Fenchel-Y oung losses are a family of loss functions, encompassing the squared, logistic and sparsemax losses, among others. They are convex w.r.t. the model

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We study the statistical-computational tradeoffs in a high dimensional single index modelY = f(X>β)+, where f is unknown,X is a Gaussian vector and β is s-sparse with unit norm. WhenCov(Y,X>β) 6= 0, [43] shows that the direction and support ofβ can be recovered using a generalized version of Lasso.

We appreciate the valuable comments from reviewers on paper presentation and typos. We will revise our work1 accordingly. Compared with these related work, we consider alarger model class.

Amongst those functions, the simplest are single-index modelsf(x) = ϕ(x θ), where the labels are generated by an arbitrary non-linear scalar link functionϕ applied to an unknown one-dimensional projectionθ of the input data.

Estimating the Riesz representer is central to debiased machine learning for causal and structural parameter estimation. We propose generalized Riesz regression, a unified framework that estimates the Riesz representer by fitting a representer model via Bregman divergence minimization. This framework includes the squared loss and the Kullback--Leibler (KL) divergence as special cases: the former recovers Riesz regression, while the latter recovers tailored loss minimization. Under suitable model specifications, the dual problems correspond to covariate balancing, which we call automatic covariate balancing. Moreover, under the same specifications, outcome averages weighted by the estimated Riesz representer satisfy Neyman orthogonality even without estimating the regression function, a property we call automatic Neyman orthogonalization. This property not only reduces the estimation error of Neyman orthogonal scores but also clarifies a key distinction between debiased machine learning and targeted maximum likelihood estimation. Our framework can also be viewed as a generalization of density ratio fitting under Bregman divergences to Riesz representer estimation, and it applies beyond density ratio estimation. We provide convergence analyses for both reproducing kernel Hilbert space (RKHS) and neural network model classes. A Python package for generalized Riesz regression is available at https://github.com/MasaKat0/grr.

Aligning large language models to preference data is commonly implemented by assuming a known link function between the distribution of observed preferences and the unobserved rewards (e.g., a logistic link as in Bradley-Terry). If the link is wrong, however, inferred rewards can be biased and policies be misaligned. We study policy alignment to preferences under an unknown and unrestricted link. We consider an $f$-divergence-constrained reward maximization problem and show that realizability of the solution in a policy class implies a semiparametric single-index binary choice model, where a scalar-valued index determined by a policy captures the dependence on demonstrations and the rest of the preference distribution is an unrestricted function thereof. Rather than focus on estimation of identifiable finite-dimensional structural parameters in the index as in econometrics, we focus on policy learning, focusing on error to the optimal policy and allowing unidentifiable and nonparametric indices. We develop a variety of policy learners based on profiling the link function, orthogonalizing the link function, and using link-agnostic bipartite ranking objectives. We analyze these and provide finite-sample policy error bounds that depend on generic functional complexity measures of the index class. We further consider practical implementations using first-order optimization suited to neural networks and batched data. The resulting methods are robust to unknown preference noise distribution and scale, while preserving the direct optimization of policies without explicitly fitting rewards.