Is Code Better Than Language for Algorithmic Reasoning

For tool-augmented language models, comparing natural-language reasoning with code-execution pipelines is difficult because the comparison changes both the intermediate representation and the execution mechanism. We separate these factors with an intermediate intervention: the model expresses its reasoning as executable code, and the language model simulates that code in context to produce an answer. On a 40-task verifiable algorithmic benchmark, deterministic code execution outperforms natural-language reasoning by +31.6pp. We observe that the intermediate intervention is not meaningfully different from natural-language reasoning (+0.15pp). These results suggest that, in our evaluated setting, changing the intermediate representation alone does not explain the tool-use advantage, providing evidence for the performance gains requiring reliable external execution. We formalize this intuition with a simple statistical decision-theoretic model that characterizes when execution dominates end-to-end risk in our disentangled trace-generation/execution regime. We validate our theory using a reconstruction intervention that leverages a proxy language model to infer natural-language reasoning traces from code representations, recovering performance comparable to the original natural-language reasoning pipeline. All experiments are at https://github.com/TerryTong-Git/ToolProj.

Less Data, Faster Training: repeating smaller datasets speeds up learning via sampling biases

This work investigates the ``small-vs-large gap'', where repeating on fewer samples can lead to compute saving during training compared to using a larger dataset. This is observed across algorithmic tasks, architectures and optimizers and cannot be explained using prior theory. We argue that the speedup comes from appropriate layer-wise growth enabled by sampling biases, which is more pronounced when the dataset size is smaller. We provide both theoretical analysis and empirical evidence from various interventions. Our results suggest that using a smaller dataset with more repetitions is not just a fallback strategy under data scarcity, but can be proactively leveraged as a favorable inductive biases for optimization, particularly in reasoning tasks.

Learning When to Stop: Selective Imitation Learning Under Arbitrary Dynamics Shift

Behavior cloning provides strong imitation learning guarantees when training and test environments share the same dynamics. However, in many deployment settings the test environment's transitions differ from training, and classical offline IL offers no recourse: the learner must commit to an action at every state, even when its demonstrations are uninformative and could lead to arbitrary degradation of performance. This motivates the study of selective imitation, where the learner may choose to stop when it cannot act reliably. We introduce a model for selective imitation under arbitrary dynamics shift: given labeled expert demonstrations from a training environment and unlabeled state trajectories from the same expert in a test environment, the learner outputs a selective policy that is complete (rarely stops in training) and sound (incurs low regret before stopping in test). Our algorithm, SeqRejectron, constructs a stopping rule using a small set of validator policies whose size is independent of the horizon or policy class. For deterministic policies, this yields horizon-free $\tilde{O}(\log|Π|/ε^2)$ sample complexity, assuming sparse costs. For stochastic policies, we obtain analogous horizon-free guarantees using a cumulative Hellinger stopping time. We extend the framework to misspecified experts and different expert policies across train and test and obtain results that gracefully degrade with the amount of misspecification.

Weight Clipping for Robust Conformal Inference under Unbounded Covariate Shifts

Conformal prediction (CP) provides powerful, distribution-free prediction sets, but its guarantees rely on the exchangeability of training and test data, which is often violated in practice due to covariate shifts. While weighted conformal prediction (WCP) is designed to handle such shifts, it can suffer from significant undercoverage when the density ratio between the distributions is unbounded and/or must be learned. This is because of both overfitting in learning the density ratio, and high variance in estimating the nonconformity score threshold. To address this, we introduce clipped least-squares importance fitting (CLISF) as a reduced-variance method for density ratio estimation. Specifically, we show that density ratios learned using CLISF, when plugged into WCP, have bounded expected undercoverage. Furthermore, we show that the undercoverage can be corrected by running WCP with a slightly inflated coverage target; crucially, we are able to estimate the required level of inflation from the data. We provide the first theoretical guarantees for weight clipping in conformal inference, achieving dataset-conditional coverage with a sample complexity that does not blow up with the higher moments of the true density ratio -- a key limitation of prior work. We verify our results on real-world benchmarks and synthetic data.

Model Agreement via Anchoring

Numerous lines of aim to control $\textit{model disagreement}$ -- the extent to which two machine learning models disagree in their predictions. We adopt a simple and standard notion of model disagreement in real-valued prediction problems, namely the expected squared difference in predictions between two models trained on independent samples, without any coordination of the training processes. We would like to be able to drive disagreement to zero with some natural parameter(s) of the training procedure using analyses that can be applied to existing training methodologies. We develop a simple general technique for proving bounds on independent model disagreement based on $\textit{anchoring}$ to the average of two models within the analysis. We then apply this technique to prove disagreement bounds for four commonly used machine learning algorithms: (1) stacked aggregation over an arbitrary model class (where disagreement is driven to 0 with the number of models $k$ being stacked) (2) gradient boosting (where disagreement is driven to 0 with the number of iterations $k$) (3) neural network training with architecture search (where disagreement is driven to 0 with the size $n$ of the architecture being optimized over) and (4) regression tree training over all regression trees of fixed depth (where disagreement is driven to 0 with the depth $d$ of the tree architecture). For clarity, we work out our initial bounds in the setting of one-dimensional regression with squared error loss -- but then show that all of our results generalize to multi-dimensional regression with any strongly convex loss.

Reliable Abstention under Adversarial Injections: Tight Lower Bounds and New Upper Bounds

We study online learning in the adversarial injection model introduced by [Goel et al. 2017], where a stream of labeled examples is predominantly drawn i.i.d.\ from an unknown distribution $\mathcal{D}$, but may be interspersed with adversarially chosen instances without the learner knowing which rounds are adversarial. Crucially, labels are always consistent with a fixed target concept (the clean-label setting). The learner is additionally allowed to abstain from predicting, and the total error counts the mistakes whenever the learner decides to predict and incorrect abstentions when it abstains on i.i.d.\ rounds. Perhaps surprisingly, prior work shows that oracle access to the underlying distribution yields $O(d^2 \log T)$ combined error for VC dimension $d$, while distribution-agnostic algorithms achieve only $\tilde{O}(\sqrt{T})$ for restricted classes, leaving open whether this gap is fundamental. We resolve this question by proving a matching $Ω(\sqrt{T})$ lower bound for VC dimension $1$, establishing a sharp separation between the two information regimes. On the algorithmic side, we introduce a potential-based framework driven by \emph{robust witnesses}, small subsets of labeled examples that certify predictions while remaining resilient to adversarial contamination. We instantiate this framework using two combinatorial dimensions: (1) \emph{inference dimension}, yielding combined error $\tilde{O}(T^{1-1/k})$ for classes of inference dimension $k$, and (2) \emph{certificate dimension}, a new relaxation we introduce. As an application, we show that halfspaces in $\mathbb{R}^2$ have certificate dimension $3$, obtaining the first distribution-agnostic bound of $\tilde{O}(T^{2/3})$ for this class. This is notable since [Blum et al. 2021] showed halfspaces are not robustly learnable under clean-label attacks without abstention.

Emergent Alignment via Competition

Aligning AI systems with human values remains a fundamental challenge, but does our inability to create perfectly aligned models preclude obtaining the benefits of alignment? We study a strategic setting where a human user interacts with multiple differently misaligned AI agents, none of which are individually well-aligned. Our key insight is that when the users utility lies approximately within the convex hull of the agents utilities, a condition that becomes easier to satisfy as model diversity increases, strategic competition can yield outcomes comparable to interacting with a perfectly aligned model. We model this as a multi-leader Stackelberg game, extending Bayesian persuasion to multi-round conversations between differently informed parties, and prove three results: (1) when perfect alignment would allow the user to learn her Bayes-optimal action, she can also do so in all equilibria under the convex hull condition (2) under weaker assumptions requiring only approximate utility learning, a non-strategic user employing quantal response achieves near-optimal utility in all equilibria and (3) when the user selects the best single AI after an evaluation period, equilibrium guarantees remain near-optimal without further distributional assumptions. We complement the theory with two sets of experiments.

Conformal Language Model Reasoning with Coherent Factuality

Language models are increasingly being used in important decision pipelines, so ensuring the correctness of their outputs is crucial. Recent work has proposed evaluating the "factuality" of claims decomposed from a language model generation and applying conformal prediction techniques to filter out those claims that are not factual. This can be effective for tasks such as information retrieval, where constituent claims may be evaluated in isolation for factuality, but is not appropriate for reasoning tasks, as steps of a logical argument can be evaluated for correctness only within the context of the claims that precede them. To capture this, we define "coherent factuality" and develop a conformal-prediction-based method to guarantee coherent factuality for language model outputs. Our approach applies split conformal prediction to subgraphs within a "deducibility" graph" that represents the steps of a reasoning problem. We evaluate our method on mathematical reasoning problems from the MATH and FELM datasets and find that our algorithm consistently produces correct and substantiated orderings of claims, achieving coherent factuality across target coverage levels. Moreover, we achieve 90% factuality on our stricter definition while retaining 80% or more of the original claims, highlighting the utility of our deducibility-graph-guided approach.

Probabilistic Stability Guarantees for Feature Attributions

Stability guarantees are an emerging tool for evaluating feature attributions, but existing certification methods rely on smoothed classifiers and often yield conservative guarantees. To address these limitations, we introduce soft stability and propose a simple, model-agnostic, and sample-efficient stability certification algorithm (SCA) that provides non-trivial and interpretable guarantees for any attribution. Moreover, we show that mild smoothing enables a graceful tradeoff between accuracy and stability, in contrast to prior certification methods that require a more aggressive compromise. Using Boolean function analysis, we give a novel characterization of stability under smoothing. We evaluate SCA on vision and language tasks, and demonstrate the effectiveness of soft stability in measuring the robustness of explanation methods.

Collaborative Prediction: Tractable Information Aggregation via Agreement

We give efficient "collaboration protocols" through which two parties, who observe different features about the same instances, can interact to arrive at predictions that are more accurate than either could have obtained on their own. The parties only need to iteratively share and update their own label predictions-without either party ever having to share the actual features that they observe. Our protocols are efficient reductions to the problem of learning on each party's feature space alone, and so can be used even in settings in which each party's feature space is illegible to the other-which arises in models of human/AI interaction and in multi-modal learning. The communication requirements of our protocols are independent of the dimensionality of the data. In an online adversarial setting we show how to give regret bounds on the predictions that the parties arrive at with respect to a class of benchmark policies defined on the joint feature space of the two parties, despite the fact that neither party has access to this joint feature space. We also give simpler algorithms for the same task in the batch setting in which we assume that there is a fixed but unknown data distribution. We generalize our protocols to a decision theoretic setting with high dimensional outcome spaces, where parties communicate only "best response actions." Our theorems give a computationally and statistically tractable generalization of past work on information aggregation amongst Bayesians who share a common and correct prior, as part of a literature studying "agreement" in the style of Aumann's agreement theorem. Our results require no knowledge of (or even the existence of) a prior distribution and are computationally efficient. Nevertheless we show how to lift our theorems back to this classical Bayesian setting, and in doing so, give new information aggregation theorems for Bayesian agreement.