Neural Information Processing Systems http://nips.cc/

We propose Partition Trees, a tree-based framework for conditional density estimation over general outcome spaces, supporting both continuous and categorical variables within a unified formulation. Our approach models conditional distributions as piecewise-constant densities on data adaptive partitions and learns trees by directly minimizing conditional negative log-likelihood. This yields a scalable, nonparametric alternative to existing probabilistic trees that does not make parametric assumptions about the target distribution. We further introduce Partition Forests, an ensemble extension obtained by averaging conditional densities. Empirically, we demonstrate improved probabilistic prediction over CART-style trees and competitive or superior performance compared to state-of-the-art probabilistic tree methods and Random Forests, along with robustness to redundant features and heteroscedastic noise.

Neural Information Processing Systems http://nips.cc/

We examine the intrinsic (within the attention head) and extrinsic (amongst the attention heads) structure of the self-attention mechanism in transformers. Theoretical evidence for invariance of the self-attention mechanism to softmax activation is obtained by appealing to paradifferential calculus, (and is supported by computational examples), which relies on the intrinsic organization of the attention heads. Furthermore, we use an existing methodology for hierarchical organization of tensors to examine network structure by constructing hierarchal partition trees with respect to the query, key, and head axes of network 3-tensors. Such an organization is consequential since it allows one to profitably execute common signal processing tasks on a geometry where the organized network 3-tensors exhibit regularity. We exemplify this qualitatively, by visualizing the hierarchical organization of the tree comprised of attention heads and the diffusion map embeddings, and quantitatively by investigating network sparsity with the expansion coefficients of individual attention heads and the entire network with respect to the bi and tri-haar bases (respectively) on the space of queries, keys, and heads of the network. To showcase the utility of our theoretical and methodological findings, we provide computational examples using vision and language transformers. The ramifications of these findings are two-fold: (1) a subsequent step in interpretability analysis is theoretically admitted, and can be exploited empirically for downstream interpretability tasks (2) one can use the network 3-tensor organization for empirical network applications such as model pruning (by virtue of network sparsity) and network architecture comparison.

Many modern methods for prediction leverage nearest neighbor search to find past training examples most similar to a test example, an idea that dates back in text to at least the 11th century and has stood the test of time. This monograph aims to explain the success of these methods, both in theory, for which we cover foundational nonasymptotic statistical guarantees on nearest-neighbor-based regression and classification, and in practice, for which we gather prominent methods for approximate nearest neighbor search that have been essential to scaling prediction systems reliant on nearest neighbor analysis to handle massive datasets. Furthermore, we discuss connections to learning distances for use with nearest neighbor methods, including how random decision trees and ensemble methods learn nearest neighbor structure, as well as recent developments in crowdsourcing and graphons. In terms of theory, our focus is on nonasymptotic statistical guarantees, which we state in the form of how many training data and what algorithm parameters ensure that a nearest neighbor prediction method achieves a user-specified error tolerance. We begin with the most general of such results for nearest neighbor and related kernel regression and classification in general metric spaces. In such settings in which we assume very little structure, what enables successful prediction is smoothness in the function being estimated for regression, and a low probability of landing near the decision boundary for classification. In practice, these conditions could be difficult to verify for a real dataset. We then cover recent guarantees on nearest neighbor prediction in the three case studies of time series forecasting, recommending products to people over time, and delineating human organs in medical images by looking at image patches. In these case studies, clustering structure enables successful prediction.

We initiate a study of a query-driven approach to designing partition trees for range-searching problems. Our model assumes that a data structure is to be built for an unknown query distribution that we can access through a sampling oracle, and must be selected such that it optimizes a meaningful performance parameter on expectation. Our first contribution is to show that a near-linear sample of queries allows the construction of a partition tree with a near-optimal expected number of nodes visited during querying. We enhance this approach by treating node processing as a classification problem, leveraging fast classifiers like shallow neural networks to obtain experimentally efficient query times. Our second contribution is to develop partition trees using sparse geometric separators. Our preprocessing algorithm, based on a sample of queries, builds a balanced tree with nodes associated with separators that minimize query stabs on expectation; this yields both fast processing of each node and a small number of visited nodes, significantly reducing query time.

Tree-based models for probability distributions are usually specified using a predetermined, data-independent collection of candidate recursive partitions of the sample space. To characterize an unknown target density in detail over the entire sample space, candidate partitions must have the capacity to expand deeply into all areas of the sample space with potential non-zero sampling probability. Such an expansive system of partitions often incurs prohibitive computational costs and makes inference prone to overfitting, especially in regions with little probability mass. Existing models typically make a compromise and rely on relatively shallow trees. This hampers one of the most desirable features of trees, their ability to characterize local features, and results in reduced statistical efficiency. Traditional wisdom suggests that this compromise is inevitable to ensure coherent likelihood-based reasoning, as a data-dependent partition system that allows deeper expansion only in regions with more observations would induce double dipping of the data and thus lead to inconsistent inference. We propose a simple strategy to restore coherency while allowing the candidate partitions to be data-dependent, using Cox's partial likelihood. This strategy parametrizes the tree-based sampling model according to the allocation of probability mass based on the observed data, and yet under appropriate specification, the resulting inference remains valid. Our partial likelihood approach is broadly applicable to existing likelihood-based methods and in particular to Bayesian inference on tree-based models. We give examples in density estimation in which the partial likelihood is endowed with existing priors on tree-based models and compare with the standard, full-likelihood approach. The results show substantial gains in estimation accuracy and computational efficiency from using the partial likelihood.

From organizing recorded videos and meetings into chapters, to breaking down large inputs in order to fit them into the context window of commoditized Large Language Models (LLMs), topic segmentation of large transcripts emerges as a task of increasing significance. Still, accurate segmentation presents many challenges, including (a) the noisy nature of the Automatic Speech Recognition (ASR) software typically used to obtain the transcripts, (b) the lack of diverse labeled data and (c) the difficulty in pin-pointing the ground-truth number of segments. In this work we present TreeSeg, an approach that combines off-the-shelf embedding models with divisive clustering, to generate hierarchical, structured segmentations of transcripts in the form of binary trees. Our approach is robust to noise and can handle large transcripts efficiently. We evaluate TreeSeg on the ICSI and AMI corpora, demonstrating that it outperforms all baselines. Finally, we introduce TinyRec, a small-scale corpus of manually annotated transcripts, obtained from self-recorded video sessions.

We introduce the Forget-me-not Process, an efficient, non-parametric metaalgorithm for online probabilistic sequence prediction for piecewise stationary, repeating sources. Our method works by taking a Bayesian approach to partitioning a stream of data into postulated task-specific segments, while simultaneously building a model for each task. We provide regret guarantees with respect to piecewise stationary data sources under the logarithmic loss, and validate the method empirically across a range of sequence prediction and task identification problems.