Understanding how semantic meaning is encoded in the representation spaces of large language models is a fundamental problem in interpretability. In this paper, we study the two foundational questions in this area. First, how are categorical concepts, such as {'mammal', 'bird', 'reptile', 'fish'}, represented? Second, how are hierarchical relations between concepts encoded? For example, how is the fact that 'dog' is a kind of 'mammal' encoded? We show how to extend the linear representation hypothesis to answer these questions. We find a remarkably simple structure: simple categorical concepts are represented as simplices, hierarchically related concepts are orthogonal in a sense we make precise, and (in consequence) complex concepts are represented as polytopes constructed from direct sums of simplices, reflecting the hierarchical structure. We validate these theoretical results on the Gemma large language model, estimating representations for 957 hierarchically related concepts using data from WordNet.

In anytime-valid sequential inference, it is known that any admissible inference procedure must be based on test martingales and their composite generalization, called e-processes, which are nonnegative processes whose expectation at any arbitrary stopping time is upper-bounded by one. An e-process quantifies the accumulated evidence against a composite null hypothesis over a sequence of outcomes. This paper studies methods for combining e-processes that are computed using different information sets, i.e., filtrations, for a null hypothesis. Even though e-processes constructed on the same filtration can be combined effortlessly (e.g., by averaging), e-processes constructed on different filtrations cannot be combined as easily because their validity in a coarser filtration does not translate to validity in a finer filtration. We discuss three concrete examples of such e-processes in the literature: exchangeability tests, independence tests, and tests for evaluating and comparing forecasts with lags. Our main result establishes that these e-processes can be lifted into any finer filtration using adjusters, which are functions that allow betting on the running maximum of the accumulated wealth (thereby insuring against the loss of evidence). We also develop randomized adjusters that can improve the power of the resulting sequential inference procedure.

Abstaining classifiers have the option to abstain from making predictions on inputs that they are unsure about. These classifiers are becoming increasingly popular in high-stakes decision-making problems, as they can withhold uncertain predictions to improve their reliability and safety. When evaluating black-box abstaining classifier(s), however, we lack a principled approach that accounts for what the classifier would have predicted on its abstentions. These missing predictions matter when they can eventually be utilized, either directly or as a backup option in a failure mode. In this paper, we introduce a novel approach and perspective to the problem of evaluating and comparing abstaining classifiers by treating abstentions as missing data. Our evaluation approach is centered around defining the counterfactual score of an abstaining classifier, defined as the expected performance of the classifier had it not been allowed to abstain. We specify the conditions under which the counterfactual score is identifiable: if the abstentions are stochastic, and if the evaluation data is independent of the training data (ensuring that the predictions are missing at random), then the score is identifiable. Note that, if abstentions are deterministic, then the score is unidentifiable because the classifier can perform arbitrarily poorly on its abstentions. Leveraging tools from observational causal inference, we then develop nonparametric and doubly robust methods to efficiently estimate this quantity under identification. Our approach is examined in both simulated and real data experiments.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

We consider two or more forecasters each making a sequence of predictions over time and tackle the problem of how to compare them -- either online or post-hoc. In fields ranging from meteorology to sports, forecasters make predictions on different events or quantities over time, and this work describes how to compare them in a statistically rigorous manner. Specifically, we design a nonasymptotic sequential inference procedure for estimating the time-varying difference in forecast quality when using a relatively large class of scoring rules (bounded scores with a linear equivalent). The resulting confidence intervals can be continuously monitored and yield statistically valid comparisons at arbitrary data-dependent stopping times ("anytime-valid"); this is enabled by adapting recent variance-adaptive confidence sequences (CS) to our setting. In the spirit of Shafer and Vovk's game-theoretic probability, the coverage guarantees for our CSs are also distribution-free, in the sense that they make no distributional assumptions whatsoever on the forecasts or outcomes. Additionally, in contrast to a recent preprint by Henzi and Ziegel, we show how to sequentially test a weak null hypothesis about whether one forecaster outperforms another on average over time, by designing different e-processes that quantify the evidence at any stopping time. We examine the validity of our methods over their fixed-time and asymptotic counterparts in synthetic experiments and demonstrate their effectiveness in real-data settings, including comparing probability forecasts on Major League Baseball (MLB) games and comparing statistical postprocessing methods for ensemble weather forecasts.

Invariant risk minimization (IRM; Arjovsky et al., 2019) is a recently proposed framework designed for learning predictors that are invariant to spurious correlations across different training environments. Because IRM does not assume that the test data is identically distributed as the training data, it can allow models to learn invariances that generalize well on unseen and out-of-distribution (OOD) samples. Yet, despite this theoretical justification, IRM has not been extensively tested across various settings. In an attempt to gain a better understanding of IRM, we empirically investigate several research questions using IRMv1, which is the first practical algorithm proposed in (Arjovsky et al., 2019) to approximately solve IRM. By extending the ColoredMNIST experiment from (Arjovsky et al., 2019) in multiple ways, we find that IRMv1 (i) performs better as the spurious correlation varies more widely between training environments, (ii) learns an approximately invariant predictor when the underlying relationship is approximately invariant, and (iii) can be extended to multiple environments, multiple outcomes, and different modalities (i.e., text). We hope that this work will shed light on the characteristics of IRM and help with applying IRM to real-world OOD generalization tasks.

Accurate prediction of drug-target interaction (DTI) is essential for in silico drug design. For the purpose, we propose a novel approach for predicting DTI using a GNN that directly incorporates the 3D structure of a protein-ligand complex. We also apply a distance-aware graph attention algorithm with gate augmentation to increase the performance of our model. As a result, our model shows better performance than docking and other deep learning methods for both virtual screening and pose prediction. In addition, our model can reproduce the natural population distribution of active molecules and inactive molecules.

Although deep convolutional networks have achieved improved performance in many natural language tasks, they have been treated as black boxes because they are difficult to interpret. Especially, little is known about how they represent language in their intermediate layers. In an attempt to understand the representations of deep convolutional networks trained on language tasks, we show that individual units are selectively responsive to specific morphemes, words, and phrases, rather than responding to arbitrary and uninterpretable patterns. In order to quantitatively analyze such an intriguing phenomenon, we propose a concept alignment method based on how units respond to the replicated text. We conduct analyses with different architectures on multiple datasets for classification and translation tasks and provide new insights into how deep models understand natural language.

Large bundles of myelinated axons, called white matter, anatomically connect disparate brain regions together and compose the structural core of the human connectome. We recently proposed a method of measuring the local integrity along the length of each white matter fascicle, termed the local connectome. If communication efficiency is fundamentally constrained by the integrity along the entire length of a white matter bundle, then variability in the functional dynamics of brain networks should be associated with variability in the local connectome. We test this prediction using two statistical approaches that are capable of handling the high dimensionality of data. First, by performing statistical inference on distance-based correlations, we show that similarity in the local connectome between individuals is significantly correlated with similarity in their patterns of functional connectivity. Second, by employing variable selection using sparse canonical correlation analysis and cross-validation, we show that segments of the local connectome are predictive of certain patterns of functional brain dynamics. These results are consistent with the hypothesis that structural variability along axon bundles constrains communication between disparate brain regions.