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1. Lower Bounds for Testing Complete Positivity and Quantum Separability

   https://doi.org/10.1007/978-3-030-61792-9_30  

   Bădescu, Costin ; O'Donnell, Ryan ( December 2020 , LATIN 2020: Theoretical Informatics)

   Kohayakawa, Y. ; Miyazawa, F.K. (Ed.)

   In this work we are interested in the problem of testing quantum entanglement. More specifically, we study the separability problem in quantum property testing, where one is given n copies of an unknown mixed quantum state ϱ on Cd⊗Cd , and one wants to test whether ϱ is separable or ϵ -far from all separable states in trace distance. We prove that n=Ω(d2/ϵ2) copies are necessary to test separability, assuming ϵ is not too small, viz. ϵ=Ω(1/d−−√) . We also study completely positive distributions on the grid [d]×[d] , as a classical analogue of separable states. We analogously prove that Ω(d/ϵ2) samples from an unknown distribution p are necessary to decide whether p is completely positive or ϵ -far from all completely positive distributions in total variation distance. 

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2. Estimating Learnability in the Sublinear Data Regime

   Kong, Weihao ; Valiant, Gregory ( January 2018 , Advances in neural information processing systems)

   We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is possible to accurately estimate this “learnability” even when given an amount of data that is too small to reliably learn any accurate model. Our first result applies to the setting where the data is drawn from a d-dimensional distribution with isotropic covariance, and the label of each datapoint is an arbitrary noisy function of the datapoint. In this setting, we show that with O(sqrt(d)) samples, one can accurately estimate the fraction of the variance of the label that can be explained via the best linear function of the data. For comparison, even if the labels are noiseless linear functions of the data, a sample size linear in the dimension, d, is required to learn any function correlated with the underlying model. Our estimation approach also applies to the setting where the data distribution has an (unknown) arbitrary covariance matrix, allowing these techniques to be applied to settings where the model class consists of a linear function applied to a nonlinear embedding of the data. In this setting we give a consistent estimator of the fraction of explainable variance that uses o(d) samples. Finally, our techniques also extend to the setting of binary classification, where we obtain analogous results under the logistic model, for estimating the classification accuracy of the best linear classifier. We demonstrate the practical viability of our approaches on synthetic and real data. This ability to estimate the explanatory value of a set of features (or dataset), even in the regime in which there is too little data to realize that explanatory value, may be relevant to the scientific and industrial settings for which data collection is expensive and there are many potentially relevant feature sets that could be collected. 

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3. Algorithm Selection for Deep Active Learning with Imbalanced Datasets

   Zhang, Jifan ; Shao, Shuai ; Verma, Saurabh ; Nowak, Robert ( September 2023 , Advances in Neural Information Processing Systems)

   Label efficiency has become an increasingly important objective in deep learning applications. Active learning aims to reduce the number of labeled examples needed to train deep networks, but the empirical performance of active learning algorithms can vary dramatically across datasets and applications. It is difficult to know in advance which active learning strategy will perform well or best in a given application. To address this, we propose the first adaptive algorithm selection strategy for deep active learning. For any unlabeled dataset, our (meta) algorithm TAILOR (Thompson ActIve Learning algORithm selection) iteratively and adap- tively chooses among a set of candidate active learning algorithms. TAILOR uses novel reward functions aimed at gathering class-balanced examples. Extensive experiments in multi-class and multi-label applications demonstrate TAILOR ’s effectiveness in achieving accuracy comparable or better than that of the best of the candidate algorithms. Our implementation of TAILOR is open-sourced at https://github.com/jifanz/TAILOR. 

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4. Testing k-Monotonicity

   Canonne, C ; Grigorescu, E. ; Guo, S ; Kumar, A ; Wimmer, K. ( January 2017 , ITCS)

   A Boolean {\em $k$-monotone} function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $1$-monotone} functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following: \begin{enumerate} \item We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; \item We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$ queries, while learning $k$-monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$ queries (Blais et al. (RANDOM 2015)). \item We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$, which makes progress on a problem left open by Berman et al. (STOC 2014). \end{enumerate} Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques. 

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5. Time/Accuracy Tradeoffs for learning a ReLU with respect to Gaussian Marginals

   Goel, Surbhi ; Karmalkar, Sushrut ; Klivans, Adam ( January 2019 , Advances in neural information processing systems)

   We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let 𝗈𝗉𝗍<1 be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss 𝗈𝗉𝗍+ϵ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time. 2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error O(𝗈𝗉𝗍^{2/3}). The algorithm uses a novel reduction to noisy halfspace learning with respect to 0/1 loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU. 

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