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Inthis section, we describe the details ofMiniRTSEnvironment and human dataset. The data do not contain any personally identifiable information or offensivecontent. Figure 1: MiniRTS [2]implements the rockpaper-scissors attack graph, each army type has some units it is effective against and vulnerableto. "swordman","spearman"and"cavalry"allare effectiveagainst"archer" Figure 2: Building units can produce different army units using resources. Resource Units: Resource units are stationary and neutral.

The design of complex machines stands as both a marker of human intelligence and a foundation of engineering practice. Given recent advances in large language models (LLMs), we ask whether they, too, can learn to create. We approach this question through the lens of compositional machine design: a task in which machines are assembled from standardized components to meet functional demands like locomotion or manipulation in a simulated physical environment. With this simplification, machine design is expressed as writing XML-like code that explicitly specifies pairwise part connections. To support this investigation, we introduce BesiegeField, a testbed built on the machine-building game Besiege, which enables part-based construction, physical simulation and reward-driven evaluation. Using BesiegeField, we benchmark state-of-the-art LLMs with agentic workflows and identify key capabilities required for success, including spatial reasoning, strategic assembly, and instruction-following. As current open-source models fall short, we explore reinforcement learning (RL) as a path to improvement: we curate a cold-start dataset, conduct RL finetuning experiments, and highlight open challenges at the intersection of language, machine design, and physical reasoning.

Notably, we discover the opening up of a "sharpness reduction" phase,

In this section, we describe the details of MiniRTS Environment and human dataset. "spearman" but is retrained by "cavarly". "swordman", "spearman" and "cavalry" all are Figure 2: Building units can produce different army units using resources. "workshop" can produce "archer", "dragon" and "catapult" while other Resource Units: Resource units are stationary and neutral. Resource units cannot be constructed by anyone and are created at the beginning of a game. Building Units: MiniRTS supports 6 different building unit types.

Although gradient descent with momentum is widely used in modern deep learning, a concrete understanding of its effects on the training trajectory still remains elusive. In this work, we empirically show that momentum gradient descent with a large learning rate and learning rate warmup displays large catapults, driving the iterates towards flatter minima than those found by gradient descent. We then provide empirical evidence and theoretical intuition that the large catapult is caused by momentum "amplifying" the self-stabilization effect (Damian et al., 2023).B.1

We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) and study the effect of learning rate $\eta$, depth $d$, and width $w$ of the neural network. By analyzing the maximum eigenvalue $\lambda^H_t$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and (iv) a late time ``edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on $\eta \equiv c / \lambda_0^H $, $d$, and $w$. We identify several critical values of $c$, which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness. Notably, we discover the opening up of a ``sharpness reduction" phase, where sharpness decreases at early times, as $d$ and $1/w$ are increased.

In this paper, we first present an explanation regarding the common occurrence of spikes in the training loss when neural networks are trained with stochastic gradient descent (SGD). We provide evidence that the spikes in the training loss of SGD are "catapults", an optimization phenomenon originally observed in GD with large learning rates in [Lewkowycz et al. 2020]. We empirically show that these catapults occur in a low-dimensional subspace spanned by the top eigenvectors of the tangent kernel, for both GD and SGD. Second, we posit an explanation for how catapults lead to better generalization by demonstrating that catapults promote feature learning by increasing alignment with the Average Gradient Outer Product (AGOP) of the true predictor. Furthermore, we demonstrate that a smaller batch size in SGD induces a larger number of catapults, thereby improving AGOP alignment and test performance.

Since I first hit start on Tears of the Kingdom two weeks ago, scarcely a minute has passed when I was not either playing it, or wishing I was playing it. I am honestly slightly annoyed to be taking time away from it to write this review. I am a grown 34-year-old woman and video games rarely get their hooks into me the way they did when I was 8, or 18, and relatively free of responsibilities. But now and then, every few years, I play something that reminds me that video games are kind of magic. They can transport you somewhere else.