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.

Theoretical Foundations of Conformal Prediction

This book is about conformal prediction and related inferential techniques that build on permutation tests and exchangeability. These techniques are useful in a diverse array of tasks, including hypothesis testing and providing uncertainty quantification guarantees for machine learning systems. Much of the current interest in conformal prediction is due to its ability to integrate into complex machine learning workflows, solving the problem of forming prediction sets without any assumptions on the form of the data generating distribution. Since contemporary machine learning algorithms have generally proven difficult to analyze directly, conformal prediction's main appeal is its ability to provide formal, finite-sample guarantees when paired with such methods. The goal of this book is to teach the reader about the fundamental technical arguments that arise when researching conformal prediction and related questions in distribution-free inference. Many of these proof strategies, especially the more recent ones, are scattered among research papers, making it difficult for researchers to understand where to look, which results are important, and how exactly the proofs work. We hope to bridge this gap by curating what we believe to be some of the most important results in the literature and presenting their proofs in a unified language, with illustrations, and with an eye towards pedagogy.

How to Evaluate Reward Models for RLHF

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 crowdsourced 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 open-source for public use and further development. Our code and evaluations can be found at https://github.com/lmarena/PPE .

Data-Adaptive Tradeoffs among Multiple Risks in Distribution-Free Prediction

Decision-making pipelines are generally characterized by tradeoffs among various risk functions. It is often desirable to manage such tradeoffs in a data-adaptive manner. As we demonstrate, if this is done naively, state-of-the art uncertainty quantification methods can lead to significant violations of putative risk guarantees. To address this issue, we develop methods that permit valid control of risk when threshold and tradeoff parameters are chosen adaptively. Our methodology supports monotone and nearly-monotone risks, but otherwise makes no distributional assumptions. To illustrate the benefits of our approach, we carry out numerical experiments on synthetic data and the large-scale vision dataset MS-COCO.

AutoEval Done Right: Using Synthetic Data for Model Evaluation

The evaluation of machine learning models using human-labeled validation data can be expensive and time-consuming. AI-labeled synthetic data can be used to decrease the number of human annotations required for this purpose in a process called autoevaluation. We suggest efficient and statistically principled algorithms for this purpose that improve sample efficiency while remaining unbiased. These algorithms increase the effective human-labeled sample size by up to 50% on experiments with GPT-4.

Wavefront Randomization Improves Deconvolution

The performance of an imaging system is limited by optical aberrations, which cause blurriness in the resulting image. Digital correction techniques, such as deconvolution, have limited ability to correct the blur, since some spatial frequencies in the scene are not measured adequately (i.e., 'zeros' of the system transfer function). We prove that the addition of a random mask to an imaging system removes its dependence on aberrations, reducing the likelihood of zeros in the transfer function and consequently decreasing the sensitivity to noise during deconvolution. In simulation, we show that this strategy improves image quality over a range of aberration types, aberration strengths, and signal-to-noise ratios.

Online conformal prediction with decaying step sizes

We introduce a method for online conformal prediction with decaying step sizes. Like previous methods, ours possesses a retrospective guarantee of coverage for arbitrary sequences. However, unlike previous methods, we can simultaneously estimate a population quantile when it exists. Our theory and experiments indicate substantially improved practical properties: in particular, when the distribution is stable, the coverage is close to the desired level for every time point, not just on average over the observed sequence.

PPI++: Efficient Prediction-Powered Inference

We present PPI++: a computationally lightweight methodology for estimation and inference based on a small labeled dataset and a typically much larger dataset of machine-learning predictions. The methods automatically adapt to the quality of available predictions, yielding easy-to-compute confidence sets -- for parameters of any dimensionality -- that always improve on classical intervals using only the labeled data. PPI++ builds on prediction-powered inference (PPI), which targets the same problem setting, improving its computational and statistical efficiency. Real and synthetic experiments demonstrate the benefits of the proposed adaptations.

Conformal Decision Theory: Safe Autonomous Decisions from Imperfect Predictions

We introduce Conformal Decision Theory, a framework for producing safe autonomous decisions despite imperfect machine learning predictions. Examples of such decisions are ubiquitous, from robot planning algorithms that rely on pedestrian predictions, to calibrating autonomous manufacturing to exhibit high throughput and low error, to the choice of trusting a nominal policy versus switching to a safe backup policy at run-time. The decisions produced by our algorithms are safe in the sense that they come with provable statistical guarantees of having low risk without any assumptions on the world model whatsoever; the observations need not be I.I.D. and can even be adversarial. The theory extends results from conformal prediction to calibrate decisions directly, without requiring the construction of prediction sets. Experiments demonstrate the utility of our approach in robot motion planning around humans, automated stock trading, and robot manufacturing.

Conformal PID Control for Time Series Prediction

We study the problem of uncertainty quantification for time series prediction, with the goal of providing easy-to-use algorithms with formal guarantees. The algorithms we present build upon ideas from conformal prediction and control theory, are able to prospectively model conformal scores in an online setting, and adapt to the presence of systematic errors due to seasonality, trends, and general distribution shifts. Our theory both simplifies and strengthens existing analyses in online conformal prediction. Experiments on 4-week-ahead forecasting of statewide COVID-19 death counts in the U.S. show an improvement in coverage over the ensemble forecaster used in official CDC communications. We also run experiments on predicting electricity demand, market returns, and temperature using autoregressive, Theta, Prophet, and Transformer models. We provide an extendable codebase for testing our methods and for the integration of new algorithms, data sets, and forecasting rules.