Jane Lange

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, Jane Lange, Ali Malik et al.

Using the framework of boosting, we prove that all impurity-based decision tree learning algorithms, including the classic ID3, C4.5, and CART, are highly noise tolerant. Our guarantees hold under the strongest noise model of nasty noise, and we provide near-matching upper and lower bounds on the allowable noise rate. We further show that these algorithms, which are simple and have long been central to everyday machine learning, enjoy provable guarantees in the noisy setting that are unmatched by existing algorithms in the theoretical literature on decision tree learning. Taken together, our results add to an ongoing line of research that seeks to place the empirical success of these practical decision tree algorithms on firm theoretical footing.

Guy Blanc, Jane Lange, Ali Malik et al.

We show how any PAC learning algorithm that works under the uniform distribution can be transformed, in a blackbox fashion, into one that works under an arbitrary and unknown distribution $\mathcal{D}$. The efficiency of our transformation scales with the inherent complexity of $\mathcal{D}$, running in $\mathrm{poly}(n, (md)^d)$ time for distributions over $\{\pm 1\}^n$ whose pmfs are computed by depth-$d$ decision trees, where $m$ is the sample complexity of the original algorithm. For monotone distributions our transformation uses only samples from $\mathcal{D}$, and for general ones it uses subcube conditioning samples. A key technical ingredient is an algorithm which, given the aforementioned access to $\mathcal{D}$, produces an optimal decision tree decomposition of $\mathcal{D}$: an approximation of $\mathcal{D}$ as a mixture of uniform distributions over disjoint subcubes. With this decomposition in hand, we run the uniform-distribution learner on each subcube and combine the hypotheses using the decision tree. This algorithmic decomposition lemma also yields new algorithms for learning decision tree distributions with runtimes that exponentially improve on the prior state of the art -- results of independent interest in distribution learning.

Sitan Chen, Jaume de Dios Pont, Jun-Ting Hsieh et al.

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $E$ and an observable $O$, we aim to learn the mapping \begin{equation*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most $ρ$ sampled from a distribution $D$. Previously, Huang, Chen, and Preskill proved a surprising result that even if $E$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/ε))}$, where $ε$ is the target prediction error. However, their guarantee applied only to input distributions $D$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $ρ$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $D$, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.

Guy Blanc, Jane Lange, Mingda Qiao et al.

The authors recently gave an $n^{O(\log\log n)}$ time membership query algorithm for properly learning decision trees under the uniform distribution (Blanc et al., 2021). The previous fastest algorithm for this problem ran in $n^{O(\log n)}$ time, a consequence of Ehrenfeucht and Haussler (1989)'s classic algorithm for the distribution-free setting. In this article we highlight the natural open problem of obtaining a polynomial-time algorithm, discuss possible avenues towards obtaining it, and state intermediate milestones that we believe are of independent interest.

Jane Lange, Arsen Vasilyan

We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$ uniformly random examples of an unknown function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$, our algorithm outputs a hypothesis $g:\{\pm 1\}^n \rightarrow \{\pm 1\}$ that is monotone and $(\mathrm{opt} + \varepsilon)$-close to $f$, where $\mathrm{opt}$ is the distance from $f$ to the closest monotone function. The running time of the algorithm (and consequently the size and evaluation time of the hypothesis) is also $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$, nearly matching the lower bound of Blais et al (RANDOM '15). We also give an algorithm for estimating up to additive error $\varepsilon$ the distance of an unknown function $f$ to monotone using a run-time of $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$. Previously, for both of these problems, sample-efficient algorithms were known, but these algorithms were not run-time efficient. Our work thus closes this gap in our knowledge between the run-time and sample complexity. This work builds upon the improper learning algorithm of Bshouty and Tamon (JACM '96) and the proper semiagnostic learning algorithm of Lange, Rubinfeld, and Vasilyan (FOCS '22), which obtains a non-monotone Boolean-valued hypothesis, then ``corrects'' it to monotone using query-efficient local computation algorithms on graphs. This black-box correction approach can achieve no error better than $2\mathrm{opt} + \varepsilon$ information-theoretically; we bypass this barrier by a) augmenting the improper learner with a convex optimization step, and b) learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the ``poset sorting'' problem of [LRV22] for functions over general posets with non-Boolean labels.

Jane Lange, Mingda Qiao

Membership queries (MQ) often yield speedups for learning tasks, particularly in the distribution-specific setting. We show that in the \emph{testable learning} model of Rubinfeld and Vasilyan [RV23], membership queries cannot decrease the time complexity of testable learning algorithms beyond the complexity of sample-only distribution-specific learning. In the testable learning model, the learner must output a hypothesis whenever the data distribution satisfies a desired property, and if it outputs a hypothesis, the hypothesis must be near-optimal. We give a general reduction from sample-based \emph{refutation} of boolean concept classes, as presented in [Vadhan17, KL18], to testable learning with queries (TL-Q). This yields lower bounds for TL-Q via the reduction from learning to refutation given in [KL18]. The result is that, relative to a concept class and a distribution family, no $m$-sample TL-Q algorithm can be super-polynomially more time-efficient than the best $m$-sample PAC learner. Finally, we define a class of ``statistical'' MQ algorithms that encompasses many known distribution-specific MQ learners, such as those based on influence estimation or subcube-conditional statistical queries. We show that TL-Q algorithms in this class imply efficient statistical-query refutation and learning algorithms. Thus, combined with known SQ dimension lower bounds, our results imply that these efficient membership query learners cannot be made testable.

Guy Blanc, Jane Lange, Mingda Qiao et al.

We give an $n^{O(\log\log n)}$-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over $\{\pm 1\}^n$. Even in the realizable setting, the previous fastest runtime was $n^{O(\log n)}$, a consequence of a classic algorithm of Ehrenfeucht and Haussler. Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of O'Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be "pruned" so that every variable in the resulting tree is influential.

Guy Blanc, Jane Lange, Li-Yang Tan

We give the first {\sl reconstruction algorithm} for decision trees: given queries to a function $f$ that is $\mathrm{opt}$-close to a size-$s$ decision tree, our algorithm provides query access to a decision tree $T$ where: $\circ$ $T$ has size $S = s^{O((\log s)^2/\varepsilon^3)}$; $\circ$ $\mathrm{dist}(f,T)\le O(\mathrm{opt})+\varepsilon$; $\circ$ Every query to $T$ is answered with $\mathrm{poly}((\log s)/\varepsilon)\cdot \log n$ queries to $f$ and in $\mathrm{poly}((\log s)/\varepsilon)\cdot n\log n$ time. This yields a {\sl tolerant tester} that distinguishes functions that are close to size-$s$ decision trees from those that are far from size-$S$ decision trees. The polylogarithmic dependence on $s$ in the efficiency of our tester is exponentially smaller than that of existing testers. Since decision tree complexity is well known to be related to numerous other boolean function properties, our results also provide a new algorithms for reconstructing and testing these properties.