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.

Deep equilibrium models (DEQs) have recently emerged as a powerful paradigm for training infinitely deep weight-tied neural networks that achieve state of the art performance across many modern machine learning tasks. Despite their practical success, theoretically understanding the gradient descent dynamics for training DEQs remains an area of active research. In this work, we rigorously study the gradient descent dynamics for DEQs in the simple setting of linear models and single-index models, filling several gaps in the literature. We prove a conservation law for linear DEQs which implies that the parameters remain trapped on spheres during training and use this property to show that gradient flow remains well-conditioned for all time. We then prove linear convergence of gradient descent to a global minimizer for linear DEQs and deep equilibrium single-index models under appropriate initialization and with a sufficiently small step size. Finally, we validate our theoretical findings through experiments.

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model $f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x})$ with hidden subspace $\boldsymbol{U}\in \mathbb{R}^{r\times d}$, which is the canonical setup to study representation learning. We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error using $\widetilde{\mathcal{O}}(d)$ samples and $\widetilde{\mathcal{O}}(d^2)$ time. The sample and time complexity both align with the information-theoretic limit up to leading order and are therefore optimal. During the first stage of gradient descent learning, the proof proceeds via showing that the inner weights can perform a power-iteration process. This process implicitly mimics a spectral start for the whole span of the hidden subspace and eventually eliminates finite-sample noise and recovers this span. It surprisingly indicates that optimal results can only be achieved if the first layer is trained for more than $\mathcal{O}(1)$ steps. This work demonstrates the ability of neural networks to effectively learn hierarchical functions with respect to both sample and time efficiency.

To understand feature learning dynamics in neural networks, recent theoretical works have focused on gradient-based learning of Gaussian single-index models, where the label is a nonlinear function of a latent one-dimensional projection of the input. While the sample complexity of online SGD is determined by the information exponent of the link function, recent works improved this by performing multiple gradient steps on the same sample with different learning rates -- yielding a non-correlational update rule -- and instead are limited by the (potentially much smaller) generative exponent. However, this picture is only valid when these learning rates are sufficiently large. In this paper, we characterize the relationship between learning rate(s) and sample complexity for a broad class of gradient-based algorithms that encapsulates both correlational and non-correlational updates. We demonstrate that, in certain cases, there is a phase transition from an "information exponent regime" with small learning rate to a "generative exponent regime" with large learning rate. Our framework covers prior analyses of one-pass SGD and SGD with batch reuse, while also introducing a new layer-wise training algorithm that leverages a two-timescales approach (via different learning rates for each layer) to go beyond correlational queries without reusing samples or modifying the loss from squared error. Our theoretical study demonstrates that the choice of learning rate is as important as the design of the algorithm in achieving statistical and computational efficiency.

All prior work either held only in the realizable setting or required the activation to be known.

Test-time training (TTT) enhances model performance by explicitly updating designated parameters prior to each prediction to adapt to the test data. While TTT has demonstrated considerable empirical success, its theoretical underpinnings remain limited, particularly for nonlinear models. In this paper, we investigate the combination of TTT with in-context learning (ICL), where the model is given a few examples from the target distribution at inference time. We analyze this framework in the setting of single-index models $y=σ_*(\langle β, \mathbf{x} \rangle)$, where the feature vector $β$ is drawn from a hidden low-dimensional subspace. For single-layer transformers trained with gradient-based algorithms and adopting TTT, we establish an upper bound on the prediction risk. Our theory reveals that TTT enables the single-layer transformers to adapt to both the feature vector $β$ and the link function $σ_*$, which vary across tasks. This creates a sharp contrast with ICL alone, which is theoretically difficult to adapt to shifts in the link function. Moreover, we provide the convergence rate with respect to the data length, showing the predictive error can be driven arbitrarily close to the noise level as the context size and the network width grow.

We consider the basic problem of learning Single-Index Models with respect to the square loss under the Gaussian distribution in the presence of adversarial label noise. Our main contribution is the first computationally efficient algorithm for this learning task, achieving a constant factor approximation, that succeeds for the class of {\em all} monotone activations with bounded moment of order $2 + ζ,$ for $ζ> 0.$ This class in particular includes all monotone Lipschitz functions and even discontinuous functions like (possibly biased) halfspaces. Prior work for the case of unknown activation either does not attain constant factor approximation or succeeds for a substantially smaller family of activations. The main conceptual novelty of our approach lies in developing an optimization framework that steps outside the boundaries of usual gradient methods and instead identifies a useful vector field to guide the algorithm updates by directly leveraging the problem structure, properties of Gaussian spaces, and regularity of monotone functions.

We study the problem of learning single-index models, where the label $y \in \mathbb{R}$ depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown one-dimensional projection $\langle \boldsymbol{w}_*,\boldsymbol{x}\rangle$. Prior work has shown that under Gaussian inputs, the statistical and computational complexity of recovering $\boldsymbol{w}_*$ is governed by the Hermite expansion of the link function. In this paper, we propose a new perspective: we argue that "spherical harmonics" -- rather than "Hermite polynomials" -- provide the natural basis for this problem, as they capture its intrinsic "rotational symmetry". Building on this insight, we characterize the complexity of learning single-index models under arbitrary spherically symmetric input distributions. We introduce two families of estimators -- based on tensor unfolding and online SGD -- that respectively achieve either optimal sample complexity or optimal runtime, and argue that estimators achieving both may not exist in general. When specialized to Gaussian inputs, our theory not only recovers and clarifies existing results but also reveals new phenomena that had previously been overlooked.