We establish the universal approximation capability of single-layer, single-head self-and cross-attention mechanisms with minimal attached structures. Our key insight is to interpret single-head attention as an input domain-partition mechanism that assigns distinct values to subregions. This allows us to engineer the attention weights such that this assignment imitates the target function. Building on this, we prove that a single self-attention layer, preceded by sum-of-linear transformations, is capable of approximating any continuous function on a compact domain under the L -norm. Furthermore, we extend this construction to approximate any Lebesgue integrable function under Lp-norm for 1 p < . Lastly, we also extend our techniques and show that, for the first time, single-head cross-attention achieves the same universal approximation guarantees.

Time series foundation models (TSFMs) demonstrate impressive zero-shot performance for time series forecasting. However, an important yet underexplored challenge is how to effectively finetune TSFMs on specific downstream tasks. While naive finetuning can yield performance gains, we argue that it falls short of fully leveraging TSFMs' capabilities, often resulting in overfitting and suboptimal performance. Given the diverse temporal patterns across sampling scales and the inherent multi-scale forecasting capabilities of TSFMs, we adopt a causal perspective to analyze finetuning process, through which we highlight the critical importance of explicitly modeling multiple scales and reveal the shortcomings of naive approaches. Focusing on encoder-based TSFMs, we propose MultiScale FineTuning (MSFT), a simple yet general framework that explicitly integrates multi-scale modeling into the finetuning process. Experimental results on three different backbones (MOIRAI, MOMENT and UNITS) demonstrate that TSFMs finetuned with MSFT not only outperform naive and typical parameter efficient finetuning methods but also surpass state-of-the-art deep learning methods. Codes are available at https://github.com/zqiao11/MSFT.

We consider how to construct state abstractions compatible with a given set of abstract actions, to obtain a well-formed abstract Markov decision process (MDP). We show that the Bellman equation suggests that abstract states should represent distributions over states in the ground MDP; we characterize the conditions under which the resulting process is Markov and approximately model-preserving, derive an algorithm for constructing the abstract MDP, and apply it to visual chain and maze tasks. We generalize these results to the factored actions case, characterize the conditions that lead to factored abstract states, and apply the resulting algorithm to a visual grid and Montezuma's Revenge. These results provide a principled, powerful framework for learning neurosymbolic abstract Markov decision processes.

Artificial-intelligence systems are becoming ubiquitous in society, yet their predictions typically inherit biases with respect to protected attributes such as race, gender, or age. Classical fairness notions, most notably Statistical Parity (SP), demand that predictions be independent of the protected attributes, but are overly restrictive when these attributes influence mediating variables that are considered business necessities. Recent causal formulations relax SP by distinguishing allowed from not-allowed causal paths and by complementing SP with Predictive Parity (PP), requiring the predictor to replicate the legitimate influence of business-necessities. Existing path-based definitions are mainly practical when applied to categorical attributes. This paper introduces a new framework for fairness in structural causal models that is tailored to continuous protected attributes. We formalize SP and PP through path-specific partial derivatives, establish conditions under which these criteria coincide with prior causal definitions, and characterize when a fair predictor, one that satisfies SP along not-allowed paths while achieving PP along allowed paths, exists. Building on this theory, we propose a fair tuning algorithm that either constructs such a predictor or, when not possible, allows for a trade-off between SP and PP. We present experiments on simulated and real data to evaluate our proposal, compare it with previously proposed methods, and show that it performs better when PP is considered.

Figure 1: A summary of several outlier statistics recorded from ImageNet validation set on ViT. We use zero-based indexing for dimensions. BERTRecall from Figure 1 that all the outliers are only present in hidden dimensions #123, #180,4 #225, #308, #381, #526, #720 (with the majority of them in #180, #720). In Figures 9 and 10 we show more6 examples of the discovered self-attention patterns for attention heads #3 and #12 ( hidden dim #1807 and #720, respectively). We also show self-attention patterns in attention heads and layers which are8 not associated with the outliers in Figures 11 and 12, respectively.9

Organization: This supplementary material is presented in a format parallel to the main paper. The section numbers and titles are consistent with the main paper. But, here we also add one new section: Section 10 where we describe the societal impacts and possible negative impacts of the paper. Similarly, the Theorem numbers are consistent with the main paper, but we also have several additional theorems and lemmas which were not included in the main paper. GAN-type Objective for KLEstimation Let f be a discriminator, f: X IR. Let p(x) and q(x) be two probability density functions defined over the space X.

In this section, we describe the details of the network architectures used in Sec. 4 and 5. We mainly used 4 GPUs (NVIDIAV100; 16GB) for the experiments in Sec. 4 and 5 and it took about 4 hours per seed (in the case of 3M steps). Actually, we conducted exhaustive evaluations through the enormous experiments, and we hope our empirical observations and recommendations help the practitioners to explore the explosive configuration space. Adam Adam Learning rate (policy) 1e-4 5e-5 3e-4 3e-4 Learning rate (value) 1e-4 1e-2 3e-4 3e-4 Weight initialization Uniform Xavier Uniform Xavier Uniform Xavier Uniform Initial output scale (policy) 1.0 1e-4 1e-2 1e-2 Target update Hard - Soft (5e-3) Soft (5e-3) Clipped Double QFalse - True True Table 7: Details of each network architecture. We refer the original implementations of each algorithm which is available online [23, 14, 48, 27, 42].