The vocabulary Vconstructed by Algorithm 1 exhibits the following advantageous properties. Prior to the proof, we first present a clear observation of the created vocabulary V: Proposition A.2. Given any F,F V, for any their instances arising on an arbitrary molecule during the extraction process, either they are not spatially intersected F F =, or they contain each other: F F or F F. Now we prove each claim in the above theorem. We prove it by contradiction. If it is the former case, then Fi1 should be firstly extracted and then merged with other fragments to yield Fi2 which means i1 < i2, conflicting with the assumption.

In this work, we address risk-averse Bayes-adaptive reinforcement learning. We pose the problem of optimising the conditional value at risk (CVaR) of the total return in Bayes-adaptive Markov decision processes (MDPs). We show that a policy optimising CVaR in this setting is risk-averse to both the epistemic uncertainty due to the prior distribution over MDPs, and the aleatoric uncertainty due to the inherent stochasticity of MDPs. We reformulate the problem as a two-player stochastic game and propose an approximate algorithm based on Monte Carlo tree search and Bayesian optimisation. Our experiments demonstrate that our approach significantly outperforms baseline approaches for this problem.

During model optimization, the expected calibration error tends to overfit earlier than classification accuracy, indicating distinct optimization objectives for classification error and calibration error. To ensure consistent optimization of both model accuracy and model calibration, we propose a novel method incorporating a probability-dependent gradient decay coefficient into loss function. This coefficient exhibits a strong correlation with the overall confidence level.

Numerical results demonstrate significant regret reductions by our method, in comparison with several baselines in a range of multi-armed and linear bandit problems.

Finally,tomaketrainingmorestable,we average the loss over the sequence of predictions for each question to weigh the questions evenly. Is = [0.5, 0.55, ..., 0.95] num_intervals = len(Is) def low_containment_mask(lowers, uppers, labels, Is): # lowers, uppers: Predicted lower and upper bounds of intervals # Is: Target confidence levels # Returns: A list of boolean values indicating which confidence level # has containment ratio below the target level within batch contained = (lowers <= labels) * (labels <= uppers) ratio_contained = contained.mean(dim=0) In total, there are nearly 10,000 questions. Gray text indicates the number of questions after augmenting true/false questions with theirnegations,aprocedureweusetobalancethe dataset. Animportanttaskfor numerical forecasting is outputting calibrated uncertainty estimates.

Theorem 2. Suppose that the predictive distribution Q has the sufficient ability to approximate the true unknown distribution P, given data is i.i.d. Lm(P,Q) = 0 if and only if P = Q when F is a unit ball in a universal RKHS [13]. Becausetheconfidencelevelp2 p1 is exactly equal to the proportion of samples {y1,,yn} covered by the two-sided prediction interval. B.1 Baselines MC-Dropout (MCD) [12]: A variant of standard dropout, named as Monte-Carlo Dropout. Heteroscedastic Neural Network (HNN) [17]: In this approach, similar to a heteroscedastic regression, the network has two outputs in the last layer, corresponding to the predicted mean and variance for each input xi.