A Simple and General Approach to top-$k$ Selection

We study the active learning problem of fixed-confidence top-$k$ identification from noisy pairwise comparisons. In this problem, an algorithm sequentially chooses pairs of items to compare, observes the outcomes, and stops when it can return the set of top-$k$ items with error probability at most $δ$. The objective is to design such a $δ$-correct procedure that minimizes the expected number of comparisons (the sample complexity). This problem falls within the broader literature on fixed-confidence pure exploration in bandit models, where a common target is asymptotic optimality: the algorithm's expected sample complexity matches the information theoretic lower bound as $δ\to 0$. Asymptotically optimal procedures have been developed for a range of fixed-confidence pure-exploration problems, however to the best of our knowledge, for top-$1$, or more generally top-$k$ identification from pairwise comparisons under latent utility models an asymptotically optimal algorithm has not been established. In this setting, we develop such an algorithm. We characterize the structure of the lower bound and formulate it as a saddle-point problem. This structure enables a computationally efficient primal-dual procedure that learns the asymptotically optimal comparison allocation online. We then construct an adaptive comparison-allocation algorithm that tracks the allocation learned by the primal-dual procedure and prove it is asymptotically optimal.

We study fixed-budget adaptive experimentation aimed at maximizing the probability of obtaining sufficient posterior evidence that a recommended design's mean performance exceeds a target. Unlike best-arm identification, this objective does not require finding the best design. For Gaussian bandits with a Gaussian prior, we propose Halve-Then-Greedy (HTG), which combines a large initial balanced allocation, successive halving, and posterior-greedy sampling. We prove instance-dependent, finite-budget exponential bounds on posterior-failure probability that capture both separation from unfavorable arms and the presence of multiple favorable arms. On one-positive-arm families, we establish an asymptotic sample-complexity lower bound that HTG matches up to a logarithmic factor. We further analyze uniform sampling and sequential halving, revealing regimes where they require more samples than HTG by linear and logarithmic factors, respectively. Simulations illustrate HTG's sample-efficiency advantages and their dependence on instance structure.

Top-k selection is a fundamental computational primitive with applications spanning databases, information retrieval, signal processing, and modern machine learning workloads, including sparse activations and attention pruning. As data sizes grow, existing approaches become inefficient: exact methods incur high memory and compute overhead, while approximate methods often rely on brittle heuristics that degrade under adversarial or heavy-tailed inputs. In this paper, we introduce Prof-K, a fast, scalable, and distribution-agnostic algorithm for exact top-k selection. Prof-K performs a single-pass filtering procedure: a small random sample estimates an adaptive threshold, the N input elements are streamed once into a compact buffer, and an exact top-k routine on this buffer recovers the true top-k elements on the first attempt with probability at least $1-\varepsilon$, where $\varepsilon>0$ is user specified. We derive high-probability guarantees for correctness and buffer size, together with an approximately optimal sample size that minimizes overhead as a function of N and k. Empirically, Prof-K achieves 1.5x-15x speedups over the highly optimized PyTorch topk and recent RadiK implementations, with the largest gains in the large-scale, small-to-moderate-k regime where prior methods struggle most. Unlike previous approaches, these guarantees hold independently of the input distribution, ensuring robustness to adversarial settings. A run-time check detects the rare failures and triggers a retry, so the returned set is always exact and $\varepsilon$ bounds only the probability of requiring an additional pass. We further demonstrate its impact on training BatchTopK Sparse Autoencoders (SAEs), where top-k selection constitutes a significant portion of the training cost.