Non-prefix KV-cache reuse encodes retrieved chunks independently and concatenates their caches. To find where a small repair budget should be spent, we vary the chunks around the answer-bearing one (the target) across three 7B-8B models and four datasets of dialogue memory, open-domain QA and API documentation. Reuse loses 19-100% of the context-dependent questions that full prefill answers, and it loses more when the other chunks are of the same kind as the target. Because the recoverable damage is spread over these other chunks, restoring a few of them recovers little, and the budget is better spent on the target. This works through contextual compatibility: recomputing the target against the reused cache beats inserting its full-prefill KV in all seven settings tested, although the recomputed KV stays far from the full-prefill KV. Question attention over reused caches, however, selects the target in at most 18.8% of development questions. A retriever finds it more often, and recomputing its top chunk after moving it next to the question significantly beats InfoFlow-style reordering at the same 5% budget in six of nine open-domain QA and API-documentation settings. On dialogue memory, though, recomputing chunks never significantly beats the token-level selector chosen for each setting.
Finite-sum nonconvex-strongly-concave minimax problems arise in adversarial training and distributionally robust learning. Their known lower bounds cover only linear-span algorithms, zero-respecting algorithms, or a restricted proximal protocol. They leave open the power of exact component proximal steps, whose responses are global saddle points of regularized component problems and can expose directions that gradients do not reveal. We show that neither unrestricted adaptivity nor exact proximal access improves the polynomial complexity. We consider $n$ components with joint mean-squared smoothness $L$ and condition number $κ$, started from a point with primal gap at most $Δ$ and joint gradient norm at most $\sqrt{2LΔ}$. For $\varepsilon^2/(LΔ)$ below a universal constant, every randomized adaptive algorithm with exact component values, gradients, and proximal responses needs $Ω(n+\min\{\sqrt n\,κ,n^{3/4}\sqrtκ\}LΔ/\varepsilon^2)$ component queries in the worst case to find an $\varepsilon$-stationary point of the value function. A method using only component gradients attains this rate up to a factor $\log^2(2+κ)$, including initialization. Zhou (arXiv:2610.04944) concurrently proves the same rate for zero-respecting first-order methods. In our hard instances, every component admits proximal steps with the same step sizes, and a response for a component reveals hidden directions only up to the first one assigned to another component.
Low curvature trace, an effective rank condition, is a common explanation for why randomized zeroth-order methods scale to high-dimensional tasks such as language model fine-tuning. Can deterministic methods achieve the same savings? We show that they cannot. We study finding $\varepsilon$-stationary points of $L$-smooth nonconvex functions on $\mathbb{R}^d$ from exact function values, assuming that below the starting value the Hessian is dominated by a fixed matrix with trace at most $LE$; this is a fixed-matrix form of the effective rank condition used to explain zeroth-order fine-tuning. When $ET\le d$, the Gaussian two-point method that such fine-tuning uses needs only $O(ET)$ calls if it rejects steps that raise $f$, where $T=LΔ/\varepsilon^2$ is the iteration count of gradient descent from initial gap $Δ$, and we prove that every randomized method needs $Ω(ET)$ calls, so this cost is optimal up to constants. In contrast, every deterministic method needs $\widetildeΩ(d\min\{T,\sqrt{ET}\})$ calls, and adaptive subspace search attains $O(d\min\{T,\sqrt{ET}\})$: a deterministic method pays about $d$ calls per discovered direction, and the trace budget allows about $\sqrt{ET}$ directions. Determinism thus multiplies the cost by a factor of order $d/(E+\sqrt{ET})$ up to logarithmic factors, which grows linearly with $d$ for fixed $E$ and $T$; once $E$ exceeds $T$ by a logarithmic factor, deterministic methods need $Ω(dT)$ calls, as without any trace budget. Our lower bounds hold for $C^\infty$ functions with the matrix bound on all of $\mathbb{R}^d$, even when each call also returns gradient projections priced by rank.
We study the last iterate of clipped SGD, and of every rule that queries its current iterate and steps along the resulting stochastic gradient, when training is stopped at a data-dependent time, so that guarantees must hold at all times simultaneously. On smooth convex objectives under heavy-tailed noise with a bounded central $p$th moment, $1<p\le2$, we determine the optimal time-uniform rate of these rules. A clipped SGD schedule without horizon or confidence inputs attains $\tilde O(t^{-β_p})$ at all times, with $β_p=2(p-1)/(3p-1)$, also almost surely, and no such rule attains $\tilde O(t^{-r})$ for any $r>β_p$, even with adaptive step sizes and thresholds. This rate lies between the optimal rate $O(T^{-α_p})$ for a known horizon $T$, $α_p=(p-1)/p$, and the previous time-uniform rate $\tilde O(t^{-2(p-1)/(3p)})$ (Chezhegov et al., ICLR 2026); under finite variance, the exponents are $1/2$, $2/5$, and $1/3$. Their conjecture that a better schedule reaches the optimal rate thus holds with a known horizon (Liu; Jiang et al.) but fails without one. Querying gradients at the running average attains $\tilde O(t^{-α_p})$ at all times, whereas heavy-ball momentum with geometric forgetting and polynomially bounded steps cannot improve on $β_p$.
Web services built on large language models (LLMs) receive many paraphrases of earlier queries, which a semantic cache answers from stored responses. A reused answer can be wrong, so wrong answers must stay below a share $δ$ of the queries. Existing caches reuse above a fixed similarity threshold or, like vCache, learn a per-entry rule that bounds each query's error by $δ$. However, fixed thresholds break the budget on new streams, and per-entry rules need labels on every entry before reusing it. We find an unspent error budget, as vCache realizes only 12-55% of $δ$ on three public query streams. Surplus spends this budget where reuse is most likely correct. It ranks queries by a risk score shared across paraphrase clusters and opens risk bins as far as an audited error estimate allows. Free labels from every LLM call and a ledger of realized errors keep this threshold safe under drift. On five streams, including ORCAS search queries, Surplus holds $δ$ in every test cell, which no baseline does. On the shuffled streams it serves 40% more hits than vCache, and its pooled hit rate beats the best baseline holding $δ$ in each cell by 4.10 points. In a vLLM testbed it cuts the mean latency by 31.59% at the highest load, against 23.67% for vCache.