"Name" is the name of the operation in our search space. "TFFunction" is the TensorFlow function that the name is mapped to when a DNA instruction is being converted to a line of TensorFlow code. "Argument Mapping" describes how the values in a DNA's argument set are mapped to the corresponding TensorFlow function arguments. This vocabulary is largely constructed from the lowest level TF operations needed to create Transformers (see Appendix A.5). We also add commonly used math primitives such as SIN and ABS. Here we provide additional implementation details. Relative Dimensions: We use relative dimensions [13] instead of absolute dimensions for each instruction's "dimension size" argument. This allows us to resize the models to fit within our parameter limits (32M to 38M parameters). The vocabulary for these relative dimensions is [1, 2, 4, 8, 12, 16, 24, 32, 48, 64].

Davis, Drusvyatskiy, and Jiang showed that gradient descent with an adaptive stepsize converges locally at a nearly-linear rate for smooth functions that grow at least quartically away from their minimizers. The argument is intricate, relying on monitoring the performance of the algorithm relative to a certain manifold of slow growth -- called the ravine. In this work, we provide a direct Lyapunov-based argument that bypasses these difficulties when the objective is in addition convex and a has a unique minimizer. As a byproduct of the argument, we obtain a more adaptive variant than the original algorithm with encouraging numerical performance.

Today's best LLMs clearly possess some reasoning processes. The paper gives evidence that they also have metacognitive knowledge, including ability to name skills and procedures to apply given a task. We explore this primarily in context of math reasoning, developing a prompt-guided interaction procedure to get a powerful LLM to assign sensible skill labels to math questions, followed by having it perform semantic clustering to obtain coarser families of skill labels. These coarse skill labels look interpretable to humans.To validate that these skill labels are meaningful and relevant to the LLM's reasoning processes we perform the following experiments.

Reasoning with LLMs increasingly unfolds inside a broader verification loop. Internally, systems use cheap checks, such as self-consistency or proxy rewards, which we call weak verification. Externally, users inspect outputs and steer the model through feedback until results are trustworthy, which we call strong verification. These signals differ sharply in cost and reliability: strong verification can establish trust but is resource-intensive, while weak verification is fast and scalable but noisy and imperfect. We formalize this tension through weak--strong verification policies, which decide when to accept or reject based on weak verification and when to defer to strong verification. We introduce metrics capturing incorrect acceptance, incorrect rejection, and strong-verification frequency. Over population, we show that optimal policies admit a two-threshold structure and that calibration and sharpness govern the value of weak verifiers. Building on this, we develop an online algorithm that provably controls acceptance and rejection errors without assumptions on the query stream, the language model, or the weak verifier.

Can't stop humming that tune? A lot goes into the successful'earworms,' including mathematical symmetry. Breakthroughs, discoveries, and DIY tips sent six days a week. While Super Bowl LX is over, the big game still echoes in the minds of many viewers. It's not your fault if a 30-second advertisement spot's melodic hook continues to keep you up at night, however.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.

Implicitly or explicitly, this data is composed of domains--varying distributions of language.

If the memory state can be addressed by its content, itisfurthermore called content-addressable.

Asymptotic constrained (1)has [6, 18, 21, 20]. R and closedsetC Rd introduced(1).

In addition, we present an alternative design that permits a near-optimal sublinear decoding timeofO(klog2k+klogn).