The standard definition of PAC learning (Valiant 1984) requires learners to succeed under all distributions - even ones that are intractable to sample from. This stands in contrast to samplable PAC learning (Blum, Furst, Kearns, and Lipton 1993), where learners only have to succeed under samplable distributions. We study this distinction and show that samplable PAC substantially expands the power of efficient learners. We first construct a concept class that requires exponential sample complexity in standard PAC but is learnable with polynomial sample complexity in samplable PAC. We then lift this statistical separation to the computational setting and obtain a separation relative to a random oracle. Our proofs center around a new complexity primitive, explicit evasive sets, that we introduce and study. These are sets for which membership is easy to determine but are extremely hard to sample from. Our results extend to the online setting to similarly show that its landscape changes when the adversary is assumed to be efficient instead of computationally unbounded.

The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and relevance. Towards addressing this, we prove a {\sl distributional-lifting theorem}: This upgrades a learner that succeeds with respect to a limited distribution family $\mathcal{D}$ to one that succeeds with respect to {\sl any} distribution $D^\star$, with an efficiency overhead that scales with the complexity of expressing $D^\star$ as a mixture of distributions in $\mathcal{D}$. Recent work of Blanc, Lange, Malik, and Tan considered the special case of lifting uniform-distribution learners and designed a lifter that uses a conditional sample oracle for $D^\star$, a strong form of access not afforded by the standard PAC model. Their approach, which draws on ideas from semi-supervised learning, first learns $D^\star$ and then uses this information to lift. We show that their approach is information-theoretically intractable with access only to random examples, thereby giving formal justification for their use of the conditional sample oracle. We then take a different approach that sidesteps the need to learn $D^\star$, yielding a lifter that works in the standard PAC model and enjoys additional advantages: it works for all base distribution families, preserves the noise tolerance of learners, has better sample complexity, and is simpler.

A central question in computer science and statistics is whether efficient algorithms can achieve the information-theoretic limits of statistical problems. Many computational-statistical tradeoffs have been shown under average-case assumptions, but since statistical problems are average-case in nature, it has been a challenge to base them on standard worst-case assumptions.In PAC learning where such tradeoffs were first studied, the question is whether computational efficiency can come at the cost of using more samples than informationtheoretically necessary. We base such tradeoffs on NP-hardness and obtain:◦ Sharp computational-statistical tradeoffs assuming NP requires exponential time: For every polynomial $p(n)$, there is an n-variate class $\mathcal{C}$ with VC dimension 1 such that the sample complexity of time-efficiently learning $\mathcal{C}$ is $\Theta(p(n))$.◦ A characterization of RP vs. NP in terms of learning: RP = NP iff every NP-enumerable class is learnable with $O(\operatorname{VCdim}(\mathcal{C}))$ samples in polynomial time. The forward implication has been known since (Pitt and Valiant, 1988); we prove the reverse implication.Notably, all our lower bounds hold against improper learners. These are the first NP-hardness results for improperly learning a subclass of polynomial-size circuits, circumventing formal barriers of Applebaum, Barak, and Xiao (2008).

We consider the task of properly PAC learning decision trees with queries. Recent work of Koch, Strassle, and Tan showed that the strictest version of this task, where the hypothesis tree T is required to be optimally small, is NP-hard. Their work leaves open the question of whether the task remains intractable if T is only required to be close to optimal, say within a factor of 2, rather than exactly optimal. We answer this affirmatively and show that the task indeed remains NP-hard even if T is allowed to be within any constant factor of optimal. More generally, our result allows for a smooth tradeoff between the hardness assumption and inapproximability factor. As Koch et al.’s techniques do not appear to be amenable to such a strengthening, we first recover their result with a new and simpler proof, which we couple with a new XOR lemma for decision trees. While there is a large body of work on XOR lemmas for decision trees, our setting necessitates parameters that are extremely sharp and are not known to be attainable by existing such lemmas. Our work also carries new implications for the related problem of Decision Tree Minimization.

We connect the problem of properly PAC learning decision trees to the parameterized Nearest Codeword Problem (k-NCP). Despite significant effort by the respective communities, algorithmic progress on both problems has been stuck: the fastest known algorithm for the former runs in quasipolynomial time (Ehrenfeucht and Haussler 1989) and the best known approximation ratio for the latter is$O$($n$/logn) (Berman and Karpinsky 2002; Alon, Panigrahy, and Yekhanin 2009). Research on both problems has thus far proceeded independently with no known connections. We show that any improvement of Ehrenfeucht and Haussler's algorithm will yield$O$(logn)-approximation algorithms for k-NCP, an exponential improvement of the current state of the art. This can be interpreted either as a new avenue for designing algorithms for k-NCP, or as one for establishing the optimality of Ehrenfeucht and Haussler's algorithm. Furthermore, our reduction along with existing inapproximability results for k - NCP already rule out polynomial-time algorithms for properly learning decision trees. A notable aspect of our hardness results is that they hold even in the setting of weak learning whereas prior ones were limited to the setting of strong learning.

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 $\varepsilon^{\prime}$-approximate junta complexity, even for $\varepsilon^{\prime} \gg \varepsilon$. 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.

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–Lavín–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.

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.

We establish new hardness results for decision tree optimization problems, adding to a line of work that dates back to Hyafil and Rivest in 1976. We prove, under the randomized exponential time hypothesis, superpolynomial runtime lower bounds for two basic problems: given an explicit representation of a function f and a generator for a distribution D