An elementary data analysis task is to count the number of unique elements occurring in a dataset.

Structural Causal Models (SCM) framework introduced by Pearl [2009].

Scaling inference compute in large language models (LLMs) through repeated sampling consistently increases the coverage (fraction of problems solved) as the number of samples increases. We conjecture that this observed improvement is partially due to the answer distribution of standard evaluation benchmarks, which is skewed towards a relatively small set of common answers. To test this conjecture, we define a baseline that enumerates answers according to their prevalence in the training set. Experiments spanning two domains -- mathematical reasoning and factual knowledge -- reveal that this baseline outperforms repeated model sampling for some LLMs, while the coverage for others is on par with that of a mixture strategy that obtains $k$ answers by using only $10$ model samples and similarly guessing the remaining $k-10$ attempts via enumeration. Our baseline enables a more accurate measurement of how much repeated sampling improves coverage in such settings beyond prompt-agnostic guessing.

Many modern applications use computer vision to detect and count objects in massive image collections. However, when the detection task is very difficult or in the presence of domain shifts, the counts may be inaccurate even with significant investments in training data and model development. We propose DISCount -- a detector-based importance sampling framework for counting in large image collections that integrates an imperfect detector with human-in-the-loop screening to produce unbiased estimates of counts. We propose techniques for solving counting problems over multiple spatial or temporal regions using a small number of screened samples and estimate confidence intervals. This enables end-users to stop screening when estimates are sufficiently accurate, which is often the goal in a scientific study. On the technical side we develop variance reduction techniques based on control variates and prove the (conditional) unbiasedness of the estimators. DISCount leads to a 9-12x reduction in the labeling costs over naive screening for tasks we consider, such as counting birds in radar imagery or estimating damaged buildings in satellite imagery, and also surpasses alternative covariate-based screening approaches in efficiency.

We introduce a new Collaborative Causal Discovery problem, through which we model a common scenario in which we have multiple independent entities each with their own causal graph, and the goal is to simultaneously learn all these causal graphs. We study this problem without the causal sufficiency assumption, using Maximal Ancestral Graphs (MAG) to model the causal graphs, and assuming that we have the ability to actively perform independent single vertex (or atomic) interventions on the entities. If the $M$ underlying (unknown) causal graphs of the entities satisfy a natural notion of clustering, we give algorithms that leverage this property and recovers all the causal graphs using roughly logarithmic in $M$ number of atomic interventions per entity. These are significantly fewer than $n$ atomic interventions per entity required to learn each causal graph separately, where $n$ is the number of observable nodes in the causal graph. We complement our results with a lower bound and discuss various extensions of our collaborative setting.

We introduce the problem Max#SAT, an extension of model counting (#SAT). Given a formula over sets of variables X, Y, and Z, the Max#SAT problem is to maximize over the variables X the number of assignments to Y that can be extended to a solution with some assignment to Z. We demonstrate that Max#SAT has applications in many areas, showing how it can be used to solve problems in probabilistic inference (marginal MAP), planning, program synthesis, and quantitative information flow analysis. We also give an algorithm which by making only polynomially many calls to an NP oracle can approximate the maximum count to within any desired multiplicative error. The NP queries needed are relatively simple, arising from recent practical approximate model counting and sampling algorithms, which allows our technique to be effectively implemented with a SAT solver. Through several experiments we show that our approach can be successfully applied to interesting problems.

Constraints over finite sequences of variables are ubiquitous in sequencing and timetabling. This led to general modelling techniques and generic propagators, often based on deterministic finite automata (DFA) and their extensions. We consider counter-DFAs (cDFA), which provide concise models for regular counting constraints, that is constraints over the number of times a regular-language pattern occurs in a sequence. We show how to enforce domain consistency in polynomial time for at-most and at-least regular counting constraints based on the frequent case of a cDFA with only accepting states and a single counter that can be increased by transitions. We also show that the satisfaction of exact regular counting constraints is NP-hard and that an incomplete propagator for exact regular counting constraints is faster and provides more pruning than the existing propagator from (Beldiceanu, Carlsson, and Petit 2004). Finally, by avoiding the unrolling of the cDFA used by COSTREGULAR, the space complexity reduces from O(n · |Σ| · |Q|) to O(n · (|Σ| + |Q|)), where Σ is the alphabet and Q the state set of the cDFA.