Online Conformal Prediction via Universal Portfolio Algorithms

Online conformal prediction (OCP) seeks prediction intervals that achieve long-run $1-α$ coverage for arbitrary (possibly adversarial) data streams, while remaining as informative as possible. Existing OCP methods often require manual learning-rate tuning to work well, and may also require algorithm-specific analyses. Here, we develop a general regret-to-coverage theory for interval-valued OCP based on the $(1-α)$-pinball loss. Our first contribution is to identify \emph{linearized regret} as a key notion, showing that controlling it implies coverage bounds for any online algorithm. This relies on a black-box reduction that depends only on the Fenchel conjugate of an upper bound on the linearized regret. Building on this theory, we propose UP-OCP, a parameter-free method for OCP, via a reduction to a two-asset portfolio selection problem, leveraging universal portfolio algorithms. We show strong finite-time bounds on the miscoverage of UP-OCP, even for polynomially growing predictions. Extensive experiments support that UP-OCP delivers consistently better size/coverage trade-offs than prior online conformal baselines.

💡 Research Summary

Online conformal prediction (OCP) aims to construct prediction intervals that, over an arbitrary (potentially adversarial) data stream, contain the true response with a prescribed long‑run probability 1 − α while keeping the intervals as narrow as possible. Existing OCP methods such as Adaptive Conformal Inference (ACI), Multi‑Valid Conformal Prediction (MVP), or aggregated expert ensembles all rely on a manually tuned learning‑rate or on problem‑specific analyses, which hampers their robustness to sudden distribution shifts and makes them difficult to deploy in practice.

The paper introduces a new theoretical bridge between online learning and OCP based on the linearized regret (LinRegret). For the (1 − α) pinball loss ℓ(b,S)=max{(1 − α)(S−b), α(b−S)} the sub‑gradient at the algorithm’s prediction b_t is g_t = 1{b_t ≥ S_t} − (1 − α). The cumulative mis‑coverage error is exactly the average of these gradients, MisCov_T = (1/T)∑_{t=1}^T g_t. LinRegret replaces the usual regret (difference of losses) with the sum of linearized terms g_t(b_t − u) for any comparator u. Because ℓ is convex, ordinary regret is always bounded by LinRegret, but the converse does not hold; LinRegret is a strictly stronger condition.

The central result (Theorem 3.1) shows that if an algorithm guarantees a bound F_T(u) on LinRegret for all u, then the Fenchel conjugate F_T^* can be used to bound the sum of gradients:
 −∑ g_t ∈ {z : F_T^*(z) ≤ (1 − α)∑ S_t}.
Thus, a tighter (smaller) LinRegret bound translates directly into a tighter bound on MisCov_T. This black‑box reduction depends only on the conjugate of the regret bound and does not require any further properties of the loss function.

Armed with this reduction, the authors seek an online algorithm that minimizes LinRegret as aggressively as possible. They observe that OCP can be reformulated as a two‑asset portfolio selection game: asset 1 corresponds to the algorithm’s current interval radius b_t, asset 2 to the optimal fixed radius u. The gain of each asset at round t is precisely the sub‑gradient g_t. Classical universal portfolio algorithms (e.g., Cover & Ordentlich, 2002) are known to achieve optimal log‑loss regret of order O(√T log T) against any adversarial sequence of returns, without any tuning parameters. By plugging this optimal regret bound into Theorem 3.1, they obtain a finite‑time mis‑coverage guarantee of order O(log T / T).

Crucially, the analysis allows the non‑conformity scores S_t to grow polynomially, S_t ≤ D t^q, a much weaker assumption than the bounded‑score condition used in prior work. The resulting bound on MisCov_T remains O(log (DT)/T) regardless of the growth exponent q, showing that the method is robust to non‑stationary environments where the scale of the data drifts over time.

The paper also revisits existing OCP algorithms (the KT method of Podkopaev et al., 2024, and Online Subgradient Descent) within the same framework, deriving comparable finite‑time guarantees without requiring bounded scores.

The proposed algorithm, UP‑OCP (Universal Portfolio OCP), implements the universal portfolio strategy on the two‑asset reduction. It is completely parameter‑free: no learning‑rate, no grid of thresholds, and no prior knowledge of the score range are needed. At each round it updates the portfolio weights based on observed sub‑gradients, and the resulting b_t is the weighted average of the two “assets”.

Empirical evaluation spans synthetic streams with abrupt distribution changes and several real‑world regression datasets (e.g., electricity demand, stock returns). UP‑OCP is compared against ACI, MVP, Aggregated‑ACI, Strongly‑Adaptive OCP, and the KT‑based method. Across all experiments, UP‑OCP achieves the smallest average interval width while maintaining empirical coverage extremely close to the target 1 − α. In scenarios with sudden shifts, methods that rely on a fixed learning‑rate either overshoot (producing overly wide intervals) or undershoot (causing severe under‑coverage), whereas UP‑OCP adapts instantly thanks to its parameter‑free nature.

The authors conclude by discussing limitations and future directions. The current theory is tied to the pinball loss; extending the framework to multi‑level conformal inference, structured outputs, or other proper scoring rules remains open. Moreover, while universal portfolio algorithms are theoretically optimal, their computational cost can be high in high‑dimensional asset spaces; developing efficient approximations for OCP with many candidate predictors is an important practical challenge.

In summary, the paper makes three major contributions: (1) it introduces linearized regret and a Fenchel‑conjugate reduction that turns any bound on this regret into a direct coverage guarantee; (2) it leverages optimal universal‑portfolio learning to obtain a parameter‑free OCP algorithm (UP‑OCP) with provably optimal finite‑time mis‑coverage rates even under polynomially growing scores; and (3) it validates the approach empirically, showing consistent superiority over the state‑of‑the‑art online conformal baselines. This work bridges a gap between online learning theory and conformal prediction, offering a robust, theoretically grounded, and practically appealing solution for real‑time uncertainty quantification.