Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.
The binary perceptron is a random constraint satisfaction problem that asks for a sign vector satisfying a system of random linear inequalities. Its solution space has a strikingly extreme geometry: at every positive constraint density, all but a vanishing fraction of the solutions are isolated. Yet polynomial-time algorithms find solutions at small enough constraint density, which raises the question of how efficient algorithms navigate such a landscape. Linear programming is a natural algorithm to study here, but its analysis has so far been limited to a single round. Li, Schramm, and Zhou (STOC 2025) analyzed one linear program followed by iterative rounding. In this work, we overcome this limitation and give a precise analysis of iterated linear programming for the binary perceptron. (1) For Gaussian matrices and any fixed number of rounds, we characterize the iterated linear programs exactly through a deterministic state-evolution recursion. (2) Adding a final rounding step, we obtain a randomized polynomial-time algorithm for the binary perceptron. Numerical evaluations of the recursion provide evidence that it succeeds at constraint density $α=0.60$ with zero margin, well beyond the range $α\le0.1$ guaranteed by the single-LP analysis and $α\approx0.327$ (Jo and Lee, 2026), the best guarantee so far. (3) We establish universality, showing that the same characterization holds for matrices with i.i.d. entries drawn from any centered, unit-variance subgaussian law. The key idea of our proof is that, although conditioning on earlier rounds makes the matrix non-Gaussian, its conditional law can be computed explicitly and remains close to a Gaussian law, so the convex Gaussian min-max theorem applies in every round. For universality, we compare Gaussian and subgaussian matrices through the Lindeberg principle, using a regularized version of each linear program.
The low-degree method predicts computational thresholds by asking whether low-degree polynomials can distinguish a planted distribution from a null one. Recent counterexamples by Buhai, Hsieh, Jain, and Kothari (FOCS 2025) and Mao (2026) show that low-degree indistinguishability does not, in general, rule out efficient algorithms, even on noisy inputs. Hsieh, Kane, Kothari, Li, Mohanty, and Tiegel (STOC 2026) asked what it does imply, and showed that for permutation-invariant distributions on $\{0,1\}^n$, low-degree indistinguishability from the uniform distribution implies statistical indistinguishability after noise. In this work, we study the same question for planted subgraphs in sparse random graphs. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $Γ_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and the treewidth of $Γ_n$ is $o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. In the critical and subcritical regimes $0<c\le 1$, we obtain the same conclusion without any treewidth assumption, provided $D_n=ω(\log n)$. Our proof has two main ingredients. First, we identify the subgraph-count and automorphism factors in the Fourier expansion with counts of isomorphism triples. Second, we cut a tree decomposition of $Γ_n$ into subtrees, which splits each large Fourier support into low-degree pieces that meet at only a few interface vertices, and we use noise to absorb the cost of reassembling them. At and below criticality, we use the low-degree assumption to rule out short cycles and noise to destroy the remaining long ones.
The binary perceptron problem asks for a length-$n$ sign vector in the intersection of $\lfloor αn\rfloor$-many independent random Gaussian half-spaces. A striking feature of this model is strong freezing: at every constraint density $α> 0$, all but a vanishing fraction of solutions are isolated -- $Ω(n)$-far from all others, in Hamming distance. Such solutions are believed to be algorithmically inaccessible. Nevertheless, efficient algorithms find solutions at some constraint densities $α$. One possible resolution of this apparent contradiction is that the solution space also contains rare well-connected structures, and that algorithms succeed by locating them. Gong, Huang, Li and Sellke (FOCS 2026) proved this for stable algorithms, by showing that such algorithms cannot locate isolated solutions with probability more than $\approx 0.84$. In this work, we study online algorithms, which receive the columns of the constraint matrix one by one and sign each irrevocably upon arrival. This is a widely used class of algorithms in discrepancy minimization, and it includes the algorithm of Jo and Lee (2026), which has one of the best provable guarantees for the binary perceptron. We prove that at any constraint density, any online algorithm outputs an isolated solution with probability $o(1)$. Our proof is based on the observation that the signs an online algorithm has already fixed can be completed to at most one isolated solution, so a successful algorithm must choose the unique correct sign for each of the last few columns. Using the exchangeability of these columns, we show that such an algorithm must match a suffix-parallel baseline, which signs each of them as if it were the next column to arrive. We then use an anti-concentration argument to show that such a baseline rarely produces a unique completion.