Properly Learning Decision Trees with Queries Is NP-Hard

We prove that it is NP-hard to properly PAC learn decision trees with queries, resolving a longstanding open problem in learning theory (Bshouty 1993; Guijarro-Lavin-Raghavan 1999; Mehta-Raghavan 2002; Feldman 2016). While there has been a long line of work, dating back to (Pitt-Valiant 1988), establishing the hardness of properly learning decision trees from random examples, the more challenging setting of query learners necessitates different techniques and there were no previous lower bounds. En route to our main result, we simplify and strengthen the best known lower bounds for a different problem of Decision Tree Minimization (Zantema-Bodlaender 2000; Sieling 2003). On a technical level, we introduce the notion of hardness distillation, which we study for decision tree complexity but can be considered for any complexity measure: for a function that requires large decision trees, we give a general method for identifying a small set of inputs that is responsible for its complexity. Our technique even rules out query learners that are allowed constant error. This contrasts with existing lower bounds for the setting of random examples which only hold for inverse-polynomial error. Our result, taken together with a recent almost-polynomial time query algorithm for properly learning decision trees under the uniform distribution (Blanc-Lange-Qiao-Tan 2022), demonstrates the dramatic impact of distributional assumptions on the problem.Comment: 41 pages, 10 figures, FOCS 202

A Strong Composition Theorem for Junta Complexity and the Boosting of Property Testers

We prove a strong composition theorem for junta complexity and show how such theorems can be used to generically boost the performance of property testers. The $\varepsilon$-approximate junta complexity of a function $f$ is the smallest integer $r$ such that $f$ is $\varepsilon$-close to a function that depends only on $r$ variables. A strong composition theorem states that if $f$ has large $\varepsilon$-approximate junta complexity, then $g \circ f$ has even larger $\varepsilon'$-approximate junta complexity, even for $\varepsilon' \gg \varepsilon$. We develop a fairly complete understanding of this behavior, proving that the junta complexity of $g \circ f$ is characterized by that of $f$ along with the multivariate noise sensitivity of $g$. For the important case of symmetric functions $g$, we relate their multivariate noise sensitivity to the simpler and well-studied case of univariate noise sensitivity. We then show how strong composition theorems yield boosting algorithms for property testers: with a strong composition theorem for any class of functions, a large-distance tester for that class is immediately upgraded into one for small distances. Combining our contributions yields a booster for junta testers, and with it new implications for junta testing. This is the first boosting-type result in property testing, and we hope that the connection to composition theorems adds compelling motivation to the study of both topics.Comment: 44 pages, 1 figure, FOCS 202

Certification with an NP Oracle

In the certification problem, the algorithm is given a function $f$ with certificate complexity $k$ and an input $x^\star$, and the goal is to find a certificate of size $\le \text{poly}(k)$ for $f$'s value at $x^\star$. This problem is in $\mathsf{NP}^{\mathsf{NP}}$, and assuming $\mathsf{P} \ne \mathsf{NP}$, is not in $\mathsf{P}$. Prior works, dating back to Valiant in 1984, have therefore sought to design efficient algorithms by imposing assumptions on $f$ such as monotonicity. Our first result is a $\mathsf{BPP}^{\mathsf{NP}}$ algorithm for the general problem. The key ingredient is a new notion of the balanced influence of variables, a natural variant of influence that corrects for the bias of the function. Balanced influences can be accurately estimated via uniform generation, and classic $\mathsf{BPP}^{\mathsf{NP}}$ algorithms are known for the latter task. We then consider certification with stricter instance-wise guarantees: for each $x^\star$, find a certificate whose size scales with that of the smallest certificate for $x^\star$. In sharp contrast with our first result, we show that this problem is $\mathsf{NP}^{\mathsf{NP}}$-hard even to approximate. We obtain an optimal inapproximability ratio, adding to a small handful of problems in the higher levels of the polynomial hierarchy for which optimal inapproximability is known. Our proof involves the novel use of bit-fixing dispersers for gap amplification.Comment: 25 pages, 2 figures, ITCS 202

Cortisol levels in rural Latina breast cancer survivors participating in a peer-delivered cognitive-behavioral stress management intervention: The Nuevo Amanecer-II RCT.

BackgroundCompared to their White counterparts, Latina breast cancer survivors have poorer survival rates and health-related quality of life, and higher rates of depression and anxiety which may be a result of chronic stress. Chronic stress impacts the hypothalamic-pituitary-adrenal (HPA) axis, resulting in cortisol dysregulation which may be associated with breast cancer survival. However, cortisol levels and cortisol profiles of Latina breast cancer survivors are poorly characterized due to their underrepresentation in biomedical research.ObjectiveThe objective of this study was to describe cortisol levels and patterns of cortisol secretions in rural Latina breast cancer survivors participating in an RCT study of Nuevo Amanecer-II, an evidence-based peer-delivered cognitive behavioral stress management intervention.MethodsParticipant-centered recruitment and collection strategies were used to obtain biospecimens for cortisol analysis. Nine saliva samples (3/day for 3 days) and a hair sample were obtained at baseline and 6-months (3-months post-intervention). We describe cortisol levels and profiles, explore correlations of biomarkers with self-report measures of stress and psychological distress, and compare women who received the intervention with a delayed intervention group on biomarkers of stress. Mean hair cortisol concentration (HCC) was used to assess chronic stress. Based on daily measures of cortisol (awakening, 30 min post-awakening, and bedtime), we calculated three summary measures of the dynamic nature of the cortisol awakening response (CAR): 1) the CAR slope, 2) whether CAR demonstrates a percent change ≥40, and 3) total daily cortisol output (AUCg). Linear and log-binomial regression, accounting for multiple samples per participant, were used to compare cortisol measures at 6-month follow-up by treatment arm.ResultsParticipants (n = 103) were from two rural California communities; 76 provided at least one saliva sample at baseline and follow-up and were included in the analysis. At baseline, mean age was 57 years, mean years since diagnosis was 2 years, 76% had a high school education or less, and 34% reported financial hardship. The overall median CAR slope was 0.10, and median cortisol AUCg (in thousands) was 11.34 (range = 0.93, 36.66). Mean hair cortisol concentration was 1751.6 pg/mg (SD = 1148.6). Forty-two percent of samples had a ≥40% change in CAR. We found no statistically significant correlations between the cortisol measures and self-reported measures of stress and psychological distress. At follow-up, no differences were seen in HCC (mean difference between intervention and control: -0.11, 95% CI -0.48, 0.25), CAR slope (0.001, 95% CI -0.005, 0.008), cortisol AUCg (-0.15, 95% CI -0.42, 0.13), or ≥40% change in CAR (prevalence ratio 0.87, 95% CI 0.42, 1.77) between treatment arms.ConclusionOur findings of flattened cortisol profiles among more than half of the sample suggest potential HPA-axis dysregulation among rural Spanish-speaking Latina breast cancer survivors that merits further study due to its implications for long-term survival.Trial registrationhttp://www.ClinicalTrials.gov identifier NCT02931552