Online Learning with Expert Advice

We study adversarial online learning with expert recommendations in a totally ordered action space, where the experts' recommendations may change and be reordered arbitrarily. Our focus is two-point feedback: on each round the learner plays two current recommendations, observes both, and pays their average. We study two problems. In the separated monotone problem, the loss is the sum of a hidden nondecreasing component and a hidden nonincreasing component, and a query reveals the two components separately. In the BV problem, the losses are bounded and have bounded variation (BV) along the order, and a query reveals only the loss. In the separated monotone problem, two paid observations achieve $$O\bigl(\log(|\cF|T)\sqrt{T\log|\cF|}\bigr)$$ regret against the best of $|\cF|$ experts, within a logarithmic factor of the classical full-information bound. For the BV problem, we obtain $\wtO(\sqrt T+|\cF|)$ regret. We give an application in dynamic pricing with expert advice, showing that two-option menus matches the full information rate up to $\log$ factors. The main technical ingredient is a marginal-preserving endpoint flow estimator. Two observations reveal a loss increment along the order; random paths supply telescoping identities that turn this local information into unbiased loss estimates for all current recommendations. A reversible pair sampler preserves the learner's intended marginal at both played actions. The estimator's second-moment and stability bounds support online updates for both feedback models.

We study simultaneous competitive and regret guarantees for TCP acknowledgment. Each acknowledgment serves all pending packets, and the objective is to minimize the total acknowledgment and waiting costs. It is well known that no deterministic online algorithm achieves a competitive ratio better than $2$. Given $k$ randomized online experts and $0<\varepsilon<1$, we construct a randomized online algorithm whose realized total cost is always at most $(2+\varepsilon)$ times the offline optimal cost $\mathrm{OPT}$ and whose expected regret against the expert with the smallest realized cost is $O(\sqrt{\mathrm{OPT}\log(k+1)}+\log(1/\varepsilon))$. The algorithm requires no advance knowledge of the number of packets or $\mathrm{OPT}$. Our algorithm controls switching costs between experts and uses the gap between the permitted competitive bound and its current cost to determine when to learn. We also prove lower bounds showing that, for $k\ge 2$, this regret bound is asymptotically optimal in $\mathrm{OPT}$, $k$, and $\varepsilon$, and that $\varepsilon>0$ is necessary for sublinear regret.

As large language models are increasingly deployed for multi-step reasoning, verifying the correctness of their outputs has become essential for maintaining reliability at scale. Verifying the correctness of large language model outputs is often done by querying a costly ground-truth oracle, which is impractical to invoke at every step in an online setting. Prior work addresses this by querying a single weak verifier on every step, and using its score to decide whether the costly strong verifier needs to be queried as well, reserving strong verification for only a small fraction of the steps. However, a single fixed weak verifier may not perform consistently well as the subject matter or difficulty of incoming queries changes over time, and committing to one in advance risks either overly costly or inaccurate verification. We introduce OMVV (Online Multi-Verifier Verification), an algorithm that maintains a pool of $K$ candidate weak verifiers with differing cost and verification performance, and adaptively routes each round's decision to a verifier selected via an online score combiner and an exponential-weights routing policy. OMVV provides a distribution-free, finite-time guarantee on false-accept and false-reject rates across the full pool of verifiers, and further achieves sublinear regret against the best fixed verifier in hindsight under a combined cost and consistency objective. Experiments on reasoning dataset benchmarks show that OMVV achieves higher accuracy at lower verification cost than any single fixed verifier, across a range of operating budgets.

Gaussian Processes (GPs) are widely used as surrogates for black-box functions in sequential decision-making problems such as Bayesian optimization (BO), level set estimation (LSE), and Bayesian active learning (BAL). GP performance critically depends on kernels, and standard kernels can lead to suboptimal decisions under misspecification. To address this, we introduce HACK GPs (Hedge Adaptive Cumulative Kernels), a method that views kernel selection as an online learning with expert advice problem. HACK treats each candidate kernel as a GP "expert" and updates a distribution over experts online using AdaHedge, based on a loss received as a proxy for their ability to fit the function and align with the task objective. We provide two variants of HACK: (i) Mixture of Gaussians (MoG) and (ii) categorical sampling. We establish general guarantees showing that, under a loss-gap condition, the weight concentrates on the best kernel and the resulting acquisition function is close to that of the best expert. Empirically, we observe robust performance across BO, LSE, and BAL compared to standard kernels such as Squared Exponential and Matern-5/2, as well as simple ensemble baselines.

Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate payment for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative payment defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact rather than upper bounds, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time, not as slack in worst-case analyses. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.

Recent advances in machine learning have produced probabilistic weather forecasting models comparable to state-of-the-art numerical weather predictors. But no model consistently dominates spatio-temporally, and relative performance is highly context-dependent. This motivates adaptive methods for combining multiple forecasts to obtain improvements and robustness. While combined forecasts have been proposed in the literature, these are achieved either through supervised learning or through prediction with expert advice methods. We introduce AdaWeather, an adaptive framework that combines many probabilistic forecasts using both machine learning as well as mixture of experts to arrive at a unified improved probabilistic forecast. While traditional expert methods develop the regret bounds with respect to the best single expert in hindsight, we extend the algorithm and analysis to show our method has logarithmic regret compared to the best static mixture of experts in hindsight. Empirically, we focus on forecasting temperature, and observe improvements over existing methods.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

Static regret to a single expert is often the wrong target for strictly online prediction under non-stationarity, where the best expert may switch repeatedly over time. We study Policy-Controlled Generalized Share (PCGS), a general strictly online framework in which the generalized-share recursion is fixed while the post-loss update controls are allowed to vary adaptively. Its principal instantiation in this paper is PCGS-TF, which uses a causal Transformer as an update controller: after round t finishes and the loss vector is observed, the Transformer outputs the controls that map w_t to w_{t+1} without altering the already committed decision w_t. Under admissible post-loss update controls, we obtain a pathwise weighted regret guarantee for general time-varying learning rates, and a standard dynamic-regret guarantee against any expert path with at most S switches under the constant-learning-rate specialization. Empirically, on a controlled synthetic suite with exact dynamic-programming switching-oracle evaluation, PCGS-TF attains the lowest mean dynamic regret in all seven non-stationary families, with its advantage increasing for larger expert pools. On a reproduced household-electricity benchmark, PCGS-TF also achieves the lowest normalized dynamic regret for S = 5, 10, and 20.