For a given base class of sequence-to-next-token generators, we consider learning prompt-to-answer mappings obtained by iterating a fixed, time-invariant generator for multiple steps, thus generating a chain-of-thought, and then taking the final token as the answer. We formalize the learning problems both when the chain-of-thought is observed and when training only on prompt-answer pairs, with the chain-of-thought latent. We analyze the sample and computational complexity both in terms of general properties of the base class (e.g. its VC dimension) and for specific base classes such as linear thresholds. We present a simple base class that allows for universal representability and computationally tractable chain-of-thought learning. Central to our development is that time invariance allows for sample complexity that is independent of the length of the chain-of-thought. Attention arises naturally in our construction.

We study the problem of exact community recovery in general, two-community block models, in the presence of node-attributed *side information*. We allow for a very general side information channel for node attributes, and for pairwise (edge) observations, consider both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, Submatrix Localization, and $\mathbb{Z}_2$-Synchronization as special cases. A recent work of Dreveton et al. 2024 characterized the information-theoretic limit of a very general exact recovery problem with side information. In this paper, we show algorithmic achievability in the above important cases by designing a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the pairwise observation matrix. Using the powerful tool of entrywise eigenvector analysis [Abbe et al. 2020], we show that our spectral algorithm can mimic the so called *genie-aided estimators*, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.

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 \textit{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.

Local SGD is a popular optimization method in distributed learning, often outperforming mini-batch SGD. Despite this practical success, proving the efficiency of local SGD has been difficult, creating a significant gap between theory and practice. We provide new lower bounds for local SGD under existing first-order data heterogeneity assumptions, showing these assumptions can not capture local SGD’s effectiveness. We also demonstrate the min-max optimality of accelerated mini-batch SGD under these assumptions. Our findings emphasize the need for improved modeling of data heterogeneity. Under higher-order assumptions, we provide new upper bounds that verify the dominance of local SGD over mini-batch SGD when data heterogeneity is low.

Understanding how overparameterized neural networks generalize despite perfect interpolation of noisy training data is a fundamental question. Mallinar et. al. (2022) noted that neural networks seem to often exhibit ``tempered overfitting'', wherein the population risk does not converge to the Bayes optimal error, but neither does it approach infinity, yielding non-trivial generalization. However, this has not been studied rigorously. We provide the first rigorous analysis of the overfiting behaviour of regression with minimum norm ($\ell_2$ of weights), focusing on univariate two-layer ReLU networks. We show overfitting is tempered (with high probability) when measured with respect to the $L_1$ loss, but also show that the situation is more complex than suggested by Mallinar et. al., and overfitting is catastrophic with respect to the $L_2$ loss, or when taking an expectation over the training set.

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$—potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.