The Sample Complexity of Online Reinforcement Learning: A Multi-model Perspective

We study the sample complexity of online reinforcement learning for nonlinear dynamical systems with continuous state and action spaces. Our analysis accommodates a large class of dynamical systems ranging from a finite set of nonlinear candidate models to models with bounded and Lipschitz continuous dynamics, to systems that are parametrized by a compact and real-valued set of parameters. In the most general setting, our algorithm achieves a policy regret of $\mathcal{O}(N \epsilon^2 + \mathrm{ln}(m(\epsilon))/\epsilon^2)$, where $N$ is the time horizon, $\epsilon$ is a user-specified discretization width, and $m(\epsilon)$ measures the complexity of the function class under consideration via its packing number. In the special case where the dynamics are parametrized by a compact and real-valued set of parameters (such as neural networks, transformers, etc.), we prove a policy regret of $\mathcal{O}(\sqrt{N p})$, where $p$ denotes the number of parameters, recovering earlier sample-complexity results that were derived for linear time-invariant dynamical systems. While this article focuses on characterizing sample complexity, the proposed algorithms are likely to be useful in practice, due to their simplicity, the ability to incorporate prior knowledge, and their benign transient behavior.

Conformal Prediction Sets with Improved Conditional Coverage using Trust Scores

Standard conformal prediction offers a marginal guarantee on coverage, but for prediction sets to be truly useful, they should ideally ensure coverage conditional on each test point. Unfortunately, it is impossible to achieve exact, distribution-free conditional coverage in finite samples. In this work, we propose an alternative conformal prediction algorithm that targets coverage where it matters most--in instances where a classifier is overconfident in its incorrect predictions. We start by dissecting miscoverage events in marginally-valid conformal prediction, and show that miscoverage rates vary based on the classifier's confidence and its deviation from the Bayes optimal classifier. Motivated by this insight, we develop a variant of conformal prediction that targets coverage conditional on a reduced set of two variables: the classifier's confidence in a prediction and a nonparametric trust score that measures its deviation from the Bayes classifier. Empirical evaluation on multiple image datasets shows that our method generally improves conditional coverage properties compared to standard conformal prediction, including class-conditional coverage, coverage over arbitrary subgroups, and coverage over demographic groups.

Gradient Equilibrium in Online Learning: Theory and Applications

We present a new perspective on online learning that we refer to as gradient equilibrium: a sequence of iterates achieves gradient equilibrium if the average of gradients of losses along the sequence converges to zero. In general, this condition is not implied by nor implies sublinear regret. It turns out that gradient equilibrium is achievable by standard online learning methods such as gradient descent and mirror descent with constant step sizes (rather than decaying step sizes, as is usually required for no regret). Further, as we show through examples, gradient equilibrium translates into an interpretable and meaningful property in online prediction problems spanning regression, classification, quantile estimation, and others. Notably, we show that the gradient equilibrium framework can be used to develop a debiasing scheme for black-box predictions under arbitrary distribution shift, based on simple post hoc online descent updates. We also show that post hoc gradient updates can be used to calibrate predicted quantiles under distribution shift, and that the framework leads to unbiased Elo scores for pairwise preference prediction.

An Optimistic Algorithm for Online Convex Optimization with Adversarial Constraints

We study Online Convex Optimization (OCO) with adversarial constraints, where an online algorithm must make repeated decisions to minimize both convex loss functions and cumulative constraint violations. We focus on a setting where the algorithm has access to predictions of the loss and constraint functions. Our results show that we can improve the current best bounds of $ O(\sqrt{T}) $ regret and $ \tilde{O}(\sqrt{T}) $ cumulative constraint violations to $ O(\sqrt{E_T(f)}) $ and $ \tilde{O}(\sqrt{E_T(g)}) $, respectively, where $ E_T(f) $ and $ E_T(g) $ represent the cumulative prediction errors of the loss and constraint functions. In the worst case, where $ E_T(f) = O(T) $ and $ E_T(g) = O(T) $ (assuming bounded loss and constraint functions), our rates match the prior $ O(\sqrt{T}) $ results. However, when the loss and constraint predictions are accurate, our approach yields significantly smaller regret and cumulative constraint violations. Notably, if the constraint function remains constant over time, we achieve $ \tilde{O}(1) $ cumulative constraint violation, aligning with prior results.

Dimension-free Private Mean Estimation for Anisotropic Distributions

We present differentially private algorithms for high-dimensional mean estimation. Previous private estimators on distributions over $\mathbb{R}^d$ suffer from a curse of dimensionality, as they require $\Omega(d^{1/2})$ samples to achieve non-trivial error, even in cases where $O(1)$ samples suffice without privacy. This rate is unavoidable when the distribution is isotropic, namely, when the covariance is a multiple of the identity matrix, or when accuracy is measured with respect to the affine-invariant Mahalanobis distance. Yet, real-world data is often highly anisotropic, with signals concentrated on a small number of principal components. We develop estimators that are appropriate for such signals$\unicode{x2013}$our estimators are $(\varepsilon,\delta)$-differentially private and have sample complexity that is dimension-independent for anisotropic subgaussian distributions. Given $n$ samples from a distribution with known covariance-proxy $\Sigma$ and unknown mean $\mu$, we present an estimator $\hat{\mu}$ that achieves error $\|\hat{\mu}-\mu\|_2\leq \alpha$, as long as $n\gtrsim\mathrm{tr}(\Sigma)/\alpha^2+ \mathrm{tr}(\Sigma^{1/2})/(\alpha\varepsilon)$. In particular, when $\pmb{\sigma}^2=(\sigma_1^2, \ldots, \sigma_d^2)$ are the singular values of $\Sigma$, we have $\mathrm{tr}(\Sigma)=\|\pmb{\sigma}\|_2^2$ and $\mathrm{tr}(\Sigma^{1/2})=\|\pmb{\sigma}\|_1$, and hence our bound avoids dimension-dependence when the signal is concentrated in a few principal components. We show that this is the optimal sample complexity for this task up to logarithmic factors. Moreover, for the case of unknown covariance, we present an algorithm whose sample complexity has improved dependence on the dimension, from $d^{1/2}$ to $d^{1/4}$.

Enhancing Feature-Specific Data Protection via Bayesian Coordinate Differential Privacy

Local Differential Privacy (LDP) offers strong privacy guarantees without requiring users to trust external parties. However, LDP applies uniform protection to all data features, including less sensitive ones, which degrades performance of downstream tasks. To overcome this limitation, we propose a Bayesian framework, Bayesian Coordinate Differential Privacy (BCDP), that enables feature-specific privacy quantification. This more nuanced approach complements LDP by adjusting privacy protection according to the sensitivity of each feature, enabling improved performance of downstream tasks without compromising privacy. We characterize the properties of BCDP and articulate its connections with standard non-Bayesian privacy frameworks. We further apply our BCDP framework to the problems of private mean estimation and ordinary least-squares regression. The BCDP-based approach obtains improved accuracy compared to a purely LDP-based approach, without compromising on privacy.

Optimal Design for Reward Modeling in RLHF

Reinforcement Learning from Human Feedback (RLHF) has become a popular approach to align language models (LMs) with human preferences. This method involves collecting a large dataset of human pairwise preferences across various text generations and using it to infer (implicitly or explicitly) a reward model. Numerous methods have been proposed to learn the reward model and align a LM with it. However, the costly process of collecting human preferences has received little attention and could benefit from theoretical insights. This paper addresses this issue and aims to formalize the reward training model in RLHF. We frame the selection of an effective dataset as a simple regret minimization task, using a linear contextual dueling bandit method. Given the potentially large number of arms, this approach is more coherent than the best-arm identification setting. We then propose an offline framework for solving this problem. Under appropriate assumptions - linearity of the reward model in the embedding space, and boundedness of the reward parameter - we derive bounds on the simple regret. Finally, we provide a lower bound that matches our upper bound up to constant and logarithmic terms. To our knowledge, this is the first theoretical contribution in this area to provide an offline approach as well as worst-case guarantees.

Active-Dormant Attention Heads: Mechanistically Demystifying Extreme-Token Phenomena in LLMs

Practitioners have consistently observed three puzzling phenomena in transformer-based large language models (LLMs): attention sinks, value-state drains, and residual-state peaks, collectively referred to as extreme-token phenomena. These phenomena are characterized by certain so-called "sink tokens" receiving disproportionately high attention weights, exhibiting significantly smaller value states, and having much larger residual-state norms than those of other tokens. These extreme tokens give rise to various challenges in LLM inference, quantization, and interpretability. We elucidate the mechanisms behind extreme-token phenomena. First, we show that these phenomena arise in very simple architectures -- transformers with one to three layers -- trained on a toy model, the Bigram-Backcopy (BB) task. In this setting, we identify an active-dormant mechanism, where attention heads become sinks for specific input domains while remaining non-sinks for others. Our theoretical analysis of the training dynamics reveals that these phenomena are driven by a mutual reinforcement mechanism. Building on these insights, we propose strategies to mitigate extreme-token phenomena during pretraining, including replacing softmax with ReLU and Adam with SGD. Next, we extend our analysis to pretrained LLMs, including Llama and OLMo, showing that many attention heads exhibit a similar active-dormant mechanism as in the BB task, and that the mutual reinforcement mechanism also governs the emergence of extreme-token phenomena during LLM pretraining. Our results reveal that many of the static and dynamic properties of extreme-token phenomena predicted by the BB task align with observations in pretrained LLMs.

Safety vs. Performance: How Multi-Objective Learning Reduces Barriers to Market Entry

Emerging marketplaces for large language models and other large-scale machine learning (ML) models appear to exhibit market concentration, which has raised concerns about whether there are insurmountable barriers to entry in such markets. In this work, we study this issue from both an economic and an algorithmic point of view, focusing on a phenomenon that reduces barriers to entry. Specifically, an incumbent company risks reputational damage unless its model is sufficiently aligned with safety objectives, whereas a new company can more easily avoid reputational damage. To study this issue formally, we define a multi-objective high-dimensional regression framework that captures reputational damage, and we characterize the number of data points that a new company needs to enter the market. Our results demonstrate how multi-objective considerations can fundamentally reduce barriers to entry -- the required number of data points can be significantly smaller than the incumbent company's dataset size. En route to proving these results, we develop scaling laws for high-dimensional linear regression in multi-objective environments, showing that the scaling rate becomes slower when the dataset size is large, which could be of independent interest.

Defection-Free Collaboration between Competitors in a Learning System

We study collaborative learning systems in which the participants are competitors who will defect from the system if they lose revenue by collaborating. As such, we frame the system as a duopoly of competitive firms who are each engaged in training machine-learning models and selling their predictions to a market of consumers. We first examine a fully collaborative scheme in which both firms share their models with each other and show that this leads to a market collapse with the revenues of both firms going to zero. We next show that one-sided collaboration in which only the firm with the lower-quality model shares improves the revenue of both firms. Finally, we propose a more equitable, *defection-free* scheme in which both firms share with each other while losing no revenue, and we show that our algorithm converges to the Nash bargaining solution.