genriesz: A Python Package for Automatic Debiased Machine Learning with Generalized Riesz Regression

genriesz : A Python P ac k age for Automatic Debiased Mac hine Learning with Generalized Riesz Regression Masahiro Kato ∗ Data Analytics Departmen t, Mizuho-DL Financial T ec hnology , Co., Ltd. F ebruary 20, 2026 Abstract Efficien t estimation of causal and structural parameters can b e automated using the Riesz represen tation theorem and debiased machine learning (DML). W e present genriesz , an op en-source Python pack age that implemen ts automatic DML and gen- er alize d R iesz r e gr ession , a unified framework for estimating Riesz representers b y minimizing empirical Bregman div ergences. This framework includes co v ariate balanc- ing, nearest-neigh b or matching, calibrated estimation, and density ratio estimation as sp ecial cases. A k ey design principle of the pack age is automatic r e gr essor b alanc- ing (ARB): given a Bregman generator g and a representer mo del class, genriesz automatically constructs a compatible link function so that the generalized Riesz re- gression estimator satisfies balancing (momen t-matching) optimalit y conditions in a user-c hosen basis. The pac k age pro vides a mo dular interface for specifying (i) the target linear functional via a blac k-b o x ev aluation oracle, (ii) the represen ter mo del via basis functions (polynomial, RKHS appro ximations, random forest leaf enco dings, neural em b eddings, and a nearest-neigh b or catchmen t basis), and (iii) the Bregman generator, with optional user-supplied deriv ativ es. It returns regression adjustment (RA), Riesz w eighting (R W), augmen ted Riesz w eighting (AR W), and TMLE-style estimators with cross-fitting, confidence interv als, and p -v alues. W e highlight represen tative w orkflows for estimation problems such as the a verage treatmen t effect (A TE), A TE on treated (A TT), and av erage marginal effect estimation. The Python pack age is a v ailable at https://github.com/MasaKat0/genriesz and on PyPI. 1 In tro duction Man y targets in causal inference and econometrics can b e expressed as linear functionals of an unkno wn regression function. Prominent examples include the a v erage treatmen t effect (A TE), the av erage treatment effect on the treated (A TT), and a verage marginal effects (AME). ∗ Email: mkato-csecon@g.ecc.u-tokyo.ac.jp 1 Automatic debiased machine learning (ADML) pro vides general tools for v alid inference on suc h targets ( Chernozh uko v et al. , 2022b ). The ADML w orkflow separates the problem in to t wo parts. First, one estimates n uisance comp onen ts, t ypically a regression function and a Riesz representer. Second, one plugs these estimates in to a Neyman orthogonal score and a verages the resulting score o v er the sample. A Donsk er condition or cross-fitting then yields v alid inference under w eak rate conditions ( Chernozh uko v et al. , 2018 ; Klaassen , 1987 ). V arious metho ds ha ve been prop osed for Riesz representer estimation, suc h as Riesz regres- sion ( Chernozh uko v et al. , 2021 ; Chen & Liao , 2015 ), co v ariate balancing ( Imai & Ratko vic , 2013 ), and density ratio estimation ( Sugiyama et al. , 2012 ). Kato ( 2026 , 2025a ) demonstrates a unifying p erspective, Riesz r epr esenter fitting under Br e gman diver genc es , which is also referred to as generalized Riesz regression, Bregman-Riesz regression, or generalized co v ariate balancing. Hereafter, w e refer to this metho d as generalized Riesz regression, a name that emphasizes the connection b etw een Riesz representer fitting and balancing w eigh ts through the c hoice of link function. In this framew ork, one can reco ver the v arious metho ds listed ab o v e by sp ecifying particular forms of the Bregman div ergence. Our genriesz pac k age implemen ts generalized Riesz regression for ADML. Users sp ecify the estimand through an ev aluation oracle and choose a Riesz representer mo del and a Bregman generator. The pac k age then constructs a generator-induced link function to deliv er automatic regressor balancing, estimates the representer b y con vex optimization with ℓ p regularization, and rep orts regression adjustmen t (RA), Riesz w eighting (R W), augmen ted Riesz w eighting (AR W), and targeted maxim um likelihoo d estimation (TMLE)-st yle estimates with confidence in terv als, optionally using cross-fitting ( Bang & Robins , 2005 ; v an der Laan , 2006 ). The flo wc hart of this automatic procedure is below, where each ob ject will b e defined in the subsequen t sections (Figure 1 ): (i) the user sp ecifies the parameter functional m ( W , γ 0 ) for the parameter of in terest θ 0 : = E [ m ( W , γ 0 )] , the Bregman generator g for Riesz represen ter estimation, and the basis functions ϕ ( X ). (ii) the pack age automatically computes the link function ζ  X , ϕ ( X ) ⊤ β  that yields regres- sor balancing, and estimates the Riesz representer α 0 and, when needed, the regression function γ 0 . (iii) the pac k age outputs RA, R W, AR W, and TMLE-style estimators with standard errors and confidence interv als. During this process, the user do es not need to specify the analytic form of the Riesz represen ter α 0 or the link function ζ . These are constructed from the chosen basis and generator. Relation to existing softw are. The genriesz pac k age complemen ts established DML libraries suc h as DoubleML ( Bac h et al. , 2022 ) and EconML ( Batto cchi et al. , 2019 ), as w ell as broader causal inference to olkits such as DoWh y ( Sharma & Kiciman , 2020 ) and CausalML ( Chen et al. , 2020 ). These libraries offer ric h sets of estimators for canonical causal models and heterogeneous treatmen t effects. In con trast, genriesz is estimand-and-b alancing-c entric : 2 Figure 1: Flow c hart of genriesz . the user provides the functional m , the represen ter mo del, and the specific form of the Bregman div ergence, while the pac k age constructs generalized Riesz representers with a link function that yields balancing w eigh ts and returns the corresponding estimators. This fo cus also mak es explicit the connection betw een Riesz representer fitting and balancing weigh ts, including stable weigh ts ( Zubizarreta , 2015 ) and entrop y balancing ( Hainm ueller , 2012 ), as w ell as density ratio estimation metho ds av ailable in sp ecialized pac k ages suc h as densratio ( Makiy ama , 2019 ). 2 Key Ingredien ts of Generalized Riesz Regression Generalized Riesz regression connects Riesz represen ter estimation, co v ariate balancing, and debiased estimation through Bregman divergence minimization. This section summarizes the metho dological and theoretical foundations that underlie the soft w are design. F ull technical details, proofs, and extensions are provided in Kato ( 2026 ). 2.1 Linear F unctionals, Riesz Representers, and Orthogonal Scores Let W : = ( X , Y ) ∼ P , where X ∈ R d is a regressor and Y ∈ R is an outcome. Let γ 0 ( x ) : = E [ Y | X = x ]. W e consider targets of the form θ 0 : = E [ m ( W , γ 0 )] , (1) where m ( W , γ ) is linear in γ . In genriesz , users supply m as a callable that ev aluates γ at mo dified inputs, for example, b y switching a treatmen t comp onent. Under standard conditions, there exists a Riesz representer α 0 suc h that E [ m ( W , γ )] = E [ α 0 ( X ) γ ( X )] for all suitable γ . (2) The Riesz representer en ters the Neyman orthogonal score ψ ( W ; θ , γ , α ) : = m ( W , γ ) + α ( X ) ( Y − γ ( X )) − θ , (3) whic h plays a cen tral role in debiased estimation and v alid inference under weak conditions, with cross-fitting ( Chernozhuk ov et al. , 2018 ). 3 T able 1: Corresp ondence among Bregman div ergence losses, density ratio (DR) estimation metho ds, and Riesz represen ter (RR) estimation. RR estimation for A TE includes prop ensit y score estimation and cov ariate balancing weigh ts. In the table, C ∈ R denotes a constant that is determined b y the problem and the loss function. The pac k age also supports user-defined generators. g ( α ) DR estimation RR estimation ( α − C ) 2 LSIF SQ-Riesz regression ( Kanamori et al. , 2009 ) ( genriesz ) KuLSIF Riesz regression (RieszNet and F orestRiesz) ( Kanamori et al. , 2012 ) ( Chernozhuk ov et al. , 2021 , 2022a ) Hyv¨ arinen score matc hing RieszBo ost ( Hyv¨ arinen , 2005 ) ( Lee & Sc huler , 2025 ) KRRR ( Singh , 2024 ) Nearest neigh bor matching ( Lin et al. , 2023 ) Causal forest and generalized random forest ( W ager & Athey , 2018 ; Athey et al. , 2019 ) Dual solution with a linear link function Kernel mean matc hing Sieve Riesz representer ( Gretton et al. , 2009 ) ( Chen & Liao , 2015 ; Chen & Pouzo , 2015 ) Stable balancing w eights ( Zubizarreta , 2015 ; Bruns-Smith et al. , 2025 ) Approximate residual balancing ( Athey et al. , 2018 ) Cov ariate balancing by SVM ( T arr & Imai , 2025 ) Distributional balancing ( Santra et al. , 2026 )  ( | α | − C )  log  ( | α | − C )  − | α | UKL div ergence minimization UKL-Riesz regression ( Nguyen et al. , 2010 ) ( genriesz ) T ailored loss minimization ( α = β = − 1) ( Zhao , 2019 ) Calibrated estimation ( T an , 2019 ) Dual solution with a logistic or log link function KLIEP Entrop y balancing weigh ts ( Sugiyama et al. , 2008 ) ( Hainmueller , 2012 ) ( | α | − C ) log  ( | α | − C )  − ( | α | + C ) log  ( | α | + C )  BKL div ergence minimization BKL-Riesz regression ( Qin , 1998 ) ( genriesz ) TRE MLE of the prop ensity score ( Rhodes et al. , 2020 ) (Standard approac h) T ailored loss minimization ( α = β = 0) ( Zhao , 2019 )  ( | α |− C )  1+ ω −  ( | α |− C )  ω − ( | α | − C ) BP div ergence minimization BP-Riesz regression for some ω ∈ (0 , ∞ ) ( Sugiyama et al. , 2011 ) ( genriesz ) C log (1 − α ) + C α (log ( α ) − log (1 − α )) PU learning PU-Riesz regression for α ∈ (0 , 1) ( du Plessis et al. , 2015 ) ( genriesz ) Nonnegative PU learning ( Kiryo et al. , 2017 ) General form ulation b y Bregman Density-ratio matching Generalized Riesz regression divergence minimization ( Sugiyama et al. , 2011 ) ( genriesz ) D3RE ( Kato & T eshima , 2021 ) 2.2 Bregman-Riesz Ob jectiv es Let g ( x, α ) b e con vex in the scalar α for eac h fixed x . The point wise Bregman div ergence is BD g ( α 0 ( x ) ∥ α ( x )) : = g ( x, α 0 ( x )) − g ( x, α ( x )) − ∂ α g ( x, α ( x )) ( α 0 ( x ) − α ( x )) . (4) 4 Generalized Riesz regression estimates α 0 b y minimizing an empirical Bregman-Riesz ob jectiv e of the form b L ( α ) : = 1 n n X i =1  − g ( X i , α ( X i )) + ∂ α g ( X i , α ( X i )) α ( X i ) − m ( W i , ∂ α g ( · , α ( · )))  + λ Ω ( α ) , (5) where Ω is a regularizer, for example an ℓ p p enalt y on a coefficient vector. Squared-loss gener- ators reco ver Riesz regression (primal, Chernozhuk o v et al. , 2021 ) and series Riesz representer (dual, Chen & Liao , 2015 ), while KL-t yp e generators reco ver tailored-loss, calibrated estima- tion form ulations (primal, Zhao , 2019 ; T an , 2019 ) and en tropy balancing (dual, Hainmueller , 2012 ). The pac k age pro vides built-in generators, squared distance (SQ), unnormalized KL (UKL) div ergence, binary KL (BKL) divergence, Basu’s p ow er (BP) div ergence, and PU families, and it also supp orts user-defined generators. Dual co ordinate and conjugate ob jective. A key iden tit y b ehind the implemen tation is the conjugacy relation g ∗ ( x, v ) : = sup α ∈A ( x ) { αv − g ( x, α ) } , α ∗ ( x, v ) : = arg max α ∈A ( x ) { αv − g ( x, α ) } , (6) where A ( x ) is the domain of α giv en x . F or differen tiable generators, α ∗ ( x, v ) = ( ∂ α g ) − 1 ( x, v ) . Using v ( x ) : = ∂ α g ( x, α ( x )) and g ∗ ( x, v ( x )) = α ( x ) v ( x ) − g ( x, α ( x )) , the ob jectiv e ( 5 ) is equiv alent, up to constan ts, to b L ∗ ( v ) : = 1 n n X i =1  g ∗ ( X i , v ( X i )) − m ( W i , v )  + λ Ω ∗ ( v ) . (7) This dual view is con venien t for optimization and for deriving the balancing optimalit y conditions. 2.3 Automatic Regressor Balancing via Generator-Induced Links T o fit α 0 in a mo del class, genriesz fo cuses on GLM-st yle parameterizations α β ( x ) = ζ − 1 ( x, f β ( x )) , f β ( x ) = ϕ ( x ) ⊤ β , (8) where ϕ is a user-c hosen basis and ζ is a link. A key choice is to set the link to the deriv ative of the generator, ζ ( x, α ) = ∂ α g ( x, α ) , ζ − 1 ( x, · ) = ( ∂ α g ( x, · )) − 1 , (9) so that the dual coordinate v ( x ) = ∂ α g ( x, α ( x )) is linear in β . This is the pac k age’s notion of ARB. In genriesz , the mo del is estimated b y solving a conv ex program in β : b β : = arg min β ∈ R p ( 1 n n X i =1  g ∗ ( X i , f β ( X i )) − m ( W i , f β )  + λ Ω ( β ) ) . (10) After estimating b β , the fitted Riesz representer is b α ( x ) = ζ − 1  x, f b β ( x )  . 5 Prop osition 2.1 (ARB implies balancing optimalit y conditions) . Consider the mo del ( 8 )–( 9 ). Fix q ∈ [1 , ∞ ] and set Ω( β ) = 1 q ∥ β ∥ q q . L et b β b e any empiric al minimizer of ( 10 ) and define b α ( x ) = α b β ( x ) . Under mild r e gularity c onditions, the KKT c onditions imply that ther e exist sc alars s 1 , . . . , s p such that 1 n n X i =1  b α ( X i ) ϕ j ( X i ) − m ( W i , ϕ j )  + λs j = 0 , s j ∈ ∂  1 q | β j | q  | β j = b β j , j = 1 , . . . , p. (11) Conse quently, the implie d b alancing c ondition takes the fol lowing explicit form: • If q = 1 , then | s j | ≤ 1 and henc e      1 n n X i =1  b α ( X i ) ϕ j ( X i ) − m ( W i , ϕ j )       ≤ λ, j = 1 , . . . , p. (12) • If q > 1 , then s j = sign  b β j     b β j    q − 1 and henc e      1 n n X i =1  b α ( X i ) ϕ j ( X i ) − m ( W i , ϕ j )       = λ    b β j    q − 1 , j = 1 , . . . , p. (13) In p articular, when λ = 0 and the c onstr aints ar e fe asible, ( 11 ) yields exact sample b alancing. Prop osition 2.1 is a soft w are-relev ant consequence of the duality theory in Kato ( 2026 ). ARB ensures that the solv er automatically enforces the correct balancing equations for the user-sp ecified basis, ev en when the primal mo del ( 8 ) is nonlinear in β , for example under KL-t yp e links. Remark (Automatic link construction in genriesz ) . Users c an supply analytic derivatives ∂ α g and ( ∂ α g ) − 1 . If they do not, genriesz appr oximates ∂ α g by finite differ enc es and c omputes ( ∂ α g ) − 1 by r o ot finding. This al lows r apid pr ototyping of new Br e gman gener ators, at the c ost of additional numeric al c ar e, such as domain c onstr aints and br anch sele ction. Remark. ℓ 1 p enalty and sp arsity The ℓ 1 choic e is useful b e c ause ( 12 ) dir e ctly c ontr ols the maximum absolute moment imb alanc e by the single tuning p ar ameter λ . In genriesz , this r ole is primarily ab out fe asibility r elaxation and stability of r epr esenter fitting, r ather than r e c overing a sp arse c o efficient ve ctor. In p articular, using ℓ 1 her e should b e understo o d as imp osing an interpr etable slack on b alancing e quations, not as a sp arsity assumption on the true r epr esenter mo del. 2.4 Sp ecial Cases and Connections to Balancing W eigh ts The ARB construction ( 9 ) mak es explicit ho w generalized Riesz regression reco v ers classical balancing-w eight estimators as dual solutions. Intuitiv ely , the dual v ariable associated with the linear moment conditions corresp onds to p er-sample w eigh ts, and different generators g induce differen t weigh t regularizers. T able 1 summarizes common c hoices implemented in genriesz , see Kato ( 2026 ) for precise statements. 6 Domain constraints and branc h selection. Some generators require constrain ts suc h as | α | > C and α ∈ (0 , 1) . genriesz exp oses a branch fn in terface that selects a v alid branch, for example treated versus control, so that the fitted represen ter resp ects the generator domain b y construction. 2.5 Estimators: RA, R W, AR W, and TMLE Giv en nuisance estimates b γ and b α , optionally cross-fitted, genriesz outputs four plug-in estimators, regression adjustment (RA), Riesz w eighting (R W), augmented Riesz weigh ting (AR W), and targeted maxim um likelihoo d estimation (TMLE)-st yle estimators, defined as follo ws: b θ RA : = 1 n n X i =1 m ( W i , b γ ) , (14) b θ R W : = 1 n n X i =1 b α ( X i ) Y i , (15) b θ AR W : = 1 n n X i =1  b α ( X i ) ( Y i − b γ ( X i )) + m ( W i , b γ )  , (16) b θ TMLE : = 1 n n X i =1 m  W i , b γ (1)  , (17) In TMLE, b γ (1) is a one-dimensional fluctuation up date of b γ along the direction b α . W e consider t wo lik eliho o d choices, Gaussian and Bernoulli with a logit link, which lead to differen t up date maps for b γ (1) . See App endix B . Prop osition 2.2 (Asymptotic normalit y) . Supp ose b α and b γ ar e obtaine d by cr oss-fitting and satisfy me an-squar e-err or r ate c onditions such as ∥ b α − α 0 ∥ L 2 ( P ) ∥ b γ − γ 0 ∥ L 2 ( P ) = o p  n − 1 / 2  along with mild moment c onditions. Then the AR W estimator in ( 17 ) is asymptotic al ly line ar with influenc e function ψ ( W ; θ 0 , γ 0 , α 0 ) in ( 3 ), so that √ n  b θ AR W − θ 0  d − → N (0 , V [ ψ ( W ; θ 0 , γ 0 , α 0 )]) . A nalo gous statements hold for the TMLE-style estimator. Inference. F or eac h estimator, genriesz computes W ald-type standard errors from the empirical v ariance of the corresp onding estimated influence-function scores. Cross-fitting is recommended in high-capacity settings ( Chernozhuk o v et al. , 2018 ). 3 API Design and Soft w are Arc hitecture The design goal of genriesz is to separate statistic al intent , the functional m and the represen ter class, from numeric al implementation , basis matrices, generator deriv atives, solv ers, and cross-fitting. 7 3.1 Core Abstractions The main entry point is grr functional : from genriesz import grr_functional res = grr_functional( X=X, Y=Y, m=m, # Functional object basis=basis, # Feature map phi(x) generator=gen, # BregmanGenerator (or g=..., grad_g=..., inv_grad_g=..., grad2_g=...) cross_fit=True, folds=5, estimators=("ra","rw","arw","tmle"), ) print(res.summary_text()) The returned result ob ject stores p oin t estimates, standard errors, confidence in terv als, p -v alues, and optional out-of-fold nuisance predictions. Estimators, cross-fitting, and outcome mo dels. The estimators argumen t selects whic h plug-in estimators to report. The built-in names follo w the genriesz naming con v ention, "ra" for regression adjustment, "rw" for Riesz weigh ting, "arw" for augmen ted Riesz weigh ting, and "tmle" for the TMLE-st yle up date. Cross-fitting is enabled b y cross fit=True with folds con trolling the n um b er of folds. F or RA, AR W, and TMLE, the pac k age needs an outcome regression mo del b γ . Users can con trol its construction b y outcome models , for example, "shared" fits a linear mo del using the same basis and regularization in terface as the Riesz mo del, while "separate" fits the same outcome regression on a user-supplied outcome basis outcome basis . Setting outcome models="none" skips outcome mo deling, then only R W is av ailable. 3.2 Built-In F unctionals and W rapp ers F or common causal estimands with a binary treatment indicator D stored in a column of X , the pac k age provides conv enience wrappers that call grr functional with predefined m : • grr ate : A TE for X = [ D , Z ] with m ( W , γ ) = γ (1 , Z ) − γ (0 , Z ). • grr att : A TT for X = [ D , Z ] . One con v enient linear functional is m ( W , γ ) = D π 1 ( γ (1 , Z ) − γ (0 , Z )) with π 1 : = E [ D ] . In the wrapp er, π 1 is estimated b y b π 1 : = 1 n P n i =1 D i . • grr did : panel difference-in-difference (DID) implemen ted as A TT on ∆ Y : = Y 1 − Y 0 , where Y 0 and Y 1 are pre and p ost outcomes for the same units. • grr ame : AME via deriv atives, requiring a basis that implements ∂ ϕ ( x ) ∂ x k . These wrapp ers reduce b oilerplate for standard w orkflows while still exp osing basis and generator c hoices. 8 Algorithm 1 Simplified w orkflo w implemen ted b y grr functional Require: Data ( X i , Y i ) n i =1 , functional m , basis ϕ , generator g , folds K . 1: for k = 1 to K (if cross-fitting; else K = 1) do 2: Fit represen ter mo del b α ( − k ) on training fold via generalized Riesz regression. 3: Fit outcome mo del b γ ( − k ) on training fold if RA, AR W, or TMLE is requested. 4: Predict b α i and b γ i on held-out fold. 5: end for 6: Compute RA, R W, AR W, and TMLE estimators. 7: Compute standard errors and confidence in terv als from influence-function scores. 8: return Estimates and inference summary . 3.3 Basis F unctions and Extensibilit y The genriesz pac k age treats the represen ter mo del as linear in a user-sp ecified feature map ϕ ( x ). The core module includes polynomial features and treatment-in teraction features for A TE and A TT w orkflo ws, as well as RKHS-st yle appro ximations via random F ourier features and Nystr¨ om features. Tw o optional mo dules expand the basis library: (i) genriesz.sklearn basis pro vides a random forest leaf one-hot basis that turns a fitted ensem ble into a sparse feature map, and (ii) genriesz.torch basis wraps a PyT orch embedding net w ork as a frozen feature map. Finally , genriesz pro vides a kNN catchmen t basis KNNCatchmentBasis and matching utilities in genriesz.matching , whic h connect nearest-neigh b or matc hing to squared-loss Riesz regression ( Kato , 2025b ; Lin et al. , 2023 ). F eature maps. Keeping the solver linear in β in the dual co ordinate preserv es con v exity and the exact ARB optimality conditions in Prop osition 2.1 . This suggests a practical pattern for neural pip elines, learn an em b edding, freeze it, and then fit the representer in the induced feature space. 3.4 Optimization and Regularization The GLM solver supports ℓ p p enalties for an y p ≥ 1. P enalty in terface. F or the Riesz model, set riesz penalty="l2" for ridge, riesz penalty="l1" for lasso, and riesz penalty="lp" with riesz p norm=p for general p ≥ 1. A shorthand suc h as riesz penalty="l1.5" is also supported. When using the default linear outcome regression, the same interface is a v ailable via outcome penalty and outcome p norm . F or p ≥ 1, genriesz uses L-BF GS-B via SciPy , with a smo oth appro ximation of the ℓ 1 subgradien t when p = 1. The solv er exp oses iteration limits and tolerances, but defaults aim to w ork out-of-the-b o x for mo derate feature dimensions. 9 4 Assumptions on Input Data genriesz is designed around a simple data interface: X is a NumPy arra y of shap e ( n, d ) and Y is a vector of shap e ( n, ) . The meaning of columns of X is left to the user and to the functional ob ject. Binary treatmen t con v entions. F or wrapp ers for A TE, A TT, and DID estimation, the treatmen t indicator D is assumed to b e binary and stored at a known column index of X . The user can freely c ho ose the remaining co v ariates Z . P anel DID. F or grr did , the pac k age exp ects tw o outcome vectors Y0 and Y1 represen ting pre and p ost outcomes for the same units and computes ∆ Y : = Y 1 − Y 0 in ternally . Sampling and inference. The default inference uses an i.i.d. appro ximation and W ald- t yp e confidence in terv als. As with standard DML softw are, clustered or dep endent data require user-supplied adaptations, for example cluster-robust v ariance estimators, whic h are not y et part of the core release. 5 Examples and Repro ducibilit y The repository includes runnable scripts and Jupyter noteb o oks illustrating the workflo ws b elo w. 5.1 A TE with P olynomial F eatures and UKL Generator Let X = [ D , Z ] and choose a polynomial basis with treatmen t in teractions: from genriesz import ( grr_ate, PolynomialBasis, TreatmentInteractionBasis, UKLGenerator ) psi = PolynomialBasis(degree=2, include_bias=True) phi = TreatmentInteractionBasis(base_basis=psi) gen = UKLGenerator(C=1.0, branch_fn=lambda x: int(x[0] == 1.0)).as_generator() res = grr_ate( X=X, Y=Y, basis=phi, generator=gen, cross_fit=True, folds=5, estimators=("ra","rw","arw","tmle"), riesz_penalty="l2", riesz_lam=1e-3, ) print(res.summary_text()) 10 5.2 A TT and P anel DID A TT and panel DID are a v ailable via wrapper functions: from genriesz import ( grr_att, grr_did, PolynomialBasis, TreatmentInteractionBasis, SquaredGenerator ) psi = PolynomialBasis(degree=2, include_bias=True) phi = TreatmentInteractionBasis(base_basis=psi) gen = SquaredGenerator().as_generator() res_att = grr_att( X=X, Y=Y, basis=phi, generator=gen, cross_fit=True, folds=5, estimators=("arw","tmle"), ) res_did = grr_did( X=X, Y0=Y0, Y1=Y1, basis=phi, generator=gen, cross_fit=True, folds=5, estimators=("arw","tmle"), ) 5.3 Av erage Marginal Effects Av erage marginal effects can b e computed when the chosen basis implements deriv ativ es: from genriesz import grr_ame, PolynomialBasis, SquaredGenerator phi = PolynomialBasis(degree=2, include_bias=True) res_ame = grr_ame( X=X, Y=Y, coordinate=2, basis=phi, generator=SquaredGenerator().as_generator(), cross_fit=True, folds=5, estimators=("ra","rw","arw","tmle"), ) 11 5.4 Nearest-Neigh b or Matc hing Nearest-neigh b or matc hing w eights can b e computed using the built-in matc hing Riesz metho d: from genriesz import grr_ate, PolynomialBasis # A basis is required by the API. When outcome_models="none", it is not used. basis = PolynomialBasis(degree=1, include_bias=True) res_match = grr_ate( X=X, Y=Y, basis=basis, riesz_method="nn_matching", M=1, cross_fit=False, outcome_models="none", estimators=("rw",), ) 6 Pro ject Dev elopmen t and Dep endencies Av ailabilit y and installation. The pac k age is a v ailable on PyPI and can b e installed b y pip install genriesz . The pack age is also a v ailable on GitHub https://github.com/ MasaKat0/genriesz . Optional extras enable scikit-learn-based and PyT orc h-based bases. The documentation is hosted at https://genriesz.readthedocs.io . Dep endencies. The core pac k age dep ends only on NumPy and SciPy ( Harris et al. , 2020 ; Virtanen et al. , 2020 ). Optional extras provide scikit-learn-based tree feature maps and PyT orch-based neural feature maps ( P edregosa et al. , 2011 ; Buitinck et al. , 2013 ; Paszk e et al. , 2019 ). Qualit y control. The pro ject includes unit tests, t yp e hin ts, and contin uous in tegration to catc h regressions. The soft ware is released under the GNU General Public License v3.0 (GPL-3.0). 7 Comparison to Related Soft w are Debiased ML and causal ML libraries. DoubleML ( Bac h et al. , 2022 , 2024 ) and EconML ( Batto cc hi et al. , 2019 ) provide pro duction-quality implemen tations of canonical DML estimators, with a fo cus on treatmen t effect estimation and, in EconML , heterogeneous effects. DoWhy ( Sharma & Kiciman , 2020 ) pro vides an end-to-end causal inference in terface cen tered on causal graphs. CausalML ( Chen et al. , 2020 ) offers a broad suite of uplift mo deling and heterogeneous effect estimators. 12 genriesz is complemen tary . It targets low-dimensional linear functionals with v alid inference via orthogonalization and emphasizes representer estimation and balancing. This estimand-cen tric API makes it conv enient to prototype new targets b y defining only the oracle m , while keeping the represen ter fitting problem explicit. Balancing-w eight and densit y ratio to olkits. Several to olkits implement sp ecific balancing-w eight estimators, such as en tropy balancing or stable weigh ts, for A TE-lik e problems. Density ratio estimation pack ages such as densratio ( Makiy ama , 2019 ) implement metho ds suc h as uLSIF, RuLSIF, and KLIEP . genriesz connects these approac hes to Riesz represen ter estimation and DML-st yle inference through a single in terface. 8 Conclusion The genriesz pac k age provides an estimand-and-balancing-cen tric implementation of ADML via generalized Riesz regression under Bregman div ergences. 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Journal of the Americ an Statistic al Asso ciation , 113(523):1228–1242, 2018. 4 Qingyuan Zhao. Cov ariate balancing propensity score b y tailored loss functions. The Annals of Statistics , 47(2):965 – 993, 2019. 4 , 5 Jos ´ e R. Zubizarreta. Stable w eigh ts that balance co v ariates for estimation with incomplete outcome data. Journal of the A meric an Statistic al Asso ciation , 110(511):910–922, 2015. 3 , 4 17 , A Built-in Bregman generators This appendix lists the built-in Bregman generators and their induced links. Squared distance generator. SquaredGenerator uses g ( α ) = ( α − C ) 2 . The link is ζ ( x, α ) = ∂ α g ( α ) = 2 ( α − C ). Unnormalized KL divergence generator. UKLGenerator uses, for | α | > C , g ( α ) = ( | α | − C ) log ( | α | − C ) − | α | . The link is ζ ( x, α ) = sign ( α ) log ( | α | − C ). Binary KL divergence generator. BKLGenerator uses, for | α | > C , g ( α ) = ( | α | − C ) log ( | α | − C ) − ( | α | + C ) log ( | α | + C ) . The corresponding link is ζ ( x, α ) = sign ( α )  log ( | α | − C ) − log ( | α | + C )  . Basu’s p ow er divergence generator. BPGenerator uses, for some ω > 0 and | α | > C , g ( α ) = ( | α | − C ) 1+ ω − ( | α | − C ) ω − ( | α | − C ) . The link is ζ ( x, α ) = sign ( α ) 1+ ω ω (( | α | − C ) ω − 1). PU learning loss generator. PUGenerator uses, for | α | ∈ (0 , 1), g ( α ) = C  | α | log ( | α | ) + (1 − | α | ) log (1 − | α | )  . The link is ζ ( x, α ) = sign ( α ) C  log ( | α | ) − log (1 − | α | )  . B Lik eliho o ds in TMLE Gaussian lik eliho o d. When Y is con tin uous, we use the additiv e fluctuation b γ (1) ( x ) : = b γ ( x ) + b ϵ b α ( x ) , b ϵ : = 1 n P n i =1 b α ( X i ) ( Y i − b γ ( X i )) 1 n P n i =1 b α ( X i ) 2 . (18) Since the functional γ 7→ m ( W, γ ) is linear, ( 17 ) can be written as b θ TMLE = b θ RA + b ϵ 1 n n X i =1 m ( W i , b α ) . (19) 18 Bernoulli lik eliho o d. When Y ∈ { 0 , 1 } , w e use a logistic fluctuation that updates b γ on the logit scale. Define Λ( t ) : = 1 / (1 + exp ( − t )) and logit ( p ) : = log  p 1 − p  . W e set b γ (1) ( x ) : = Λ (logit ( b γ ( x )) + b ϵ b α ( x )) , (20) where b ϵ is defined as a solution to the one-dimensional score equation 1 n n X i =1 b α ( X i )  Y i − b γ (1) ( X i )  = 0 . (21) The TMLE-st yle estimator is then computed by plugging b γ (1) in to ( 17 ). In con trast to the Gaussian case, the represen tation ( 19 ) do es not generally hold b ecause the map ϵ 7→ b γ (1) is nonlinear under the logit fluctuation. 19