arXiv subjects

Nathan Srebro

Publications and source records attributed to Nathan Srebro.

At least 19 recordsLinked to original sources

arXiv2026

Hierarchical Domain Generalization

We study hierarchical domain generalization as a problem of extrapolation from finite observed regions to an entire instance space, replacing i.i.d. sampling with arbitrary domain hierarchies. We show that the central obstruction is not only the complexity of the hypothesis class, but the train/test domain partition through which evidence is revealed. In particular, no matter how small the class or how large the training size, some partition makes generalization fail for some target. These results suggest that modern generalization theory must treat domain structure as a first-class object.

cs.LG↗

arXiv2026

On Incentivized Exploration beyond Bayesianism and Full-Information

We extend Incentive Compatible Exploration beyond the Bayesian full-information setting of Kremer et al. [2014]. We consider agents that may possess external information unknown to the principal. We show such settings require new notions of incentivized exploration, as well as going beyond a Bayesian perspective, and we introduce a definition where agents choose any reasonable (undominated) action. Furthermore, our framework provides for a more robust treatment of ties, and extends to settings where agents lack a single common prior and instead only know that reward distributions belong to a collection of potential priors.

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arXiv2026

Tight Sample Complexity of Transformers

We tightly characterize the VC dimension of depth-$L$ Transformers with a total of $W$ parameters, mapping an input sequence of length $T$ to a single output, establishing an upper bound of $O(L W \log (T W))$ and a nearly matching lower bound of $\Omega(L W \log (T W / L))$. We further tightly characterize the sample complexity of chain-of-thought learning using such a Transformer, showing teacher forcing (i.e. selecting a predictor consistent with the entire chain-of-thought on training data) learns with sample complexity $O\left(L W \log \left(\left(T+T^{\prime}\right) W\right)\right)$ and that any learning rule that uses chain-of-thought data requires at least $\Omega\left(L W \log \left(\left(T+T^{\prime}\right) W / L\right)\right)$ examples, where $T$ is the input length and $T^{\prime}$ is the number of autoregressive steps.

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arXiv2026

Learning to Think from Multiple Thinkers

We study learning with Chain-of-Thought (CoT) supervision from multiple thinkers, all of whom provide correct but possibly systematically different solutions, e.g., step-by-step solutions to math problems written by different thinkers, or step-by-step execution traces of different programs solving the same problem. We consider classes that are computationally easy to learn using CoT supervision from a single thinker, but hard to learn with only end-result supervision, i.e., without CoT (Joshi et al. 2025). We establish that, under cryptographic assumptions, learning can be hard from CoT supervision provided by two or a few different thinkers, in passive data-collection settings. On the other hand, we provide a generic computationally efficient active learning algorithm that learns with a small amount of CoT data per thinker that is completely independent of the target accuracy $\varepsilon$, a moderate number of thinkers that scales as $\log \frac{1}{\varepsilon}\log \log \frac{1}{\varepsilon}$, and sufficient passive end-result data that scales as $\frac{1}{\varepsilon}\cdot poly\log\frac{1}{\varepsilon}$.

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arXiv2026

Overfitting and Generalizing with (PAC) Bayesian Prediction in Noisy Binary Classification

We consider a PAC-Bayes type learning rule for binary classification, balancing the training error of a randomized ''posterior'' predictor with its KL divergence to a pre-specified ''prior''. This can be seen as an extension of a modified two-part-code Minimum Description Length (MDL) learning rule, to continuous priors and randomized predictions. With a balancing parameter of $\lambda=1$ this learning rule recovers an (empirical) Bayes posterior and a modified variant recovers the profile posterior, linking with standard Bayesian prediction (up to the treatment of the single-parameter noise level). However, from a risk-minimization prediction perspective, this Bayesian predictor overfits and can lead to non-vanishing excess loss in the agnostic case. Instead a choice of $\lambda \gg 1$, which can be seen as using a sample-size-dependent-prior, ensures uniformly vanishing excess loss even in the agnostic case. We precisely characterize the effect of under-regularizing (and over-regularizing) as a function of the balance parameter $\lambda$, understanding the regimes in which this under-regularization is tempered or catastrophic. This work extends previous work by Zhu and Srebro [2025] that considered only discrete priors to PAC Bayes type learning rules and, through their rigorous Bayesian interpretation, to Bayesian prediction more generally.

stat.ML↗

arXiv2026

Recursive Models for Long-Horizon Reasoning

Modern language models reason within bounded context, an inherent constraint that poses a fundamental barrier to long-horizon reasoning. We identify recursion as a core principle for overcoming this barrier, and propose recursive models as a minimal realization, where the model can recursively invoke itself to solve subtasks in isolated contexts. We prove that any computable problem admits a recursive decomposition of reasoning in which each subtask requires only exponentially smaller active context than standard autoregressive models; this strictly surpasses any context management approach confined to a single sequence, such as summarization. We further generalize our framework to modern agentic systems with arbitrary context processing and control flows, and prove that recursive models can achieve optimal power within this broader class. Experimentally, we test two settings: fine-tuning a pretrained base model for recursive SAT solving, and training a small model from scratch on Go traces generated by exact game-tree search. Both show improved long-horizon accuracy with small active contexts.

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arXiv2026

Positive Distribution Shift as a Framework for Understanding Tractable Learning

We study a setting where the goal is to learn a target function f(x) with respect to a target distribution D(x), but training is done on i.i.d. samples from a different training distribution D'(x), labeled by the true target f(x). Such a distribution shift (here in the form of covariate shift) is usually viewed negatively, as hurting or making learning harder, and the traditional distribution shift literature is mostly concerned with limiting or avoiding this negative effect. In contrast, we argue that with a well-chosen D'(x), the shift can be positive and make learning easier -- a perspective called Positive Distribution Shift (PDS). Such a perspective is central to contemporary machine learning, where much of the innovation is in finding good training distributions D'(x), rather than changing the training algorithm. We further argue that the benefit is often computational rather than statistical, and that PDS allows computationally hard problems to become tractable even using standard gradient-based training. We formalize different variants of PDS, show how certain hard classes are easily learnable under PDS, and make connections with membership query learning.

cs.LG↗

arXiv2025

Research Program: Theory of Learning in Dynamical Systems

Modern learning systems increasingly interact with data that evolve over time and depend on hidden internal state. We ask a basic question: when is such a dynamical system learnable from observations alone? This paper proposes a research program for understanding learnability in dynamical systems through the lens of next-token prediction. We argue that learnability in dynamical systems should be studied as a finite-sample question, and be based on the properties of the underlying dynamics rather than the statistical properties of the resulting sequence. To this end, we give a formulation of learnability for stochastic processes induced by dynamical systems, focusing on guarantees that hold uniformly at every time step after a finite burn-in period. This leads to a notion of dynamic learnability which captures how the structure of a system, such as stability, mixing, observability, and spectral properties, governs the number of observations required before reliable prediction becomes possible. We illustrate the framework in the case of linear dynamical systems, showing that accurate prediction can be achieved after finite observation without system identification, by leveraging improper methods based on spectral filtering. We survey the relationship between learning in dynamical systems and classical PAC, online, and universal prediction theories, and suggest directions for studying nonlinear and controlled systems.

cs.LG↗

arXiv2025

Shift is Good: Mismatched Data Mixing Improves Test Performance

We consider training and testing on mixture distributions with different training and test proportions. We show that in many settings, and in some sense generically, distribution shift can be beneficial, and test performance can improve due to mismatched training proportions, even if the components are unrelated and with no transfer between components. In a variety of scenarios, we identify the optimal training proportions and the extent to which such distribution shift can be beneficial. We show how the same analysis applies also to a compositional setting with differing distribution of component "skills'' at training and test.

cs.LG↗

arXiv2025

Learning to Answer from Correct Demonstrations

We study the problem of learning to generate an answer (or completion) to a question (or prompt), where there could be multiple correct answers, any one of which is acceptable at test time. Learning is based on demonstrations of some correct answer to each training question, as in Supervised Fine Tuning (SFT). We formalize the problem as imitation learning (i.e., apprenticeship learning) in contextual bandits, with offline demonstrations from some expert (optimal, or very good) policy, without explicitly observed rewards. In contrast to prior work, which assumes the demonstrator belongs to a bounded-complexity policy class, we propose relying only on the underlying reward model (i.e., specifying which answers are correct) being in a bounded-complexity class, which we argue is a strictly weaker assumption. We show that likelihood-maximization methods can fail in this setting, and instead present an approach that learns to answer nearly as well as the demonstrator, with sample complexity logarithmic in the cardinality of the reward class. Our method is similar to Syed and Schapire 2007, when adapted to a contextual bandit (i.e., single step) setup, but is a simple one-pass online approach that enjoys an "optimistic rate" (i.e., $1/\varepsilon$ when the demonstrator is optimal, versus $1/\varepsilon^2$ in Syed and Schapire), and works even with arbitrarily adaptive demonstrations.

cs.LG↗

arXiv2025

On the Hardness of Learning Regular Expressions

Despite the theoretical significance and wide practical use of regular expressions, the computational complexity of learning them has been largely unexplored. We study the computational hardness of improperly learning regular expressions in the PAC model and with membership queries. We show that PAC learning is hard even under the uniform distribution on the hypercube, and also prove hardness of distribution-free learning with membership queries. Furthermore, if regular expressions are extended with complement or intersection, we establish hardness of learning with membership queries even under the uniform distribution. We emphasize that these results do not follow from existing hardness results for learning DFAs or NFAs, since the descriptive complexity of regular languages can differ exponentially between DFAs, NFAs, and regular expressions.

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arXiv2025

Learning single-index models via harmonic decomposition

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.

cs.LG↗

arXiv2025

Temperature is All You Need for Generalization in Langevin Dynamics and other Markov Processes

We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution $\theta_0 \sim p_0$. We focus on Langevin dynamics with a positive temperature $\beta^{-1}$, i.e. gradient descent on a training loss $L$ with infinitesimal step size, perturbed with $\beta^{-1}$-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, at any time during training, by $\sqrt{(\beta\mathbb{E} L (\theta_0) + \log(1/\delta))/N}$ with probability $1-\delta$ over the dataset, where $N$ is the sample size, and $\mathbb{E} L (\theta_0) =O(1)$ with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.

cs.LG↗

arXiv2025

From Continual Learning to SGD and Back: Better Rates for Continual Linear Models

We study the common continual learning setup where an overparameterized model is sequentially fitted to a set of jointly realizable tasks. We analyze forgetting, defined as the loss on previously seen tasks, after $k$ iterations. For continual linear models, we prove that fitting a task is equivalent to a single stochastic gradient descent (SGD) step on a modified objective. We develop novel last-iterate SGD upper bounds in the realizable least squares setup and leverage them to derive new results for continual learning. Focusing on random orderings over $T$ tasks, we establish universal forgetting rates, whereas existing rates depend on problem dimensionality or complexity and become prohibitive in highly overparameterized regimes. In continual regression with replacement, we improve the best existing rate from $O((d-\bar{r})/k)$ to $O(\min(1/\sqrt[4]{k}, \sqrt{(d-\bar{r})}/k, \sqrt{T\bar{r}}/k))$, where $d$ is the dimensionality and $\bar{r}$ the average task rank. Furthermore, we establish the first rate for random task orderings without replacement. The resulting rate $O(\min(1/\sqrt[4]{T},\, (d-\bar{r})/T))$ shows that randomization alone, without task repetition, prevents catastrophic forgetting in sufficiently long task sequences. Finally, we prove a matching $O(1/\sqrt[4]{k})$ forgetting rate for continual linear classification on separable data. Our universal rates extend to broader methods, such as block Kaczmarz and POCS, illuminating their loss convergence under i.i.d. and single-pass orderings.

cs.LG↗

arXiv2025

PENCIL: Long Thoughts with Short Memory

While state-of-the-art LLMs have demonstrated great promise of using long Chains-of-Thought (CoT) to boost reasoning, scaling it up to more challenging problems at test-time is fundamentally limited by suboptimal memory usage -- intermediate computations accumulate indefinitely in context even when no longer needed for future thoughts. We introduce PENCIL, which incorporates a novel reduction mechanism into the autoregressive generation process that recursively cleans up intermediate thoughts based on patterns learned from training. By iteratively generating and erasing thoughts, PENCIL can think deeper to solve harder problems using shorter context and less compute. Empirically, we observe PENCIL is significantly more effective and efficient than CoT. For example, we demonstrate PENCIL with a small 25M-parameter transformer and 2048 context length solves Einstein's puzzle -- a task that challenges much larger models like GPT-4. Theoretically, we prove PENCIL can perform universal efficient computation by simulating any Turing machines with optimal time and space complexity, and thus can solve arbitrary computable tasks that are otherwise intractable for vanilla CoT.

cs.LG↗

arXiv2025

A Theory of Learning with Autoregressive Chain of Thought

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.

stat.ML↗

arXiv2025

Quantifying Overfitting along the Regularization Path for Two-Part-Code MDL in Supervised Classification

We provide a complete characterization of the entire regularization curve of a modified two-part-code Minimum Description Length (MDL) learning rule for binary classification, based on an arbitrary prior or description language. Grunwald and Langford [2004] previously established the lack of asymptotic consistency, from an agnostic PAC (frequentist worst case) perspective, of the MDL rule with a penalty parameter of $λ=1$, suggesting that it underegularizes. Driven by interest in understanding how benign or catastrophic under-regularization and overfitting might be, we obtain a precise quantitative description of the worst case limiting error as a function of the regularization parameter $λ$ and noise level (or approximation error), significantly tightening the analysis of Grunwald and Langford for $λ=1$ and extending it to all other choices of $λ$.

stat.ML↗

arXiv2025

Weak-to-Strong Generalization Even in Random Feature Networks, Provably

Weak-to-Strong Generalization (Burns et al., 2024) is the phenomenon whereby a strong student, say GPT-4, learns a task from a weak teacher, say GPT-2, and ends up significantly outperforming the teacher. We show that this phenomenon does not require a strong learner like GPT-4. We consider student and teacher that are random feature models, described by two-layer networks with a random and fixed bottom layer and a trained top layer. A "weak" teacher, with a small number of units (i.e. random features), is trained on the population, and a "strong" student, with a much larger number of units (i.e. random features), is trained only on labels generated by the weak teacher. We demonstrate, prove, and understand how the student can outperform the teacher, even though trained only on data labeled by the teacher. We also explain how such weak-to-strong generalization is enabled by early stopping. Importantly, we also show the quantitative limits of weak-to-strong generalization in this model.

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