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We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo m and does not require factorization of m. For smooth m, i.e., when the prime factors of m are of size poly(log m), the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general m, the algorithm of Mosca and Zalka is available. Even for m=3 and k=1 our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as m. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.

In high-dimensional regression, we attempt to estimate a parameter vector $\beta_0\in\mathbb{R}^p$ from $n\lesssim p$ observations $\{(y_i,x_i)\}_{i\leq n}$ where $x_i\in\mathbb{R}^p$ is a vector of predictors and $y_i$ is a response variable. A well-established approach uses convex regularizers to promote specific structures (e.g. sparsity) of the estimate $\widehat{\beta}$, while allowing for practical algorithms. Theoretical analysis implies that convex penalization schemes have nearly optimal estimation properties in certain settings. However, in general the gaps between statistically optimal estimation (with unbounded computational resources) and convex methods are poorly understood. We show that when the statistican has very simple structural information about the distribution of the entries of $\beta_0$, a large gap frequently exists between the best performance achieved by any convex regularizer satisfying a mild technical condition and either (i) the optimal statistical error or (ii) the statistical error achieved by optimal approximate message passing algorithms. Remarkably, a gap occurs at high enough signal-to-noise ratio if and only if the distribution of the coordinates of $\beta_0$ is not log-concave. These conclusions follow from an analysis of standard Gaussian designs. Our lower bounds are expected to be generally tight, and we prove tightness under certain conditions.

It is common to split a dataset into training and testing sets before fitting a statistical or machine learning model. However, there is no clear guidance on how much data should be used for training and testing. In this article we show that the optimal splitting ratio is $\sqrt{p}:1$, where $p$ is the number of parameters in a linear regression model that explains the data well.

We give the first almost optimal polynomial-time proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. For $s$-sparse polynomial over $n$ variables and $\epsilon=1/s^\beta$, $\beta>1$, our algorithm makes $$q_U=\left(\frac{s}{\epsilon}\right)^{\frac{\log \beta}{\beta}+O(\frac{1}{\beta})}+ \tilde O\left(s\right)\left(\log\frac{1}{\epsilon}\right)\log n$$ queries. Notice that our query complexity is sublinear in $1/\epsilon$ and almost linear in $s$. All previous algorithms have query complexity at least quadratic in $s$ and linear in $1/\epsilon$. We then prove the almost tight lower bound $$q_L=\left(\frac{s}{\epsilon}\right)^{\frac{\log \beta}{\beta}+\Omega(\frac{1}{\beta})}+ \Omega\left(s\right)\left(\log\frac{1}{\epsilon}\right)\log n,$$ Applying the reduction in~\cite{Bshouty19b} with the above algorithm, we give the first almost optimal polynomial-time tester for $s$-sparse polynomial. Our tester, for $\beta>3.404$, makes $$\tilde O\left(\frac{s}{\epsilon}\right)$$ queries.

In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the sampling strategy affects the sample complexity -- that is, the number of samples that suffice for accurate and stable recovery -- and to use this insight to obtain optimal or near-optimal sampling procedures. We consider two settings. First, when a target sparse representation is known, in which case we present a near-complete answer based on drawing independent random samples from carefully-designed probability measures. Second, we consider the more challenging scenario when such representation is unknown. In this case, while not giving a full answer, we describe a general construction of sampling measures that improves over standard Monte Carlo sampling. We present examples using algebraic and trigonometric polynomials, and for the former, we also introduce a new procedure for function approximation on irregular (i.e., nontensorial) domains. The effectiveness of this procedure is shown through numerical examples. Finally, we discuss a number of structured sparsity models, and how they may lead to better approximations.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.

Train machine learning models on sensitive user data has raised increasing privacy concerns in many areas. Federated learning is a popular approach for privacy protection that collects the local gradient information instead of real data. One way to achieve a strict privacy guarantee is to apply local differential privacy into federated learning. However, previous works do not give a practical solution due to three issues. First, the noisy data is close to its original value with high probability, increasing the risk of information exposure. Second, a large variance is introduced to the estimated average, causing poor accuracy. Last, the privacy budget explodes due to the high dimensionality of weights in deep learning models. In this paper, we proposed a novel design of local differential privacy mechanism for federated learning to address the abovementioned issues. It is capable of making the data more distinct from its original value and introducing lower variance. Moreover, the proposed mechanism bypasses the curse of dimensionality by splitting and shuffling model updates. A series of empirical evaluations on three commonly used datasets, MNIST, Fashion-MNIST and CIFAR-10, demonstrate that our solution can not only achieve superior deep learning performance but also provide a strong privacy guarantee at the same time.

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.

In this paper we discuss policy iteration methods for approximate solution of a finite-state discounted Markov decision problem, with a focus on feature-based aggregation methods and their connection with deep reinforcement learning schemes. We introduce features of the states of the original problem, and we formulate a smaller "aggregate" Markov decision problem, whose states relate to the features. The optimal cost function of the aggregate problem, a nonlinear function of the features, serves as an architecture for approximation in value space of the optimal cost function or the cost functions of policies of the original problem. We discuss properties and possible implementations of this type of aggregation, including a new approach to approximate policy iteration. In this approach the policy improvement operation combines feature-based aggregation with reinforcement learning based on deep neural networks, which is used to obtain the needed features. We argue that the cost function of a policy may be approximated much more accurately by the nonlinear function of the features provided by aggregation, than by the linear function of the features provided by deep reinforcement learning, thereby potentially leading to more effective policy improvement.