OpenAI is facing another wrongful death lawsuit . Leila Turner-Scott and Angus Scott filed a lawsuit against the company, alleging that it designed and distributed a defective product that led to the death of their son Sam Nelson from an accidental overdose. Specifically, they're alleging that Sam died following the exact medical advice GPT-4o had provided and approved. In the lawsuit, the plaintiffs described how Sam, a 19-year-old junior at the University of California, Merced, started using ChatGPT in 2023 when he was in high school to help with homework and to troubleshoot computer problems. Sam then started asking the chatbot about safe drug use, but ChatGPT initially refused to answer his question, telling him that it couldn't assist him and warning him that taking drugs can have serious consequences for his health and well-being.

Elon Musk Had'Hair-Raising' Idea of Passing OpenAI Onto His Kids, Sam Altman Says Musk's lawyers questioned Altman over allegations of deception and his network of financial investments, but the OpenAI CEO painted a picture of Musk as obsessed with controlling the company. Sam Altman took to the witness stand to defend his reputation in the trial on Tuesday, as Elon Musk's lawyers peppered the OpenAI CEO with hours of questions regarding his alleged history of deceptive behavior . The cross examination was a much needed win for Musk, who has so far struggled to make a convincing case. Tuesday's testimony included several heated exchanges in which the OpenAI CEO had to respond to allegations from former colleagues suggesting he's untrustworthy . Highlighting this evidence is not only important for Musk winning over a jury, but also for beating OpenAI in the court of public opinion.

Batch normalization (BN) is central to modern deep networks, but its effect on the realized function during training remains less understood than its optimization benefits. We study training-time BN in continuous piecewise-affine (CPA) networks through the geometry of switching hyperplanes and the induced affine-region partition. Conditioned on a mini-batch, we show that BN defines for each neuron a reference hyperplane through the batch centroid, and that breakpoint-switching hyperplanes are parallel translates whose offsets are expressed in batch-standardized coordinates and are independent of the raw bias. This yields an exact criterion for when a switching hyperplane intersects a local $\ell_\infty$ window and motivates a local region-density functional based on exact affine-region counts. Under explicit sufficient conditions, we show that BN increases expected local partition refinement in ReLU and more general piecewise-affine networks, and that this mechanism transfers locally through depth inside parent affine regions where the upstream representation map is an affine embedding. These results provide a function-level geometric account of training-time BN as a batch-conditional recentering mechanism near the data.

Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This problem has traditionally been addressed in machine learning by directly learning the conditional distribution of a fixed joint distribution. This paper introduces a novel perspective: we propose to solve the conditioning problem by identifying a single operator that maps any joint density to its conditional, thus amortizing over joint-conditional pairs. We establish that the conditioning operator can be approximated to arbitrary accuracy by neural operators. Our proof relies on new results establishing continuity of the conditioning operator over suitable classes of densities. Finally, we learn the conditioning map for a class of Gaussian mixtures using neural operators, illustrating the promise of our framework. This work provides the theoretical underpinnings for general-purpose, amortized methods for probabilistic conditioning, such as foundation models for Bayesian inference.

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(θ,x)=u(t+θ,x)$, $θ\in[-τ,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

We study the large-depth limit of transformers trained with AdamW, by modelling the hidden-state dynamics as an interacting particle system (IPS) coupled through the attention mechanism. Under appropriate scaling of the attention heads, we prove that the joint dynamics of the hidden states and backpropagated variables converge in $L^2$, uniformly over the initial condition, to the solution of a forward--backward system of ODEs at rate $\mathcal O(L^{-1}+L^{-1/3}H^{-1/2})$. Here, $L$ and $H$ denote the depth and number of heads of the transformer, respectively. The limiting system of ODEs can be identified with a McKean--Vlasov ODE (MVODE) when the attention heads do not incorporate causal masking. By using the flow maps associated with this MVODE and applying concentration of measure techniques, we obtain bounds on the difference between the discrete and continuous models that are uniform over compact sets of initial conditions. As this is achieved without resorting to a covering argument, the constants in our bounds are independent of the number of tokens. Furthermore, under a suitable adaptation to AdamW, the bounds become independent of the token embedding dimension.

The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon's performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

Long-context inference is increasingly a memory-traffic problem. The culprit is the key--value (KV) cache: it grows with context length, batch size, layers, and heads, and it is read at every decoding step. Rotation-based scalar codecs meet this systems constraint by storing a norm, applying a shared random rotation, and quantizing one coordinate at a time. They are universal and random-access, but they discard the geometry created by the normalization step. After a Haar rotation, a block of $k$ consecutive coordinates is not a product source; it is a spherical-Beta source on the unit ball. We introduce \textsc{FibQuant}, a universal fixed-rate vector quantizer that keeps the same normalize--rotate--store interface while replacing scalar tables by a shared radial--angular codebook matched to this canonical source. The codebook combines Beta-quantile radii, Fibonacci\,/\,Roberts--Kronecker quasi-uniform directions, and multi-restart Lloyd--Max refinement. We prove that the resulting vector code strictly improves on its scalar product specialization at matched rate, with a high-rate gain that separates into a cell-shaping factor and a density-matching factor. The same construction gives a dense rate axis, including fractional-bit and sub-one-bit operating points, without calibration or variable-length addresses. On GPT-2 small KV caches, \textsc{FibQuant} traces a memory--fidelity frontier from $5\times$ compression at $0.99$ attention cosine similarity to $34\times$ at $0.95$. End-to-end on TinyLlama-1.1B, it is within $0.10$ perplexity of fp16 at $4\times$ compression and has $3.6\times$ lower perplexity than scalar \textsc{TurboQuant} at $b = 2$ ($8\times$ compression), where scalar random-access quantization begins to fail.

Activation functions play a central role in neural networks by shaping internal representations. Recently, learning binary activation representations has attracted significant attention due to their advantages in computational and memory efficiency, as well as interpretability. However, training neural networks with Heaviside activations remains challenging, as their non-differentiability obstructs standard gradient-based optimization. In this paper, we propose Heavy Tailed Activation Function (HTAF), a smooth approximation to the Heaviside function that enables stable training with gradient-based optimization. We construct HTAF as a sigmoid hyperbolic tangent composite function and theoretically show that it maintains a large gradient mass around zero inputs while exhibiting slower gradient decay in the tail regions. We show that Spiking Neural Networks, Binary Neural Networks and Deep Heaviside neural Networks can be trained stably using HTAF with gradient-based optimization. Finally, we introduce Implicit Concept Bottleneck Models (ICBMs), an interpretable image model that leverages HTAF to induce discrete feature representations. Extensive experiments across various architectures and image datasets demonstrate that ICBM enables stable discretization while achieving prediction performance comparable to or better than standard models.

Standard Bradley--Terry (BT) reward models are limited when human preferences are pluralistic. Although soft preference labels preserve disagreement information, BT can only express it by shrinking reward margins. Gaussian reward models provide an alternative by jointly predicting a reward mean and a reward variance, but suffer from a fundamental non-identifiability from pairwise preferences alone. We propose Anchor-guided Variance-aware Reward Modeling, a framework that resolves this non-identifiability by augmenting preference data with two coarse response-level anchor labels. Building on this, we prove that two anchors are sufficient for identification, develop a joint training objective and establish a non-asymptotic convergence rate for both the estimated reward mean and variance functions. Across simulation studies and four real-world diverging-preference datasets, our method consistently improves reward modeling performance and downstream RLHF, including PPO training and best-of-$N$ selection.