Finally, we provide mathematical bounds on the interference between superposition channels in MIMOFormer.

Obtaining reliable inferences with traditional difference-in-differences (DiD) methods can be difficult. Problems can arise when both outcomes and errors are serially correlated, when there are few clusters or few treated clusters, when cluster sizes vary greatly, and in various other cases. In recent years, recognition of the ``staggered adoption'' problem has shifted the focus away from inference towards consistent estimation of treatment effects. One of the most popular new estimators is the CSDID procedure of Callaway and Sant'Anna (2021). We find that the issues of over-rejection with few clusters and/or few treated clusters are at least as severe for CSDID as for traditional DiD methods. We also propose using a cluster jackknife for inference with CSDID, which simulations suggest greatly improves inference. We provide software packages in Stata csdidjack and R didjack to calculate cluster-jackknife standard errors easily.

We propose ResidualPlanner, a matrix mechanism for marginals with Gaussian noise that is both optimal and scalable.

Since the switching costs introduce coupling across all stages, multi-step-ahead (long-term) predictions areincorporated toimprovethe online performance.

For example, theleader aims tooptimize some traffic-networkparameters (e.g., road width values) so that players can spend less traveling time at equilibrium.

Spectral gradient methods, such as the recently popularized Muon optimizer, are a promising alternative to standard Euclidean gradient descent for training deep neural networks and transformers, but it is still unclear in which regimes they are expected to perform better. We propose a simple layerwise condition that predicts when a spectral update yields a larger decrease in the loss than a Euclidean gradient step. This condition compares, for each parameter block, the squared nuclear-to-Frobenius ratio of the gradient to the stable rank of the incoming activations. To understand when this condition may be satisfied, we first prove that post-activation matrices have low stable rank at Gaussian initialization in random feature regression, feedforward networks, and transformer blocks. In spiked random feature models we then show that, after a short burn-in, the Euclidean gradient's nuclear-to-Frobenius ratio grows with the data dimension while the stable rank of the activations remains bounded, so the predicted advantage of spectral updates scales with dimension. We validate these predictions in synthetic regression experiments and in NanoGPT-scale language model training, where we find that intermediate activations have low-stable-rank throughout training and the corresponding gradients maintain large nuclear-to-Frobenius ratios. Together, these results identify conditions for spectral gradient methods, such as Muon, to be effective in training deep networks and transformers.

Multi-agent multi-objective systems (MAMOS) have emerged as powerful frameworks for modelling complex decision-making problems across various real-world domains, such as robotic exploration, autonomous traffic management, and sensor network optimisation. MAMOS offers enhanced scalability and robustness through decentralised control and more accurately reflects inherent trade-offs between conflicting objectives. In MAMOS, each agent uses utility functions that map return vectors to scalar values. Existing MAMOS optimisation methods face challenges in handling heterogeneous objective and utility function settings, where training non-stationarity is intensified due to private utility functions and the associated policies. In this paper, we first theoretically prove that direct access to, or structured modeling of, global utility functions is necessary for the Bayesian Nash Equilibrium under decentralised execution constraints. To access the global utility functions while preserving the decentralised execution, we propose an Agent-Attention Multi-Agent Multi-Objective Reinforcement Learning (AA-MAMORL) framework. Our approach implicitly learns a joint belief over other agents' utility functions and their associated policies during centralised training, effectively mapping global states and utilities to each agent's policy. In execution, each agent independently selects actions based on local observations and its private utility function to approximate a BNE, without relying on inter-agent communication. We conduct comprehensive experiments in both a custom-designed MAMO Particle environment and the standard MOMALand benchmark. The results demonstrate that access to global preferences and our proposed AA-MAMORL significantly improve performance and consistently outperform state-of-the-art methods.

Weight decay is a standard regularization technique for training large language models (LLMs). While it is common to assign a uniform decay rate to every layer, this approach overlooks the structural diversity of LLMs and the varying spectral properties across modules. In this paper, we introduce AlphaDecay, a simple yet effective method that adaptively assigns different weight decay strengths to each module of an LLM. Our approach is guided by Heavy-Tailed Self-Regularization (HT-SR) theory, which analyzes the empirical spectral density (ESD) of weight correlation matrices to quantify "heavy-tailedness." Modules exhibiting more pronounced heavy-tailed ESDs, reflecting stronger feature learning, are assigned weaker decay, while modules with lighter-tailed spectra receive stronger decay. Our method leverages tailored weight decay assignments to balance the module-wise differences in spectral properties, leading to improved performance. Extensive pre-training tasks with various model sizes from 60M to 1B demonstrate that AlphaDecay achieves better perplexity and generalization than conventional uniform decay and other adaptive decay baselines. Our code is available at https://github.com/hed-ucas/AlphaDecay.