We present a brief proof about the local optimality of one-hot encodings in the decision-theoretic framework presented in Section 3.2. We seek to prove that, under assumptions of an identity reward matrix, tokens constrained to a unit hypercube, and gaussian additive noise, one-hot tokens are an optimally robust communication strategy. We only seek to prove local optimality, as one many trivially generate multiple, equally optimal tokens by, for example, flipping all bits. The following derivation uses Karush-Kuhn-Tucker (KKT) conditions, a generalization of Lagrange multipliers [17]. We maximize the function, subject to constraints. T>j Ti Ti + ||Tj||2 Ti # ~µi + ~λi = ~0 (13) (14) We seek to show that one-hot vectors are an optimum, so we now show that one-hot vectors indeed respect the constraints and set the derivatives to zero.

How can we make use of information parallelism in online decision making problems while efficiently balancing the exploration-exploitation trade-off? In this paper, we introduce a batch Thompson Sampling framework for two canonical online decision making problems, namely, stochastic multi-arm bandit and linear contextual bandit with finitely many arms. Over a time horizon T, our batch Thompson Sampling policy achieves the same (asymptotic) regret bound of a fully sequential one while carrying out only O(log T) batch queries. To achieve this exponential reduction, i.e., reducing the number of interactions from T to O(log T), our batch policy dynamically determines the duration of each batch in order to balance the exploration-exploitation trade-off. We also demonstrate experimentally that dynamic batch allocation dramatically outperforms natural baselines such as static batch allocations.

Increasingly, copious amounts of consumer data are used to learn high-revenue mechanisms via machine learning. Existing research on mechanism design via machine learning assumes that there is a distribution over the buyers' values for the items for sale and that the learning algorithm's input is a training set sampled from this distribution. This setup makes the strong assumption that no noise is introduced during data collection. In order to help place mechanism design via machine learning on firm foundations, we investigate the extent to which this learning process is robust to noise. Optimizing revenue using noisy data is challenging because revenue functions are extremely volatile: an infinitesimal change in the buyers' values can cause a steep drop in revenue. Nonetheless, we provide guarantees when arbitrarily correlated noise is added to the training set; we only require that the noise has bounded magnitude or is sub-Gaussian. We conclude with an application of our guarantees to multi-task mechanism design, where there are multiple distributions over buyers' values and the goal is to learn a high-revenue mechanism per distribution. To our knowledge, we are the first to study mechanism design via machine learning with noisy data as well as multi-task mechanism design.

Increasingly, copious amounts of consumer data are used to learn high-revenue mechanisms via machine learning. Existing research on mechanism design via machine learning assumes that there is a distribution over the buyers' values for the items for sale and that the learning algorithm's input is a training set sampled from this distribution. This setup makes the strong assumption that no noise is introduced during data collection. In order to help place mechanism design via machine learning on firm foundations, we investigate the extent to which this learning process is robust to noise. Optimizing revenue using noisy data is challenging because revenue functions are extremely volatile: an infinitesimal change in the buyers' values can cause a steep drop in revenue. Nonetheless, we provide guarantees when arbitrarily correlated noise is added to the training set; we only require that the noise has bounded magnitude or is sub-Gaussian. We conclude with an application of our guarantees to multi-task mechanism design, where there are multiple distributions over buyers' values and the goal is to learn a high-revenue mechanism per distribution. To our knowledge, we are the first to study mechanism design via machine learning with noisy data as well as multi-task mechanism design.

Since the attention mechanism is rarely used within the GP literature, we provide a brief review of the topic in this section. Below we follow the description of attention as given by Vaswani et al. [8], including extensions to self-attention and multi-head self-attention. The dot-product attention mechanism [8] takes as input a set of queries, keys and values. The queries and keys have dimension Dz and the values have dimension Dv which may differ from Dz. The operation of dot-product attention then generates weights from the queries and keys which are used to produce a linear mapping of the input values.

Kernel selection plays a central role in determining the performance of Gaussian Process (GP) models, as the chosen kernel determines both the inductive biases and prior support of functions under the GP prior. This work addresses the challenge of constructing custom kernel functions for high-dimensional GP regression models. Drawing inspiration from recent progress in deep learning, we introduce a novel approach named KITT: Kernel Identification Through Transformers. KITT exploits a transformer-based architecture to generate kernel recommendations in under 0.1 seconds, which is several orders of magnitude faster than conventional kernel search algorithms. We train our model using synthetic data generated from priors over a vocabulary of known kernels. By exploiting the nature of the selfattention mechanism, KITT is able to process datasets with inputs of arbitrary dimension. We demonstrate that kernels chosen by KITT yield strong performance over a diverse collection of regression benchmarks.