machine learning tool developers

Abstract: We study the geometry and closure of spaces of functions realized by ReLU neural networks of fixed architecture. We first establish a general result about limits of continuous piecewise-linear (CPWL) functions: the minimum number of convex sets on which a function is affine is lower semicontinuous under pointwise convergence. The proof uses a connection to hypergraph coloring and a compactness principle, and yields quantitative bounds on the polyhedral complexity of continuous limits. We then characterize the functions realizable by shallow ReLU networks with vector-valued outputs, extending a known scalar-valued characterization. Using this and the geometry of limiting hyperplane arrangements, we prove that fixed-width shallow scalar ReLU realization spaces are closed in the weighted supremum norm on the whole input space. In contrast, we construct an explicit sequence of networks with architecture $(2,3,2\mid1)$ whose limit is not realizable by that architecture. Non-closedness holds both globally in weighted supremum norm and under uniform convergence on compact sets with nonempty interior, disproving a conjecture of Petersen, Raslan, and Voigtlaender. Finally, we show that non-closedness for a fixed scalar-output ReLU architecture necessarily involves a \emph{blow-up residue}, in which diverging output coefficients amplify an emerging linear dependence among hidden features. Together, these results provide geometric and functional-analytic tools for studying the boundaries and degenerations of ReLU realization spaces.

Abstract: Layer normalization (LN) and root mean square normalization (RMSNorm) depend on their inputs and remain nonlinear at inference. Inserting either next to an activation may therefore change the function a network represents. We study when shallow activation-normalized (AN) and preactivation-normalized (PN) networks can approximate a prescribed width-$N$ one-hidden-layer MLP. Viewing RMSNorm as input-dependent scaling, we use width augmentation to locally linearize normalization on compact feature sets. For AN networks, an unbounded activation range allows width $N+1$, while every continuous nonconstant activation admits width $N+r$ with error $O_f(r^{-1})$. For PN networks, affine parameters or, without them, degree-one positive homogeneity recover width $N+1$; general continuous activations admit width $N+r$ with error tending to zero. The corresponding LN constructions use centered coordinate pairs, yielding widths $N+2$ and $N+2r$, respectively. We prove that $N+1$ for RMSNorm and $N+2$ for LN are also the worst-case bound for approximations, and show that applicable universal-approximation theorems for ordinary MLPs transfer to these normalized architectures. Numerical experiments recover the predicted approximation rates for both normalizations across all five settings and illustrate the shifted-ReLU finite-width obstruction.

Abstract: Attention sinks are usually studied after they have formed. We ask whether a measurable hidden-state change appears beforehand and whether changing that geometry alters what happens later. To remove network depth as a shortcut, we compare prompts in the same model and at the same layer. When sink timing varies across prompts, simple spectral summaries of the hidden states add information beyond current attention, with the clearest prompt-specific signal in a short window about two to three layers before sink formation. The predictive strength varies across cohorts, so we treat the spectrum as a readout of a pre-sink state change rather than a universal warning statistic. The signal is broader than one measured extreme activation. We then modify only the singular values of the post-layer hidden states after current attention has already been computed. These interventions change later sink behavior in Qwen2.5, GLM-Z1, and GPT-OSS relative to norm-matched random spectral perturbations. In Qwen2.5, the immediate suppressive effect is accompanied by a reduction in next-layer query--key preference for position~0, and the learned spectral directions remain stable when estimated from substantially less training data. Together, the results show that attention sinks are preceded by a short spectral reorganization of hidden states, and directly perturbing this geometry can causally alter subsequent sink formation.

Abstract: Reinforcement learning (RL) algorithms frequently compare probability distributions, such as state visitation distributions induced by policies and experts, action distributions from learned policies and offline datasets, or transition distributions from learned models and environments. However, commonly used divergences may become ineffective when these distributions overlap weakly, which is frequently encountered in imitation learning, offline RL, and deployment under distribution shift. Optimal transport (OT) offers an alternative by measuring the cost of \emph{moving} probability mass from one distribution to another under a ground cost that encodes task geometry. This survey covers how OT is used inside RL objectives and algorithms. For each method, we identify: the role OT plays, the distributions compared, the OT formulation used, and the treatment of temporal structure. Beyond categorising existing methods, we discuss the motivations behind different OT choices, practical considerations such as cost design and computational challenges, and highlight open problems including scalable trajectory-level transport, principled handling of mass mismatch, and theoretical analysis for OT-regularised RL.