Chain-of-thought reasoning, where language models expend additional computation by producing thinking tokens prior to final responses, has driven significant advances in model capabilities. However, training these reasoning models is extremely costly in terms of both data and compute, as it involves collecting long traces of reasoning behavior from humans or synthetic generators and further post-training the model via reinforcement learning. Are these costs fundamental, or can they be reduced through better algorithmic design? We show that autocurriculum, where the model uses its own performance to decide which problems to focus training on, provably improves upon standard training recipes for both supervised fine-tuning (SFT) and reinforcement learning (RL). For SFT, we show that autocurriculum requires exponentially fewer reasoning demonstrations than non-adaptive fine-tuning, by focusing teacher supervision on prompts where the current model struggles. For RL fine-tuning, autocurriculum decouples the computational cost from the quality of the reference model, reducing the latter to a burn-in cost that is nearly independent of the target accuracy. These improvements arise purely from adaptive data selection, drawing on classical techniques from boosting and learning from counterexamples, and requiring no assumption on the distribution or difficulty of prompts.

Compositional generalization, the ability to solve complex problems by combining solutions to simpler sub-problems, is a fundamental capability of both natural and artificial intelligence, and a key mechanism underlying chain-of-thought reasoning. However, the theoretical underpinnings of compositional generalization remain poorly understood: when and why does decomposing a problem into parts yield more efficient learning than solving it directly? We study this question through the canonical problem of learning to simulate semiautomata (predicting the outcome of T steps of sequential computation), a model that captures state tracking, regular language recognition, and modular arithmetic. We show that an autocurriculum-based approach building on Part I of this series, recursively decomposing longer sequences into shorter sub-problems, learning to solve them, and composing the solutions, achieves dramatically better statistical complexity than direct methods. (i) For a setting inspired by supervised fine-tuning (SFT) where the learner receives interactive feedback on intermediate states of the computation, curriculum facilitates learning from only 2^𝒪(√(log T)) tokens of supervision; i.e., subpolynomial in the sequence length T, overcoming the Ω(T) token barrier required by direct simulation. (ii) For a setting inspired by reinforcement learning with verifiable rewards (RLVR), where the learner improves a pre-trained reference model using an outcome verifier, we show that curriculum reduces the requirement on the reference model from coverage at the full sequence length T to coverage at a shorter block length B ≪ T, an exponentially weaker condition.

The problem of contextual dueling bandits is central to reinforcement learning with human feedback (RLHF), a widely used approach in AI alignment for incorporating human preferences into learning systems. Despite its importance, existing methods are constrained either by strong preference modeling assumptions or by applicability only to finite action spaces. Moreover, prior algorithms typically rely on online optimization oracles, which are computationally infeasible for complex function classes, limiting their practical effectiveness. In this work, we present the first fundamental theoretical study of general contextual dueling bandits over continuous action spaces. Our key contribution is a novel algorithm based on a regularized min-max optimization framework that achieves a regret bound of $\tilde{O}(\sqrt{dT})$—the first such guarantee for this general setting. By leveraging offline oracles instead of online ones, our method further improves computational efficiency. Empirical evaluations validate our theoretical findings, with our approach significantly outperforming existing baselines in terms of regret.

We study the task of bandit learning, also known as best-arm identification, under the assumption that the true reward function f belongs to a known, but arbitrary, function class F. We seek a general theory of bandit learnability, akin to the PAC framework for classification. Our investigation is guided by the following two questions: (1) which classes F are learnable, and (2) how they are learnable. For example, in the case of binary PAC classification, learnability is fully determined by a combinatorial dimension - the VC dimension- and can be attained via a simple algorithmic principle, namely, empirical risk minimization (ERM). In contrast to classical learning-theoretic results, our findings reveal limitations of learning in structured bandits, offering insights into the boundaries of bandit learnability. First, for the question of "which", we show that the paradigm of identifying the learnable classes via a dimension-like quantity fails for bandit learning. We give a simple proof demonstrating that no combinatorial dimension can characterize bandit learnability, even in finite classes, following a standard definition of dimension introduced by Ben-David et al. (2019). For the question of "how", we prove a computational hardness result: we construct a reward function class for which at most two queries are needed to find the optimal action, yet no algorithm can do so in polynomial time unless RP=NP. We also prove that this class admits efficient algorithms for standard algorithmic operations often considered in learning theory, such as an ERM. This implies that computational hardness is in this case inherent to the task of bandit learning. Beyond these results, we investigate additional themes such as learning under noise, trade-offs between noise models, and the relationship between query complexity and regret minimization.

We study interactive learning in a setting where the agent has to generate a response (e.g., an action or trajectory) given a context and an instruction. In contrast, to typical approaches that train the system using reward or expert supervision on response, we study learning with hindsight instruction where a teacher provides an instruction that is most suitable for the agent's generated response. This hindsight labeling of instruction is often easier to provide than providing expert supervision of the optimal response which may require expert knowledge or can be impractical to elicit. We initiate the theoretical analysis of interactive learning with hindsight labeling. We first provide a lower bound showing that in general, the regret of any algorithm must scale with the size of the agent's response space. We then study a specialized setting where the underlying instruction-response distribution can be decomposed as a low-rank matrix. We introduce an algorithm called LORIL for this setting and show that its regret scales as $\sqrt{T}$ where $T$ is the number of rounds and depends on the intrinsic rank but does not depend on the size of the agent's response space. We provide experiments in two domains showing that LORIL outperforms baselines even when the low-rank assumption is violated.

A lexicographic maximum of a set X subset of R-n is a vector in X whose smallest component is as large as possible, and subject to that requirement, whose second smallest component is as large as possible, and so on for the third smallest component, etc. Lexicographic maximization has numerous practical and theoretical applications, including fair resource allocation, analyzing the implicit regularization of learning algorithms, and characterizing refinements of game-theoretic equilibria. We prove that a minimizer in X of the exponential loss function L-c(x) = Sigma(i) exp(-cx(i)) converges to a lexicographic maximum of X as c -> infinity, provided that X is stable in the sense that a well-known iterative method for finding a lexicographic maximum of X cannot be made to fail simply by reducing the required quality of each iterate by an arbitrarily tiny degree. Our result holds for both near and exact minimizers of the exponential loss, while earlier convergence results made much stronger assumptions about the set X and only held for the exact minimizer. We are aware of no previous results showing a connection between the iterative method for computing a lexicographic maximum and exponential loss minimization. We show that every convex polytope is stable, but that there exist compact, convex sets that are not stable. We also provide the first analysis of the convergence rate of an exponential loss minimizer (near or exact) and discover a curious dichotomy: While the two smallest components of the vector converge to the lexicographically maximum values very quickly (at roughly the rate log n/c), all other components can converge arbitrarily slowly.

We study contextual search, a generalization of binary search in higher dimensions, which captures settings such as feature-based dynamic pricing. Standard formulations of this problem assume that agents act in accordance with a specific homogeneous response model. In practice, however, some responses may be adversarially corrupted. Existing algorithms heavily depend on the assumed responsemodel being (approximately) accurate for all agents and have poor performance in the presence of even a few such arbitrary misspecifications. We initiate the study of contextual search when some of the agents can behave in ways inconsistent with the underlying response model. In particular, we provide two algorithms, one based on multidimensional binary search methods and one based on gradient descent. We show that these algorithms attain near-optimal regret in the absence of adversarial corruptions and their performance degrades gracefully with the number of such agents, providing the first results for contextual search in any adversarial noise model. Our techniques draw inspiration from learning theory, game theory, high-dimensional geometry, and convex analysis.

We study an abstract framework for interactive learning called interactive estimation in which the goal is to estimate a target from its "similarity” to points queried by the learner. We introduce a combinatorial measure called dissimilarity dimension which largely captures learnability in our model. We present a simple, general, and broadly-applicable algorithm, for which we obtain both regret and PAC generalization bounds that are polynomial in the new dimension. We show that our framework subsumes and thereby unifies two classic learning models: statistical-query learning and structured bandits. We also delineate how the dissimilarity dimension is related to well-known parameters for both frameworks, in some cases yielding significantly improved analyses.

We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversarially. This is a considerably harder problem, compared to both the stochastic version and the "classic" adversarial bandits, in that regret minimization is no longer feasible. Instead, the objective is to minimize the competitive ratio: the ratio of the benchmark reward to algorithm's reward. We design an algorithm with competitive ratio O(log T) relative to the best fixed distribution over actions, where T is the time horizon; we also prove a matching lower bound. The key conceptual contribution is a new perspective on the stochastic version of the problem. We suggest a new algorithm for the stochastic version, which builds on the framework of regret minimization in repeated games and admits a substantially simpler analysis compared to prior work. We then analyze this algorithm for the adversarial version, and use it as a subroutine to solve the latter. Our algorithm is the first "black-box reduction" from bandits to BwK: it takes an arbitrary bandit algorithm and uses it as a subroutine. We use this reduction to derive several extensions.

Not all convex functions on $\mathbb{R}^n$ have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study astral space, a compact extension of $\mathbb{R}^n$ to which such points at infinity have been added. Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space. Although astral space includes all of $\mathbb{R}^n$, it is not a vector space, nor even a metric space. However, it is sufficiently well-structured to allow useful and meaningful extensions of concepts of convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.

We study reinforcement learning (RL) in settings where observations are high-dimensional, but where an RL agent has access to abstract knowledge about the structure of the state space, as is the case, for example, when a robot is tasked to go to a specific room in a building using observations from its own camera, while having access to the floor plan. We formalize this setting as transfer reinforcement learning from an abstract simulator, which we assume is deterministic (such as a simple model of moving around the floor plan), but which is only required to capture the target domain's latent-state dynamics approximately up to unknown (bounded) perturbations (to account for environment stochasticity). Crucially, we assume no prior knowledge about the structure of observations in the target domain except that they can be used to identify the latent states (but the decoding map is unknown). Under these assumptions, we present an algorithm, called TASID, that learns a robust policy in the target domain, with sample complexity that is polynomial in the horizon, and independent of the number of states, which is not possible without access to some prior knowledge. In synthetic experiments, we verify various properties of our algorithm and show that it empirically outperforms transfer RL algorithms that require access to "full simulators" (i.e., those that also simulate observations).

Thompson sampling and other Bayesian sequential decision-making algorithms are among the most popular approaches to tackle explore/exploit trade-offs in (contextual) bandits. The choice of prior in these algorithms offers flexibility to encode domain knowledge but can also lead to poor performance when misspecified. In this paper, we demonstrate that performance degrades gracefully with misspecification. We prove that the expected reward accrued by Thompson sampling (TS) with a misspecified prior differs by at most $\tilde{\mathcal{O}}(H^2 \epsilon)$ from TS with a well specified prior, where $\epsilon$ is the total-variation distance between priors and $H$ is the learning horizon. Our bound does not require the prior to have any parametric form. For priors with bounded support, our bound is independent of the cardinality or structure of the action space, and we show that it is tight up to universal constants in the worst case. Building on our sensitivity analysis, we establish generic PAC guarantees for algorithms in the recently studied Bayesian meta-learning setting and derive corollaries for various families of priors. Our results generalize along two axes: (1) they apply to a broader family of Bayesian decision-making algorithms, including a Monte-Carlo implementation of the knowledge gradient algorithm (KG), and (2) they apply to Bayesian POMDPs, the most general Bayesian decision-making setting, encompassing contextual bandits as a special case. Through numerical simulations, we illustrate how prior misspecification and the deployment of one-step look-ahead (as in KG) can impact the convergence of meta-learning in multi-armed and contextual bandits with structured and correlated priors.

We study contextual search, a generalization of binary search in higher dimensions, which captures settings such as feature-based dynamic pricing. Standard game-theoretic formulations of this problem assume that agents act in accordance with a specific behavioral model. In practice, some agents may not subscribe to the dominant behavioral model or may act in ways that are seemingly arbitrarily irrational . Existing algorithms heavily depend on the behavioral model being (approximately) accurate for all agents and have poor performance even with a few arbitrarily irrational agents. We initiate the study of contextual search when some of the agents can behave in ways inconsistent with the underlying behavioral model. In particular, we provide two algorithms, one based on multidimensional binary search methods and one based on gradient descent. Our techniques draw inspiration from learning theory, game theory, high-dimensional geometry, and convex analysis.

Boosting is an algorithmic approach which is based on the idea of combining weak and moderately inaccurate hypotheses to a strong and accurate one. In this work we study multiclass boosting with a possibly large number of classes or categories. Multiclass boosting can be formulated in various ways. Here, we focus on an especially natural formulation in which the weak hypotheses are assumed to belong to an ''easy-to-learn'' base class, and the weak learner is an agnostic PAC learner for that class with respect to the standard classification loss. This is in contrast with other, more complicated losses as have often been considered in the past. The goal of the overall boosting algorithm is then to learn a combination of weak hypotheses by repeatedly calling the weak learner. We study the resources required for boosting, especially how they depend on the number of classes $k$, for both the booster and weak learner. We find that the boosting algorithm itself only requires $O(\log k)$ samples, as we show by analyzing a variant of AdaBoost for our setting. In stark contrast, assuming typical limits on the number of weak-learner calls, we prove that the number of samples required by a weak learner is at least polynomial in $k$, exponentially more than the number of samples needed by the booster. Alternatively, we prove that the weak learner's accuracy parameter must be smaller than an inverse polynomial in $k$, showing that the returned weak hypotheses must be nearly the best in their class when $k$ is large. We also prove a trade-off between number of oracle calls and the resources required of the weak learner, meaning that the fewer calls to the weak learner the more that is demanded on each call.

We present a novel interactive learning protocol that enables training request-fulfilling agents by verbally describing their activities. Unlike imitation learning (IL), our protocol allows the teaching agent to provide feedback in a language that is most appropriate for them. Compared with reward in reinforcement learning (RL), the description feedback is richer and allows for improved sample complexity. We develop a probabilistic framework and an algorithm that practically implements our protocol. Empirical results in two challenging request-fulfilling problems demonstrate the strengths of our approach: compared with RL baselines, it is more sample-efficient; compared with IL baselines, it achieves competitive success rates without requiring the teaching agent to be able to demonstrate the desired behavior using the learning agent's actions. Apart from empirical evaluation, we also provide theoretical guarantees for our algorithm under certain assumptions about the teacher and the environment.

We propose a general framework for interactively learning combinatorial structures, such as (binary or non-binary) classifiers, orderings/rankings of items, or clusterings, when the underlying structure changes over time. Inspired by Angluin's equivalence query model, the algorithm proposes a structure in each round, and it either learns that its proposal is the true structure in this round, or it observes a specific mistake in the proposal. The feedback is correct only with probability 1 - p, and adversarially incorrect with probability p. The algorithm's goal is to minimize its number of mistakes over the course of R rounds. Our general framework is based on a graph representation of the structures and feedback in a static environment, proposed by Emamjomeh-Zadeh and Kempe (2017). To be able to learn efficiently, it is sufficient that there be a graph G whose nodes are the candidate structures and whose (weighted) edges capture the possible feedback, satisfying a certain natural shortest paths property. To model the evolution of the underlying structure, we consider two natural models, which we term the Shifting Target model and Drifting Target model. In the former, the true structure always belongs to a small pool of candidate structures. In the latter, the structure can change only by transitioning along the edges of a known evolution graph. In order to achieve non-trivial results, we bound the total number of times the underlying structure can change, denoted by B. We provide upper and lower bounds on the number of mistakes, which depend on the total number of changes B, the total number of structures n, and natural measures of complexity of the dynamic models: the size of the pool of candidate structures in the Shifting Target model, and the maximum degree of the evolution graph in the Drifting Target model.

Recent work across many machine learning disciplines has highlighted that standard descent methods, even without explicit regularization, do not merely minimize the training error, but also exhibit an implicit bias. This bias is typically towards a certain regularized solution, and relies upon the details of the learning process, for instance the use of the cross-entropy loss. In this work, we show that for empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses, if the risk does not attain its infimum, then the gradient-descent path and the algorithm-independent regularization path converge to the same direction (whenever either converges to a direction). Using this result, we provide a justification for the widely-used exponentially-tailed losses (such as the exponential loss or the logistic loss): while this convergence to a direction for exponentially-tailed losses is necessarily to the maximum-margin direction, other losses such as polynomially-tailed losses may induce convergence to a direction with a poor margin.

We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Follow-the-Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren, who showed that oracle-efficient algorithms do not exist in general and asked whether one can identify properties under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) s-level auctions of Morgenstern and Roughgarden for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders' valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting. Finally, we derive various extensions, including: (1) oracle-efficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding in simultaneous auctions, resolving an open problem of Daskalakis and Syrgkanis.

In standard reinforcement learning (RL), a learning agent seeks to optimize the overall reward. However, many key aspects of a desired behavior are more naturally expressed as constraints. For instance, the designer may want to limit the use of unsafe actions, increase the diversity of trajectories to enable exploration, or approximate expert trajectories when rewards are sparse. In this paper, we propose an algorithmic scheme that can handle a wide class of constraints in RL tasks: specifically, any constraints that require expected values of some vector measurements (such as the use of an action) to lie in a convex set. This captures previously studied constraints (such as safety and proximity to an expert), but also enables new classes of constraints (such as diversity). Our approach comes with rigorous theoretical guarantees and only relies on the ability to approximately solve standard RL tasks. As a result, it can be easily adapted to work with any model-free or model-based RL. In our experiments, we show that it matches previous algorithms that enforce safety via constraints, but can also enforce new properties that these algorithms do not incorporate, such as diversity.

We study the computational tractability of provably sample-efficient (PAC) reinforcement learning in episodic environments with rich observations. We present new sample-efficient algorithms for environments with deterministic hidden state dynamics and stochastic rich observations. These methods operate in an oracle model of computation — accessing policy and value function classes exclusively through standard optimization primitives — and therefore represent computationally efficient alternatives to prior algorithms that require enumeration. In the more general stochastic transition setting, we prove that the only known sample-efficient algorithm, Olive [1], cannot be implemented in our oracle model. We also present several examples that illustrate fundamental challenges of tractable PAC reinforcement learning in such general settings.