To produce accurate predictions, language models (LMs) must balance between generalization and memorization. Yet, little is known about the mechanism by which transformer LMs employ their memorization capacity. When does a model decide to output a memorized phrase, and how is this phrase then retrieved from memory? In this work, we offer the first methodological framework for probing and characterizing recall of memorized sequences in transformer LMs. First, we lay out criteria for detecting model inputs that trigger memory recall, and propose idioms as inputs that typically fulfill these criteria. Next, we construct a dataset of English idioms and use it to compare model behavior on memorized vs. non-memorized inputs. Specifically, we analyze the internal prediction construction process by interpreting the model's hidden representations as a gradual refinement of the output probability distribution. We find that across different model sizes and architectures, memorized predictions are a two-step process: early layers promote the predicted token to the top of the output distribution, and upper layers increase model confidence. This suggests that memorized information is stored and retrieved in the early layers of the network. Last, we demonstrate the utility of our methodology beyond idioms in memorized factual statements. Overall, our work makes a first step towards understanding memory recall, and provides a methodological basis for future studies of transformer memorization.

Radial basis function is derived from Cover's Theorem of Separability of Patterns: "A complex pattern classification problem, cast in a high dimensional space non linearly is more likely to be linearly separable then in a low dimensional space provided that the space is not densely populated". Hidden neuron activations in RBFN are computed using an exponential of a distance measure (Euclidean distance) between input vectors and prototype vectors associated with hidden neurons. RBFN was originally introduced for the purpose of interpolation of data points on a finite training set T {Xk, dk } k 1 Q . Then solving the exact interpolation problem, we have to search for a map "f" such that f(x) dk k 1 to Q. There are 3 functions for RBF used in ML: 1. Gaussian functions 2. Multiquadric 3. Inverse multiquadric When deciding whether to use an RBF network or an MLP, there are several factors to consider.

This manuscript investigates the one-pass stochastic gradient descent (SGD) dynamics of a two-layer neural network trained on Gaussian data and labels generated by a similar, though not necessarily identical, target function. We rigorously analyse the limiting dynamics via a deterministic and low-dimensional description in terms of the sufficient statistics for the population risk. Our unifying analysis bridges different regimes of interest, such as the classical gradient-flow regime of vanishing learning rate, the high-dimensional regime of large input dimension, and the overparameterised "mean-field" regime of large network width, covering as well the intermediate regimes where the limiting dynamics is determined by the interplay between these behaviours. In particular, in the high-dimensional limit, the infinite-width dynamics is found to remain close to a low-dimensional subspace spanned by the target principal directions. Our results therefore provide a unifying picture of the limiting SGD dynamics with synthetic data.

Unsupervised/self-supervised graph neural networks (GNN) are vulnerable to inherent randomness in the input graph data which greatly affects the performance of the model in downstream tasks. In this paper, we alleviate the interference of graph randomness and learn appropriate representations of nodes without label information. To this end, we propose USER, an unsupervised robust version of graph neural networks that is based on structural entropy. We analyze the property of intrinsic connectivity and define intrinsic connectivity graph. We also identify the rank of the adjacency matrix as a crucial factor in revealing a graph that provides the same embeddings as the intrinsic connectivity graph. We then introduce structural entropy in the objective function to capture such a graph. Extensive experiments conducted on clustering and link prediction tasks under random-noises and meta-attack over three datasets show USER outperforms benchmarks and is robust to heavier randomness.

Wind energy resource assessment typically requires numerical models, but such models are too computationally intensive to consider multi-year timescales. Increasingly, unsupervised machine learning techniques are used to identify a small number of representative weather patterns to simulate long-term behaviour. Here we develop a novel wind energy workflow that for the first time combines weather patterns derived from unsupervised clustering techniques with numerical weather prediction models (here WRF) to obtain efficient and accurate long-term predictions of power and downstream wakes from an entire wind farm. We use ERA5 reanalysis data clustering not only on low altitude pressure but also, for the first time, on the more relevant variable of wind velocity. We also compare the use of large-scale and local-scale domains for clustering. A WRF simulation is run at each of the cluster centres and the results are aggregated using a novel post-processing technique. By applying our workflow to two different regions, we show that our long-term predictions agree with those from a year of WRF simulations but require less than 2% of the computational time. The most accurate results are obtained when clustering on wind velocity. Moreover, clustering over the Europe-wide domain is sufficient for predicting wind farm power output, but downstream wake predictions benefit from the use of smaller domains. Finally, we show that these downstream wakes can affect the local weather patterns. Our approach facilitates multi-year predictions of power output and downstream farm wakes, by providing a fast, accurate and flexible methodology that is applicable to any global region. Moreover, these accurate long-term predictions of downstream wakes provide the first tool to help mitigate the effects of wind energy loss downstream of wind farms, since they can be used to determine optimum wind farm locations.

Gradient descent methods have long been the de facto standard for training deep neural networks. Millions of training samples are fed into models with billions of parameters, which are slowly updated over hundreds of epochs. Recently, it's been shown that large, randomly initialized neural networks contain subnetworks that perform as well as fully trained models. This insight offers a promising avenue for training future neural networks by simply pruning weights from large, random models. However, this problem is combinatorically hard and classical algorithms are not efficient at finding the best subnetwork. In this paper, we explore how quantum algorithms could be formulated and applied to this neuron selection problem. We introduce several methods for local quantum neuron selection that reduce the entanglement complexity that large scale neuron selection would require, making this problem more tractable for current quantum hardware.

Marketers employ various online advertising channels to reach customers, and they are particularly interested in attribution for measuring the degree to which individual touchpoints contribute to an eventual conversion. The availability of individual customer-level path-to-purchase data and the increasing number of online marketing channels and types of touchpoints bring new challenges to this fundamental problem. We aim to tackle the attribution problem with finer granularity by conducting attribution at the path level. To this end, we develop a novel graphical point process framework to study the direct conversion effects and the full relational structure among numerous types of touchpoints simultaneously. Utilizing the temporal point process of conversion and the graphical structure, we further propose graphical attribution methods to allocate proper path-level conversion credit, called the attribution score, to individual touchpoints or corresponding channels for each customer's path to purchase. Our proposed attribution methods consider the attribution score as the removal effect, and we use the rigorous probabilistic definition to derive two types of removal effects. We examine the performance of our proposed methods in extensive simulation studies and compare their performance with commonly used attribution models. We also demonstrate the performance of the proposed methods in a real-world attribution application.

A wide spectrum of design and decision problems, including parameter tuning, A/B testing and drug design, intrinsically are instances of black-box optimization. Bayesian optimization (BO) is a powerful tool that models and optimizes such expensive "black-box" functions. However, at the beginning of optimization, vanilla Bayesian optimization methods often suffer from slow convergence issue due to inaccurate modeling based on few trials. To address this issue, researchers in the BO community propose to incorporate the spirit of transfer learning to accelerate optimization process, which could borrow strength from the past tasks (source tasks) to accelerate the current optimization problem (target task). This survey paper first summarizes transfer learning methods for Bayesian optimization from four perspectives: initial points design, search space design, surrogate model, and acquisition function. Then it highlights its methodological aspects and technical details for each approach. Finally, it showcases a wide range of applications and proposes promising future directions.

In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via e.g., the test log predictive density, and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduced in \citet{bhattacharya_evidence_2020} which is theoretically appealing but practically intractable. Our proposal takes shape as a normalizing flow where the base distribution is a carefully-initialized generalized gamma. We conduct experiments comparing this to the canonical Gaussian base distribution and show improvements in terms of variational free energy and variational generalization error.

Deep neural networks have been successfully applied in various fields such as image analysis, natural language processing, protein structure prediction, etc.[40, 22, 35]. Since the number of parameters appeared in deep neural networks is often ten times or hundred times larger than the sample size of data, the successes of neural network methods have challenged the traditional bias variances trade-off principle, one of the primary doctrines in the classical statistical learning theories [61]. For example, many influential experiments [9, 67, 8, 48, 7] suggested that if one trains a neural network till it overfits the data, the resulting network can still generalize well. This observation, often referred to as the "benign overfitting phenomenon" [4, 53, 26, 45], actually reshaped the landscape of the studies in neural networks. For example, some researchers built giant neural networks in practice which can easily achieve nearly zero training error and possess the state-of-the-art performances [31, 50, 21]. Inspired by these experiments and observations, researchers proposed various new theories to explain why overfitted neural networks do generalize well on certain data [9, 43, 26, 47]. Several groups of statisticians tried to explain the generalization ability of neural networks from statistical decision theory with various carefully designed nonparametric regression frameworks. For example, assuming that the regression function belongs to a carefully designed sub-class of the Hölder continuous functions, [5] proved that there exists a neural network with sigmoid activation function achieving the corresponding minimax rate; [54] further established similar results for ReLU neural networks based on the approximation theory from [66]; [59] then extended these results to regression functions in Besov space and its variants.