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.

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.

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).

We show that this method can solve SDPs in polynomial time in a smoothed analysis setting.

ThepolytopeP(X,L) is in fact a "twisted sum" of a finite number of lattice polytopes fibering overP(F,L| F) .

High-quality, diverse pre-training corpora form the cornerstone for developing powerful foundation models, enabling AI assistants like ChatGPT [47] to exhibit balanced competencies across a broad spectrum of tasks [11].