovian decision processes abstract fitting the value function in a markovian decision process by a linear superposition of m basis functions reduces the problem dimensionality from the number of states down to m with good accuracy retained if the value function is a smooth function of its argument the state vector this paper provides for both the discounted and undiscounted cases three algorithms for computing the coefficients in the linear superposition linear programming policy iteration and least squares

fourier transform and the global gan gross prasad conjecture for unitary groups by the relative trace formula approach of jacquet rallis we prove the global gan gross prasad conjecture for unitary groups under some local restrictions for the automorphic representations

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

characteristic cycles of discrete series for rank one groups we determine the characteristic cycles of the discrete series representations for connected r rank one linear groups the computation is made through the moment maps we determine their fibers and the cohomology in question case by case the multiplicity of the discrete series in terms of their harish chandra parameters is given by recursive formulae for groups of type a and b closed formulae are obtained

from laplace to langlands via representations of orthogonal groups in the late 1960s robert langlands proposed a new and far reaching connection between the representation theory of lie groups over real and p adic fields and the structure of the galois groups of these fields 24 even though this local langlands correspondence remains largely conjectural the relation that it predicts between representation theory and number theory has profoundly changed our views of both fields moreover we now know enough about the correspondence to address and sometimes solve traditional problems in representation theory that were previously inaccessible roughly speaking the local langlands correspondence predicts that complex irreducible representations of a reductive group g over a local field k should be

finite reductive dual pairs in g2 abstract this paper classifies dual pairs f g in complex g2 where f is finite and g is infinite complex reductive

on special values of certain l functions we prove an algebraicity result concerning special values at critical points in the sense of deligne of tensor product l functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite and cuspidal representations of rm gl 2 corresponding to primitive cusp forms over totally real number fields we also prove the reciprocity law i e the equivariance under the action of rm gal overline bbb q bbb q for the special values in the appendix the second author calculates the deligne periods for such l functions assuming the existence of corresponding motives and the automorphic transfer to a quasi split form of the special orthogonal group our result conforms with the celebrated conjecture of deligne on special values of motivic l functions

correspondence of models under local theta lifting abstract we study correspondence of models of representations under local theta lifting between general linear groups and unitary groups over a non archimedean local field of characteristic 0

the local theta correspondence and the local gan gross prasad conjecture for the symplectic metaplectic case we prove the local gan gross prasad conjecture for the symplectic metaplectic case under some assumptions this is the last case of the local gan gross prasad conjectures which has not been established we also prove two of prasad s conjectures on the local theta correspondence in the almost equal rank case

least squares policy iteration we propose a new approach to reinforcement learning for control problems which combines value function approximation with linear architectures and approximate policy iteration this new approach is motivated by the least squares temporal difference learning algorithm lstd for prediction problems which is known for its efficient use of sample experiences compared to pure temporal difference algorithms heretofore lstd has not had a straightforward application to control problems mainly because lstd learns the state value function of a fixed policy which cannot be used for action selection and control without a model of the underlying process our new algorithm least squares policy iteration lspi learns the state action value function which allows for action selection without a model and for incremental policy improvement within a policy iteration framework lspi is a model free off policy method which can use efficiently and reuse in each iteration sample experiences collected in any manner by separating the sample collection method the choice of the linear approximation architecture and the solution method lspi allows for focused attention on the distinct elements that contribute to practical reinforcement learning lspi is tested on the simple task of balancing an inverted pendulum and the harder task of balancing and riding a bicycle to a target location in both cases lspi learns to control the pendulum or the bicycle by merely observing a relatively small number of trials where actions are selected randomly lspi is also compared against q learning both with and without experience replay using the same value function architecture while lspi achieves good performance fairly consistently on the difficult bicycle task q learning variants were rarely able to balance for more than a small fraction of the time needed to reach the target location

manifold regularized reinforcement learning this paper introduces a novel manifold regularized reinforcement learning scheme for continuous markov decision processes smooth feature representations for value function approximation can be automatically learned using the unsupervised manifold regularization method the learned features are data driven and can be adapted to the geometry of the state space furthermore the scheme provides a direct basis representation extension for novel samples during policy learning and control the performance of the proposed scheme is evaluated on two benchmark control tasks i e the inverted pendulum and the energy storage problem simulation results illustrate the concepts of the proposed scheme and show that it can obtain excellent performance

knapp stein type intertwining operators for symmetric pairs abstract for a symmetric pair g h of reductive groups we construct a family of intertwining operators between spherical principal series representations of g and h that are induced from parabolic subgroups satisfying certain compatibility conditions the operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation in particular this establishes lower bounds for multiplicities of intertwining operators we further discuss uniqueness of intertwining operators and for the rank one cases g h su 1 n f s u 1 m f u n m f with f r c h o and for the pair g h gl 4 n r gl 2 n c we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners

uniqueness of fourier jacobi models the archimedean case we prove uniqueness of fourier jacobi models for general linear groups unitary groups symplectic groups and metaplectic groups over an archimedean local field

on central critical values of the degree four l functions for text gsp left 4 right a simple trace formula we establish a simple relative trace formula for text gsp 4 and inner forms with respect to bessel subgroups to obtain a certain bessel identity from such an identity one can hope to prove a formula relating central values of degree four spinor l functions to squares of bessel periods as conjectured by bocherer under some local assumptions we obtain nonvanishing results i e a global gross prasad conjecture for text so 5 text so 2

rates of convergence for empirical processes of stationary mixing sequences classical empirical process theory for vapnik cervonenkis classes deals mainly with sequences of independent variables this paper extends the theory to stationary sequences of dependent variables it establishes rates of convergence for mixing and mixing empirical processes indexed by classes of functions the method of proof depends on a coupling of the dependent sequence with sequences of independent blocks to which the classical theory can be applied a uniform o ns 1 s rate of convergence over v c classes is established for sequences whose mixing coefficients decay slightly faster than o n s