Caleb Koch

Caleb Koch, Carmen Strassle, Li-Yang Tan

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 randomized ETH, superpolynomial lower bounds for two basic problems: given an explicit representation of a function $f$ and a generator for a distribution $\mathcal{D}$, construct a small decision tree approximator for $f$ under $\mathcal{D}$, and decide if there is a small decision tree approximator for $f$ under $\mathcal{D}$. Our results imply new lower bounds for distribution-free PAC learning and testing of decision trees, settings in which the algorithm only has restricted access to $f$ and $\mathcal{D}$. Specifically, we show: $n$-variable size-$s$ decision trees cannot be properly PAC learned in time $n^{\tilde{O}(\log\log s)}$, and depth-$d$ decision trees cannot be tested in time $\exp(d^{\,O(1)})$. For learning, the previous best lower bound only ruled out $\text{poly}(n)$-time algorithms (Alekhnovich, Braverman, Feldman, Klivans, and Pitassi, 2009). For testing, recent work gives similar though incomparable bounds in the setting where $f$ is random and $\mathcal{D}$ is nonexplicit (Blais, Ferreira Pinto Jr., and Harms, 2021). Assuming a plausible conjecture on the hardness of Set-Cover, we show our lower bound for learning decision trees can be improved to $n^{Ω(\log s)}$, matching the best known upper bound of $n^{O(\log s)}$ due to Ehrenfeucht and Haussler (1989). We obtain our results within a unified framework that leverages recent progress in two lines of work: the inapproximability of Set-Cover and XOR lemmas for query complexity. Our framework is versatile and yields results for related concept classes such as juntas and DNF formulas.

Caleb Koch, Carmen Strassle, Li-Yang Tan

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.

Guy Blanc, Caleb Koch, Jane Lange et al.

We design an algorithm for finding counterfactuals with strong theoretical guarantees on its performance. For any monotone model $f : X^d \to \{0,1\}$ and instance $x^\star$, our algorithm makes \[ {S(f)^{O(Δ_f(x^\star))}\cdot \log d}\] queries to $f$ and returns {an {\sl optimal}} counterfactual for $x^\star$: a nearest instance $x'$ to $x^\star$ for which $f(x')\ne f(x^\star)$. Here $S(f)$ is the sensitivity of $f$, a discrete analogue of the Lipschitz constant, and $Δ_f(x^\star)$ is the distance from $x^\star$ to its nearest counterfactuals. The previous best known query complexity was $d^{\,O(Δ_f(x^\star))}$, achievable by brute-force local search. We further prove a lower bound of $S(f)^{Ω(Δ_f(x^\star))} + Ω(\log d)$ on the query complexity of any algorithm, thereby showing that the guarantees of our algorithm are essentially optimal.

Guy Blanc, Alexandre Hayderi, Caleb Koch et al.

Smooth boosters generate distributions that do not place too much weight on any given example. Originally introduced for their noise-tolerant properties, such boosters have also found applications in differential privacy, reproducibility, and quantum learning theory. We study and settle the sample complexity of smooth boosting: we exhibit a class that can be weak learned to $γ$-advantage over smooth distributions with $m$ samples, for which strong learning over the uniform distribution requires $\tildeΩ(1/γ^2)\cdot m$ samples. This matches the overhead of existing smooth boosters and provides the first separation from the setting of distribution-independent boosting, for which the corresponding overhead is $O(1/γ)$. Our work also sheds new light on Impagliazzo's hardcore theorem from complexity theory, all known proofs of which can be cast in the framework of smooth boosting. For a function $f$ that is mildly hard against size-$s$ circuits, the hardcore theorem provides a set of inputs on which $f$ is extremely hard against size-$s'$ circuits. A downside of this important result is the loss in circuit size, i.e. that $s' \ll s$. Answering a question of Trevisan, we show that this size loss is necessary and in fact, the parameters achieved by known proofs are the best possible.

Guy Blanc, Caleb Koch, Jane Lange et al.

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 how its landscape changes when the adversary is assumed to be efficient instead of computationally unbounded.

Caleb Koch, Carmen Strassle, Li-Yang Tan

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 the 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 XOR lemmas. Our work also carries new implications for the related problem of Decision Tree Minimization.

Caleb Koch, Carmen Strassle, Li-Yang Tan

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 $O(n/\log n)$ (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 $\textit{any}$ improvement of Ehrenfeucht and Haussler's algorithm will yield $O(\log n)$-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 $\textit{weak}$ learning whereas prior ones were limited to the setting of strong learning.

Guy Blanc, Caleb Koch, Carmen Strassle et al.

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 information-theoretically necessary. We base such tradeoffs on $\mathsf{NP}$-hardness and obtain: $\circ$ Sharp computational-statistical tradeoffs assuming $\mathsf{NP}$ requires exponential time: For every polynomial $p(n)$, there is an $n$-variate class $C$ with VC dimension $1$ such that the sample complexity of time-efficiently learning $C$ is $Θ(p(n))$. $\circ$ A characterization of $\mathsf{RP}$ vs. $\mathsf{NP}$ in terms of learning: $\mathsf{RP} = \mathsf{NP}$ iff every $\mathsf{NP}$-enumerable class is learnable with $O(\mathrm{VCdim}(C))$ 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 $\mathsf{NP}$-hardness results for improperly learning a subclass of polynomial-size circuits, circumventing formal barriers of Applebaum, Barak, and Xiao (2008).

Lorenzo Beretta, Nathaniel Harms, Caleb Koch

For a function $f \colon \{0,1\}^n \to \{0,1\}$, the junta testing problem asks whether $f$ depends on only $k$ variables. If $f$ depends on only $k$ variables, the feature selection problem asks to find those variables. We prove that these two tasks are statistically equivalent. Specifically, we show that the ``brute-force'' algorithm, which checks for any set of $k$ variables consistent with the sample, is simultaneously sample-optimal for both problems, and the optimal sample size is \[ Θ\left(\frac 1 \varepsilon \left( \sqrt{2^k \log {n \choose k}} + \log {n \choose k}\right)\right). \]

Andrew Crutcher, Caleb Koch, Kyle Coleman et al.

In recent studies, researchers have developed various computation offloading frameworks for bringing cloud services closer to the user via edge networks. Specifically, an edge device needs to offload computationally intensive tasks because of energy and processing constraints. These constraints present the challenge of identifying which edge nodes should receive tasks to reduce overall resource consumption. We propose a unique solution to this problem which incorporates elements from Knowledge-Defined Networking (KDN) to make intelligent predictions about offloading costs based on historical data. Each server instance can be represented in a multidimensional feature space where each dimension corresponds to a predicted metric. We compute features for a "hyperprofile" and position nodes based on the predicted costs of offloading a particular task. We then perform a k-Nearest Neighbor (kNN) query within the hyperprofile to select nodes for offloading computation. This paper formalizes our hyperprofile-based solution and explores the viability of using machine learning (ML) techniques to predict metrics useful for computation offloading. We also investigate the effects of using different distance metrics for the queries. Our results show various network metrics can be modeled accurately with regression, and there are circumstances where kNN queries using Euclidean distance as opposed to rectilinear distance is more favorable.