Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability $p \in (0,1/2)$. As our main result, we show that it is sufficient to use $(1+o(1)) \frac{n\log \frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $m\triangleq \min\{k,n-k+1\}$ and $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Conversely, we show that any algorithm achieving an error probability of $\delta = o(1)$ necessitates at least $(1-o(1))\frac{(n-m)\log\frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation. The upper and lower bounds are tight when $m=o(n)$, and are within a multiplicative factor of $\frac{n}{n-m}$ when $m=\Theta(n)$. In particular, when $k=n/2$, the $\mathsf{TH}_k$ function corresponds to the $\mathsf{MAJORITY}$ function, in which case the upper and lower bounds are tight up to a multiplicative factor of two. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.

We introduce a new benchmark for reward models that quantifies their ability to produce strong language models through RLHF (Reinforcement Learning from Human Feedback). The gold-standard approach is to run a full RLHF training pipeline and directly probe downstream LLM performance. However, this process is prohibitively expensive. To address this, we build a predictive model of downstream LLM performance by evaluating the reward model on proxy tasks. These proxy tasks consist of a large-scale human preference and a verifiable correctness preference dataset, in which we measure 12 metrics across 12 domains. To investigate which reward model metrics are most correlated to gold-standard RLHF outcomes, we launch an end-to-end RLHF experiment on a large-scale crowd-sourced human preference platform to view real reward model downstream performance as ground truth. Ultimately, we compile our data and findings into Preference Proxy Evaluations (PPE), the first reward model benchmark explicitly linked to post-RLHF real-world human preference performance, which we opensource for public use and further development at https://github.com/lmarena/PPE.

Language model calibration refers to the alignment between the confidence of the model and the actual performance of its responses. While previous studies point out the overconfidence phenomenon in Large Language Models (LLMs) and show that LLMs trained with Reinforcement Learning from Human Feedback (RLHF) are overconfident with a more sharpened output probability, in this study, we reveal that RLHF tends to lead models to express verbalized overconfidence in their own responses. We investigate the underlying cause of this overconfidence and demonstrate that reward models used for Proximal Policy Optimization (PPO) exhibit inherent biases towards high-confidence scores regardless of the actual quality of responses. Building upon this insight, we propose two PPO variants: PPO-M: $\underline{PPO}$ with Calibrated Reward $\underline{M}$odeling and PPO-C: $\underline{PPO}$ with Calibrated Reward $\underline{C}$alculation. PPO-M integrates explicit confidence scores in reward model training, which calibrates reward models to better capture the alignment between response quality and verbalized confidence. PPO-C adjusts the reward score during PPO based on the difference between the current reward and the moving average of past rewards. Both PPO-M and PPO-C can be seamlessly integrated into the current PPO pipeline and do not require additional golden labels. We evaluate our methods on both $\texttt{Llama3-8B}$ and $\texttt{Mistral-7B}$ across six diverse datasets including multiple-choice and open-ended generation. Experiment results demonstrate that both of our methods can reduce calibration error and maintain performance comparable to standard PPO. We further show that they do not compromise model capabilities in open-ended conversation settings.

The rapid evolution of Large Language Models (LLMs) has outpaced the development of model evaluation, highlighting the need for continuous curation of new, challenging benchmarks. However, manual curation of high-quality, human-aligned benchmarks is expensive and time-consuming. To address this, we introduce BenchBuilder, an automated pipeline that leverages LLMs to curate high-quality, open-ended prompts from large, crowd-sourced datasets, enabling continuous benchmark updates without human in the loop. We apply BenchBuilder to datasets such as Chatbot Arena and WildChat-1M, extracting challenging prompts and utilizing LLM-as-a-Judge for automatic model evaluation. To validate benchmark quality, we propose new metrics to measure a benchmark’s alignment with human preferences and ability to separate models. We release Arena-Hard-Auto, a benchmark consisting 500 challenging prompts curated by BenchBuilder. Arena-Hard-Auto provides 3x higher separation of model performances compared to MT-Bench and achieves 98.6% correlation with human preference rankings, all at a cost of $20. Our work sets a new framework for the scalable curation of automated benchmarks from extensive data.

Reinforcement learning (RL) theory has largely focused on proving minimax sample complexity bounds. These require strategic exploration algorithms that use relatively limited function classes for representing the policy or value function. Our goal is to explain why deep RL algorithms often perform well in practice, despite using random exploration and much more expressive function classes like neural networks. Our work arrives at an explanation by showing that many stochastic MDPs can be solved by performing only a few steps of value iteration on the random policy’s Q function and then acting greedily. When this is true, we find that it is possible to separate the exploration and learning components of RL, making it much easier to analyze. We introduce a new RL algorithm, SQIRL, that iteratively learns a near-optimal policy by exploring randomly to collect rollouts and then performing a limited number of steps of fitted-Q iteration over those roll- outs. We find that any regression algorithm that satisfies basic in-distribution generalization properties can be used in SQIRL to efficiently solve common MDPs. This can explain why deep RL works with complex function approximators like neural networks, since it is empirically established that neural networks generalize well in-distribution. Furthermore, SQIRL explains why random exploration works well in practice, since we show many environments can be solved by effectively estimating the random policy’s Q-function and then applying zero or a few steps of value iteration. We leverage SQIRL to derive instance-dependent sample complexity bounds for RL that are exponential only in an “effective horizon” of lookahead—which is typically much smaller than the full horizon—and on the complexity of the class used for function approximation. Empirically, we also find that SQIRL performance strongly correlates with PPO and DQN performance in a variety of stochastic environments, supporting that our theoretical analysis is predictive of practical performance. Our code and data are available at https://github.com/cassidylaidlaw/effective-horizon.

We characterize the statistical efficiency of knowledge transfer through $n$ samples from a teacher to a probabilistic student classifier with input space $\mathcal{S}$ over labels $\mathcal{A}$. We show that privileged information at three progressive levels accelerates the transfer. At the first level, only samples with hard labels are known, via which the maximum likelihood estimator attains the minimax rate $\sqrt{{|\mathcal{S}||\mathcal{A}|}/{n}}$. The second level has the teacher probabilities of sampled labels available in addition, which turns out to boost the convergence rate lower bound to ${{|\mathcal{S}||\mathcal{A}|}/{n}}$. However, under this second data acquisition protocol, minimizing a naive adaptation of the cross-entropy loss results in an asymptotically biased student. We overcome this limitation and achieve the fundamental limit by using a novel empirical variant of the squared error logit loss. The third level further equips the student with the soft labels (complete logits) on $\mathcal{A}$ given every sampled input, thereby provably enables the student to enjoy a rate ${|\mathcal{S}|}/{n}$ free of $|\mathcal{A}|$. We find any Kullback-Leibler divergence minimizer to be optimal in the last case. Numerical simulations distinguish the four learners and corroborate our theory.

Large Language Models (LLMs) have unlocked new capabilities and applications; however, evaluating the alignment with human preferences still poses significant challenges. To address this issue, we introduce Chatbot Arena, an open platform for evaluating LLMs based on human preferences. Our methodology employs a pairwise comparison approach and leverages input from a diverse user base through crowdsourcing. The platform has been operational for several months, amassing over 240K votes. This paper describes the platform, analyzes the data we have collected so far, and explains the tried-and-true statistical methods we are using for efficient and accurate evaluation and ranking of models. We confirm that the crowdsourced questions are sufficiently diverse and discriminating and that the crowd-sourced human votes are in good agreement with those of expert raters. These analyses collectively establish a robust foundation for the credibility of Chatbot Arena. Because of its unique value and openness, Chatbot Arena has emerged as one of the most referenced LLM leaderboards, widely cited by leading LLM developers and companies. The platform is publicly available at https://chat.lmsys.org.

Reinforcement Learning from Human Feedback (RLHF) is a pivotal technique that aligns language models closely with human-centric values. The initial phase of RLHF involves learning human values using a reward model from ranking data. It is observed that the performance of the reward model degrades after one epoch of training, and optimizing too much against the learned reward model eventually hinders the true objective. This paper analyzes potential reasons behind the issues, and designs improved reward learning algorithm termed 'Iterative Data Smoothing' (IDS). The core idea is that during each training epoch, we not only update the model with the data, but also update the date using the model, replacing hard labels with soft labels. Our empirical findings highlight the superior performance of this approach over the traditional methods.

Safe Reinforcement Learning (RL) aims to find a policy that achieves high rewards while satisfying cost constraints. When learning from scratch, safe RL agents tend to be overly conservative, which impedes exploration and restrains the overall performance. In many realistic tasks, e.g. autonomous driving, large-scale expert demonstration data are available. We argue that extracting expert policy from offline data to guide online exploration is a promising solution to mitigate the conserveness issue. Large-capacity models, e.g. decision transformers (DT), have been proven to be competent in offline policy learning. However, data collected in realworld scenarios rarely contain dangerous cases (e.g., collisions), which makes it prohibitive for the policies to learn safety concepts. Besides, these bulk policy networks cannot meet the computation speed requirements at inference time on real-world tasks such as autonomous driving. To this end, we propose Guided Online Distillation (GOLD), an offline-to-online safe RL framework. GOLD distills an offline DT policy into a lightweight policy network through guided online safe RL training, which outperforms both the offline DT policy and online safe RL algorithms. Experiments in both benchmark safe RL tasks and real-world driving tasks based on the Waymo Open Motion Dataset (WOMD) [1] demonstrate that GOLD can successfully distill lightweight policies and solve decision-making problems in challenging safety-critical scenarios.

We propose Byzantine-robust federated learning protocols with nearly optimal statistical rates based on recent progress in high dimensional robust statistics. In contrast to prior work, our proposed protocols improve the dimension dependence and achieve a near-optimal statistical rate for strongly convex losses. We also provide statistical lower bound for the problem. For experiments, we benchmark against competing protocols and show the empirical superiority of the proposed protocols.

Reinforcement learning (RL) provides a theoretical framework for continuously improving an agent’s behavior via trial and error. However, efficiently learning policies from scratch can be very difficult, particularly for tasks that present exploration challenges. In such settings, it might be desirable to initialize RL with an existing policy, offline data, or demonstrations. However, naively performing such initialization in RL often works poorly, especially for value-based methods. In this paper, we present a meta algorithm that can use offline data, demonstrations, or a pre-existing policy to initialize an RL policy, and is compatible with any RL approach. In particular, we propose Jump-Start Reinforcement Learning (JSRL), an algorithm that employs two policies to solve tasks: a guide-policy, and an exploration-policy. By using the guide-policy to form a curriculum of starting states for the exploration-policy, we are able to efficiently improve performance on a set of simulated robotic tasks. We show via experiments that it is able to significantly outperform existing imitation and reinforcement learning algorithms, particularly in the small-data regime. In addition, we provide an upper bound on the sample complexity of JSRL and show that with the help of a guide-policy, one can improve the sample complexity for non-optimism exploration methods from exponential in horizon to polynomial.

We study the problem of online learning in a two-player decentralized cooperative Stackelberg game. In each round, the leader first takes an action, followed by the follower who takes their action after observing the leader’s move. The goal of the leader is to learn to minimize the cumulative regret based on the history of interactions. Differing from the traditional formulation of repeated Stackelberg games, we assume the follower is omniscient, with full knowledge of the true reward, and that they always best-respond to the leader’s actions. We analyze the sample complexity of regret minimization in this repeated Stackelberg game. We show that depending on the reward structure, the existence of the omniscient follower may change the sample complexity drastically, from constant to exponential, even for linear cooperative Stackelberg games. This poses unique challenges for the learning process of the leader and the subsequent regret analysis.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). We show that when the underlying true reward is linear, under both Bradley-Terry-Luce (BTL) model (pairwise comparison) and Plackett-Luce (PL) model ($K$-wise comparison), MLE converges under certain semi-norm for the family of linear reward. On the other hand, when training a policy based on the learned reward model, we show that MLE fails while a pessimistic MLE provides policies with good performance under certain coverage assumption. We also show that under the PL model, both the true MLE and a different MLE which splits the $K$-wise comparison into pairwise comparisons converge, while the true MLE is asymptotically more efficient. Our results validate the empirical success of the existing RLHF algorithms, and provide new insights for algorithm design. Our analysis can also be applied for the problem of online RLHF and inverse reinforcement learning.

In this work, we study the problem of sorting n elements with pairwise comparisons under the presence of observation noise. We consider variable-length algorithms with a random number of queries M, and attempt to characterize the noisy sorting capacity defined as the maximal ratio $\frac{{n\log n}}{{{\text{E}}[M]}}$ such that the ordering can be correctly estimated with a vanishing error probability. This can be viewed as a generalization of the framework introduced in [1] to allow variable-length algorithms. We provide upper and lower bounds for the noisy sorting capacity. The proposed algorithm attaining the lower bound is based on the insertion sort algorithm for the sorting problem in the noiseless case and the variable-length version of the Burnashev–Zigangirov algorithm for coding over channels with feedback. Moreover, we also derive an upper bound on the maximal ratio that can be achieved by noisy sorting algorithms that are based on insertion sort.

We revisit the problem of computing with noisy information considered in Feige et al. [1], which includes computing the OR function from noisy queries, and computing the MAX, SEARCH, and SORT functions from noisy pairwise comparisons. For K given elements, the goal is to correctly recover the desired function with probability at least 1 – δ when the outcome of each query is flipped with probability p. We consider both the adaptive sampling setting where each query can be adaptively designed based on past outcomes, and the non-adaptive sampling setting where the query cannot depend on past outcomes. The prior work provides tight bounds on the worst-case query complexity in terms of the dependence on K. However, the upper and lower bounds do not match in terms of the dependence on δ and p. We improve the lower bounds for all the four functions under both adaptive and non-adaptive query models. Most of our lower bounds match the upper bounds up to constant factors when either p or δ is bounded away from 0, while the ratio between the best prior upper and lower bounds go to infinity when p → 0 or p → 1/2. On the other hand, we also provide matching upper and lower bounds for the number of queries in expectation, improving both the upper and lower bounds for variable-length query model.

Large Language Models (LLMs) and other large foundation models have achieved impressive results, but their size exacerbates existing resource consumption and latency challenges. In particular, the large-scale deployment of these models is hindered by the significant resource requirements during inference. In this paper, we study two approaches for mitigating these challenges: employing a cache to store previous queries and learning a model selector to choose from an ensemble of models for query processing. Theoretically, we provide an optimal algorithm for jointly optimizing both approaches to reduce the inference cost in both offline and online tabular settings. By combining a caching algorithm, namely Greedy Dual Size with Frequency (GDSF) or Least Expected Cost (LEC), with a model selector, we achieve optimal rates in both offline and online settings. Empirically, simulations show that our caching and model selection algorithm greatly improves over the baselines, with up to $50\times$ improvement over the baseline when the ratio between the maximum cost and minimum cost is $100$. Experiments on real datasets show a $4.3\times$ improvement in FLOPs over the baseline when the ratio for FLOPs is $10$, and a $1.8\times$ improvement in latency when the ratio for average latency is $1.85$.

Self-training is a well-established technique in semi-supervised learning, which leverages unlabeled data by generating pseudo-labels and incorporating them with a limited labeled dataset for training. The effectiveness of self-training heavily relies on the accuracy of these pseudo-labels. In this paper, we introduce doubly-robust self-training, an innovative semi-supervised algorithm that provably balances between two extremes. When pseudo-labels are entirely incorrect, our method reduces to a training process solely using labeled data. Conversely, when pseudo-labels are completely accurate, our method transforms into a training process utilizing all pseudo-labeled data and labeled data, thus increasing the effective sample size. Through empirical evaluations on both the ImageNet dataset for image classification and the nuScenes autonomous driving dataset for 3D object detection, we demonstrate the superiority of the doubly-robust loss over the self-training baseline.

Contract theory studies the interactions between a principal and an agent when the two parties transact in the presence of private information [Bolton and Dewatripont, 2004, Faure-Grimaud et al., 2001, Salanié, 2005]. The principal would like to achieve her desired outcomes by hiring agents to work for her. The agent wishes to make money by working for the principal. They develop agreements in the form of a contract, which specifies how much the principal would pay under the different possible outcomes of the agent's work.

We provide a general framework for designing Generative Adversarial Networks (GANs) to solve high-dimensional robust statistics problems, which aim at estimating unknown parameter of the true distribution given adversarially corrupted samples. Prior work [1], [2] focus on the problem of robust mean and covariance estimation when the true distribution lies in the family of Gaussian distributions or elliptical distributions, and analyze depth or scoring rule based GAN losses for the problem. Our work extend these to robust mean estimation, second-moment estimation, and robust linear regression when the true distribution only has bounded Orlicz norms, which includes the broad family of sub-Gaussian, sub-exponential and bounded moment distributions. We also provide a different set of sufficient conditions for the GAN loss to work: we only require its induced distance function to be a cumulative density function of some light-tailed distribution, which is easily satisfied by neural networks with sigmoid activation. In terms of techniques, our proposed GAN losses can be viewed as a smoothed and generalized Kolmogorov-Smirnov distance, which overcomes the computational intractability of the original Kolmogorov-Smirnov distance used in the weaken the distance approach in [3].

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger—relative to the oracle estimator equipped with the knowledge of the behavior policy— by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings.

Offline reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main methods are used: imitation learning which is suitable for expert datasets, and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown. To bridge this gap, we present a new offline RL framework, called single-policy concentrability, that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. Under this new framework, we ask: can one develop an algorithm that achieves a minimax optimal rate adaptive to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in both contextual bandits and RL, LCB achieves a fast convergence rate for nearly-expert datasets, analogous to the one achieved by imitation learning, contrary to the slow rate achieved in offline RL. In contextual bandits, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main methods are used: imitation learning which is suitable for expert datasets, and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation of the behavior policy from the expert policy alone. Under this new framework, we ask: can one develop an algorithm that achieves a minimax optimal rate adaptive to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in both contextual bandits and RL, LCB achieves a faster rate of $1/N$ for nearly-expert datasets compared to the usual rate of $1/\sqrt{N}$ in offline RL, where $N$ is the batch dataset sample size. In contextual bandits with at least two contexts, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

We analyze the performance of the Tukey median estimator under total variation (TV) distance corruptions. Previous results show that under Huber’s additive corruption model, the breakdown point is 1/3 for high-dimensional halfspace- symmetric distributions. We show that under TV corruptions, the breakdown point reduces to 1/4 for the same set of distributions. We also show that a certain projection algorithm can attain the optimal breakdown point of 1/2. Both the Tukey median estimator and the projection algorithm achieve sample complexity linear in dimension.

Generative Adversarial Networks (GANs) are a thriving unsupervised machine learning technique that has led to significant advances in various fields such as computer vision, natural language processing, among others. However, GANs are known to be difficult to train and usually suffer from mode collapse and the discriminator winning problem. To interpret the empirical observations of GANs and design better ones, we deconstruct the study of GANs into three components and make the following contributions. Formulation: we propose a perturbation view of the population target of GANs. Building on this interpretation, we show that GANs can be connected to the robust statistics framework, and propose a novel GAN architecture, termed as Cascade GANs, to provably recover meaningful low-dimensional generator approximations when the real distribution is high-dimensional and corrupted by outliers. Generalization: given a population target of GANs, we design a systematic principle, projection under admissible distance, to design GANs to meet the population requirement using only finite samples. We implement our principle in three cases to achieve polynomial and sometimes near-optimal sample complexities: (1) learning an arbitrary generator under an arbitrary pseudonorm; (2) learning a Gaussian location family under total variation distance, where we utilize our principle to provide a new proof for the near-optimality of the Tukey median viewed as GANs; (3) learning a low-dimensional Gaussian approximation of a high-dimensional arbitrary distribution under Wasserstein distance. We demonstrate a fundamental trade-off in the approximation error and statistical error in GANs, and demonstrate how to apply our principle in practice with only empirical samples to predict how many samples would be sufficient for GANs in order not to suffer from the discriminator winning problem. Optimization: we demonstrate alternating gradient descent is provably not locally asymptotically stable in optimizing the GAN formulation of PCA. We found that the minimax duality gap being non-zero might be one of the causes, and propose a new GAN architecture whose duality gap is zero, where the value of the game is equal to the previous minimax value (not the maximin value). We prove the new GAN architecture is globally asymptotically stable in solving PCA under alternating gradient descent.

Deep learning has a wide application in the area of natural language processing and image processing due to its strong ability of generalization. In this paper, we propose a novel neural network structure for jointly optimizing the transmitter and receiver in communication physical layer under fading channels. We build up a convolutional autoencoder to simultaneously conduct the role of modulation, equalization, and demodulation. The proposed system is able to design different mapping scheme from input bit sequences of arbitrary length to constellation symbols according to different channel environments. The simulation results show that the performance of neural network-based system is superior to traditional modulation and equalization methods in terms of time complexity and bit error rate under fading channels. The proposed system can also be combined with other coding techniques to further improve the performance. Furthermore, the proposed system network is more robust to channel variation than traditional communication methods.