Adapting to a priori unknown noise level is a very important but challenging
problem in sequential decision-making as efficient exploration typically
requires knowledge of the noise level, which is often loosely specified. We
report significant progress in addressing this issue in linear bandits in two
respects. First, we propose a novel confidence set that is `semi-adaptive' to
the unknown sub-Gaussian parameter $\sigma_*^2$ in the sense that the
(normalized) confidence width scales with $\sqrt{d\sigma_*^2 + \sigma_0^2}$
where $d$ is the dimension and $\sigma_0^2$ is the specified sub-Gaussian
parameter (known) that can be much larger than $\sigma_*^2$. This is a
significant improvement over $\sqrt{d\sigma_0^2}$ of the standard confidence
set of Abbasi-Yadkori et al. (2011), especially when $d$ is large. We show that
this leads to an improved regret bound in linear bandits. Second, for bounded
rewards, we propose a novel variance-adaptive confidence set that has a much
improved numerical performance upon prior art. We then apply this confidence
set to develop, as we claim, the first practical variance-adaptive linear
bandit algorithm via an optimistic approach, which is enabled by our novel
regret analysis technique. Both of our confidence sets rely critically on
`regret equality' from online learning. Our empirical evaluation in Bayesian
optimization tasks shows that our algorithms demonstrate better or comparable
performance compared to existing methods.

在线学习中通过与现有方法相比展示出更好或相当的性能，解决了适应未知噪声水平的问题，并提出了一种新的置信度集合与方差自适应方法。

线性赌博机的噪声自适应置信区间及其在贝叶斯优化中的应用

Noise-Adaptive Confidence Sets for Linear Bandits and Application to  Bayesian Optimization

When one observes a sequence of variables $(x_1, y_1), \ldots, (x_n, y_n)$,
Conformal Prediction (CP) is a methodology that allows to estimate a confidence
set for $y_{n+1}$ given $x_{n+1}$ by merely assuming that the distribution of
the data is exchangeable. CP sets have guaranteed coverage for any finite
population size $n$. While appealing, the computation of such a set turns out
to be infeasible in general, e.g. when the unknown variable $y_{n+1}$ is
continuous. The bottleneck is that it is based on a procedure that readjusts a
prediction model on data where we replace the unknown target by all its
possible values in order to select the most probable one. This requires
computing an infinite number of models, which often makes it intractable. In
this paper, we combine CP techniques with classical algorithmic stability
bounds to derive a prediction set computable with a single model fit. We
demonstrate that our proposed confidence set does not lose any coverage
guarantees while avoiding the need for data splitting as currently done in the
literature. We provide some numerical experiments to illustrate the tightness
of our estimation when the sample size is sufficiently large, on both synthetic
and real datasets.

本论文将 CP 技术与经典算法稳定性界限相结合，提出了一种置信区间集合，可用单一模型拟合计算，并证明了该方法能够保证精度，避免了数据分割的需求，成功解决了传统方法无法处理连续未知变量 y_n+1 的瓶颈问题。

稳定的共形预测集

Stable Conformal Prediction Sets

Modern deep learning based classifiers show very high accuracy on test data
but this does not provide sufficient guarantees for safe deployment, especially
in high-stake AI applications such as medical diagnosis. Usually, predictions
are obtained without a reliable uncertainty estimate or a formal guarantee.
Conformal prediction (CP) addresses these issues by using the classifier's
predictions, e.g., its probability estimates, to predict confidence sets
containing the true class with a user-specified probability. However, using CP
as a separate processing step after training prevents the underlying model from
adapting to the prediction of confidence sets. Thus, this paper explores
strategies to differentiate through CP during training with the goal of
training model with the conformal wrapper end-to-end. In our approach,
conformal training (ConfTr), we specifically "simulate" conformalization on
mini-batches during training. Compared to standard training, ConfTr reduces the
average confidence set size (inefficiency) of state-of-the-art CP methods
applied after training. Moreover, it allows to "shape" the confidence sets
predicted at test time, which is difficult for standard CP. On experiments with
several datasets, we show ConfTr can influence how inefficiency is distributed
across classes, or guide the composition of confidence sets in terms of the
included classes, while retaining the guarantees offered by CP.

本文探讨了在深度学习基础上进行置信区间预测的方法，并提出了一种基于置信区间训练的深度学习模型，该方法可有效地缩小置信边界，同时保证对医学诊断等高风险场景的准确性。