Search for: All records

We study local filters for the Lipschitz property of real-valued functions f : V → [0,r], where the Lipschitz property is defined with respect to an arbitrary undirected graph G = (V, E ). We give nearly optimal local Lipschitz filters both with respect to ℓ1-distance and ℓ0-distance. Previous work only considered unbounded- range functions over [n]d. Jha and Raskhodnikova (SICOMP ‘13) gave an algorithm for such functions with lookup complexity exponential in d, which Awasthi et al. (ACM Trans. Comput. Theory) showed was necessary in this setting. We demonstrate that important applications of local Lipschitz filters can be accomplished with filters for functions whose range is bounded in [0,r]. For functions f : [n]d → [0,r], we achieve running time (dr log n )O (log r ) for the ℓ1-respecting filter and dO(r) polylog n for the ℓ0-respecting filter, thus circumventing the lower bound. Our local filters provide a novel Lipschitz extension that can be implemented locally. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on r for the domain {0,1}d, an important special case of the domain [n]d. In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing and a new technique for proving hardness for adaptive algorithms. Finally, we provide two applications of our local filters to real-valued functions, with no restrictions on the range. In the first application, we use them in conjunction with the Laplace mechanism for differential privacy and noisy binary search to provide mechanisms for privately releasing outputs of black-box functions, even in the presence of malicious clients. In particular, our differentially private mechanism for arbitrary real-valued functions runs in time 2polylog min(r,nd ) and, for honest clients, has accuracy comparable to the Laplace mechanism for Lipschitz functions, up to a factor of O (log min(r,nd )). In the second application, we use our local filters to obtain the first nontrivial tolerant tester for the Lipschitz property. Our tester works for functions of the form f : {0,1}d → ℝ, makes queries, and has tolerance ratio 2.01. Our applications demonstrate that local filters for bounded-range functions can be applied to construct efficient algorithms for arbitrary real-valued functions. 

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 ρ↦Tr(OE[ρ]) 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 nO(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. 

We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given 2O~(n√/ε) uniformly random examples of an unknown function f:{±1}n→{±1}, our algorithm outputs a hypothesis g:{±1}n→{±1} that is monotone and (opt +ε)-close to f, where 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 2O~(n√/ε), nearly matching the lower bound of [13]. We also give an algorithm for estimating up to additive error ε the distance of an unknown function f to monotone using a run-time of 2O~(n√/ε). 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 [17] and the proper semiagnostic learning algorithm of [40], 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 opt +ε information-theoretically; we bypass this barrier bya)augmenting the improper learner with a convex optimization step, andb)learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the “poset sorting” problem of [40] for functions over general posets with non-Boolean labels. 

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 ‍D. The efficiency of our transformation scales with the inherent complexity of ‍D, running in (n, (md)d) time for distributions over 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 ‍D, and for general ones it uses subcube conditioning samples. A key technical ingredient is an algorithm which, given the aforementioned access to D, produces an optimal decision tree decomposition of D: an approximation of 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. 

In the certification problem, the algorithm is given a function f with certificate complexity k and an input x^⋆, and the goal is to find a certificate of size ≤ poly(k) for f’s value at x^⋆. This problem is in NP^NP, and assuming 𝖯 ≠ NP, is not in 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 BPP^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 BPP^NP algorithms are known for the latter task. We then consider certification with stricter instance-wise guarantees: for each x^⋆, find a certificate whose size scales with that of the smallest certificate for x^⋆. In sharp contrast with our first result, we show that this problem is NP^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 give an nO(log log n)-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1}n. Even in the realizable setting, the previous fastest runtime was nO(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.