Transformers have emerged as a powerful neural network architecture capable of tackling a wide range of learning tasks. In this work, we provide a theoretical analysis of their ability to automatically extract structure from data in an unsupervised setting. In particular, we demonstrate their suitability for clustering when the input data is generated from a Gaussian mixture model. To this end, we study a simplified two-head attention layer and define a population risk whose minimization with unlabeled data drives the head parameters to align with the true mixture centroids. This phenomenon highlights the ability of attention-based layers to capture underlying distributional structure. We further examine an attention layer with key, query, and value matrices fixed to the identity, and show that, even without any trainable parameters, it can perform in-context quantization, revealing the surprising capacity of transformer-based methods to adapt dynamically to input-specific distributions.

Many emerging applications--such as adversarial training, AI alignment, and robust optimization--can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions--spanning initialization, training dynamics, and network width--that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PL) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.

We study whether visual embedding models capture continuous, ordinal attributes along linear directions, which we term rank axes. We define a model as rankable for an attribute if projecting embeddings onto such an axis preserves the attribute's order. Across 7 popular encoders and 9 datasets with attributes like age, crowd count, head pose, aesthetics, and recency, we find that many embeddings are inherently rankable. Surprisingly, a small number of samples, or even just two extreme examples, often suffice to recover meaningful rank axes, without full-scale supervision. These findings open up new use cases for image ranking in vector databases and motivate further study into the structure and learning of rankable embeddings.

AI will lead to more need for workers rather than make people redundant, Amazon founder Jeff Bezos predicted during an appearance at a tech conference in Paris. Bezos pushed back against growing concerns that AI will replace large numbers of workers. Instead he argued that the tech will unlock new opportunities and increase demand for human labour. This is in contradiction to some other tech and political figures - including former UK prime minister Rishi Sunak, now an adviser to Microsoft and AI firm Anthropic, who recently said AI was having an impact on young people's job prospects . I know there's a lot of concern that many people have, including many smart people, that AI is going to make humans redundant and so on, Bezos said.

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph H = (V,E), we prove that the naive 2O(|V|)-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a 1-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of Hand a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters.

Consider the task of locating an unknown target point using approximate distance queries: in each round, a reconstructor selects a reference point and receives a noisy version of its distance to the target. This problem arises naturally in various contexts--ranging from localization in GPS and sensor networks to privacy-aware data access--and spans a wide variety of metric spaces. It is relevant from the perspective of both the reconstructor (seeking accurate recovery) and the responder (aiming to limit information disclosure, e.g., for privacy or security reasons). We study this reconstruction game through a learning-theoretic lens, focusing on the rate and limits of the best possible reconstruction error. Our first result provides a tight geometric characterization of the optimal error in terms of the Chebyshev radius, a classical concept from geometry. This characterization applies to all compact metric spaces (in fact, even to all totally bounded spaces) and yields explicit formulas for natural metric spaces. Our second result addresses the asymptotic behavior of reconstruction, distinguishing between pseudo-finite spaces--where the optimal error is attained after finitely many queries--and spaces where the approximation curve exhibits a nontrivial decay. We characterize pseudo-finiteness for convex Euclidean spaces.

Deep Neural Networks (DNN) have emerged as an effective approach to implementing challenging subproblems. They are increasingly being used as components in critical transportation, medical, and military systems. However, like human-written software, DNNs may have flaws that can lead to unsafe system performance. To confidently deploy DNNs in such systems, strong evidence is needed that they do not contain such flaws. This has led researchers to explore the adaptation and customization of software verification approaches to the problem of neural network verification (NNV). Many dozens of NNV tools have been developed in recent years and as a field these techniques have matured to the point where realistic networks can be analyzed to detect flaws and to prove conformance with specifications. NNV tools are highly-engineered and complex may harbor flaws that cause them to produce unsound results. We identify commonalities in algorithmic approaches taken by NNV tools to define a verifier independent proof format--activation pattern tree proofs (APTP)--and design an algorithm for checking those proofs that is proven correct and optimized to enable scalable checking. We demonstrate that existing verifiers can efficiently generate APTP proofs, and that an APTPcheckersignificantly outperforms prior work on a benchmark of 16 neural networks and 400 NNV problems, and that it is robust to variation in APTP proof structure arising from different NNV tools.

Recent work suggests that large language models (LLMs) can perform multi-hop reasoning implicitly--producing correct answers without explicitly verbalizing intermediate steps--but the underlying mechanisms remain poorly understood. In this paper, we study how such implicit reasoning emerges by training transformers from scratch in a controlled symbolic environment. Our analysis reveals a threestage developmental trajectory: early memorization, followed by in-distribution generalization, and eventually cross-distribution generalization. We find that training with atomic triples is not necessary but accelerates learning, and that second-hop generalization relies on query-level exposure to specific compositional structures. To interpret these behaviors, we introduce two diagnostic tools: cross-query semantic patching, which identifies semantically reusable intermediate representations, and a cosine-based representational lens, which reveals that successful reasoning correlates with the cosine-base clustering in hidden space. This clustering phenomenon in turn provides a coherent explanation for the behavioral dynamics observed across training, linking representational structure to reasoning capability. These findings provide new insights into the interpretability of implicit multi-hop reasoning in LLMs, helping to clarify how complex reasoning processes unfold internally and offering pathways to enhance the transparency of such models.

Bandit Convex Optimization is a fundamental class of sequential decision-making problems, where the learner selects actions from a continuous domain and observes a loss (but not its gradient) at only one point per round. We study this problem in non-stationary environments, and aim to minimize the regret under three standard measures of non-stationarity: the number of switches S in the comparator sequence, the total variation! of the loss functions, and the path-length P of the comparator sequence. We propose a polynomial-time algorithm, Tilted Exponentially Weighted Average with Sleeping Experts (TEWA-SE), which adapts the sleeping experts framework from online convex optimization to the bandit setting. For strongly convex losses, we prove that TEWA-SE is minimax-optimal with respect to known S and! by establishing matching upper and lower bounds. By equipping TEWA-SE with the Bandit-over-Bandit framework, we extend our analysis to environments with unknown non-stationarity measures. For general convex losses, we introduce a second algorithm, clipped Exploration by Optimization (cExO), based on exponential weights over a discretized action space. While not polynomial-time computable, this method achieves minimax-optimal regret with respect to known S and!, and improves on the best existing bounds with respect to P.

Neural networks are often described as black boxes, reflecting the significant challenge of understanding their internal workings and interactions. We propose a different perspective that challenges the prevailing view: rather than being inscrutable, neural networks exhibit patterns in their raw population activity that mirror regularities in the training data. We refer to this as the Reflection Hypothesis and provide evidence for this phenomenon in both simple recurrent neural networks (RNNs) and complex large language models (LLMs). Building on this insight, we propose to leverage our cognitive tendency of chunking to segment high-dimensional neural population dynamics into interpretable units that reflect underlying concepts. We propose three methods to extract recurring chunks on a neural population level, complementing each other based on label availability and neural data dimensionality.