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In this paper we apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good $\ell_p$-approximation for $p<2$. It follows that the first $n$ functions of Walsh system can be approximated with an error $n^{-\delta}$ by a linear space of dimension $n^{1-\delta}$: $$ d_{n^{1-\delta}}(\{w_1,\ldots,w_n\}, L_p[0,1]) \le n^{-\delta},\quad p\in[1,2),\;\delta=\delta(p)>0. $$ We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon--Frankl--R\"odl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be $\ell_1$-approximated by low-rank tensors. This implies lower bounds for $\Theta_m$~ -- the error of $m$-term approximation of multivariate functions by sums of tensor products $u^1(x_1)\cdots u^d(x_d)$. In particular, for the set of trigonometric polynomials with spectrum in $\prod_{j=1}^d[-n_j,n_j]$ and of norm $\|t\|_\infty\le 1$ we have $$ \Theta_m(\mathcal T(n_1,\ldots,n_d)_\infty,L_1[-\pi,\pi]^d) \ge c_1(d)>0,\quad m\le c_2(d)\frac{\prod n_j}{\max\{n_j\}}. $$ Sharp bounds follow for classes of dominated mixed smoothness: $$ \Theta_m(W^{(r,r,\ldots,r)}_p,L_q[0,1]^d)\asymp m^{-\frac{rd}{d-1}},\quad\mbox 2\le p\le\infty,\; 1\le q\le 2. $$

Estimating the quantiles of a large dataset is a fundamental problem in both the streaming algorithms literature and the differential privacy literature. However, all existing private mechanisms for distribution-independent quantile computation require space at least linear in the input size $n$. In this work, we devise a differentially private algorithm for the quantile estimation problem, with strongly sublinear space complexity, in the one-shot and continual observation settings. Our basic mechanism estimates any $\alpha$-approximate quantile of a length-$n$ stream over a data universe $\mathcal{X}$ with probability $1-\beta$ using $O\left( \frac{\log (|\mathcal{X}|/\beta) \log (\alpha \epsilon n)}{\alpha \epsilon} \right)$ space while satisfying $\epsilon$-differential privacy at a single time point. Our approach builds upon deterministic streaming algorithms for non-private quantile estimation instantiating the exponential mechanism using a utility function defined on sketch items, while (privately) sampling from intervals defined by the sketch. We also present another algorithm based on histograms that is especially suited to the multiple quantiles case. We implement our algorithms and experimentally evaluate them on synthetic and real-world datasets.

We consider a platform's problem of collecting data from privacy sensitive users to estimate an underlying parameter of interest. We formulate this question as a Bayesian-optimal mechanism design problem, in which an individual can share her (verifiable) data in exchange for a monetary reward or services, but at the same time has a (private) heterogeneous privacy cost which we quantify using differential privacy. We consider two popular differential privacy settings for providing privacy guarantees for the users: central and local. In both settings, we establish minimax lower bounds for the estimation error and derive (near) optimal estimators for given heterogeneous privacy loss levels for users. Building on this characterization, we pose the mechanism design problem as the optimal selection of an estimator and payments that will elicit truthful reporting of users' privacy sensitivities. Under a regularity condition on the distribution of privacy sensitivities we develop efficient algorithmic mechanisms to solve this problem in both privacy settings. Our mechanism in the central setting can be implemented in time $\mathcal{O}(n \log n)$ where $n$ is the number of users and our mechanism in the local setting admits a Polynomial Time Approximation Scheme (PTAS).

We consider the general problem of online convex optimization with time-varying additive constraints in the presence of predictions for the next cost and constraint functions. A novel primal-dual algorithm is designed by combining a Follow-The-Regularized-Leader iteration with prediction-adaptive dynamic steps. The algorithm achieves $\mathcal O(T^{\frac{3-\beta}{4}})$ regret and $\mathcal O(T^{\frac{1+\beta}{2}})$ constraint violation bounds that are tunable via parameter $\beta\!\in\![1/2,1)$ and have constant factors that shrink with the predictions quality, achieving eventually $\mathcal O(1)$ regret for perfect predictions. Our work extends the FTRL framework for this constrained OCO setting and outperforms the respective state-of-the-art greedy-based solutions, without imposing conditions on the quality of predictions, the cost functions or the geometry of constraints, beyond convexity.

The goal of regression is to recover an unknown underlying function that best links a set of predictors to an outcome from noisy observations. In nonparametric regression, one assumes that the regression function belongs to a pre-specified infinite-dimensional function space (the hypothesis space). In the online setting, when the observations come in a stream, it is computationally-preferable to iteratively update an estimate rather than refitting an entire model repeatedly. Inspired by nonparametric sieve estimation and stochastic approximation methods, we propose a sieve stochastic gradient descent estimator (Sieve-SGD) when the hypothesis space is a Sobolev ellipsoid. We show that Sieve-SGD has rate-optimal mean squared error (MSE) under a set of simple and direct conditions. The proposed estimator can be constructed with a low computational (time and space) expense: We also formally show that Sieve-SGD requires almost minimal memory usage among all statistically rate-optimal estimators.

Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is the asymptotic behavior of data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions. We analyze the properties of the new estimators and we establish strong law of large numbers, consistency, and bias reduction potential under fairly general assumptions. Our results are germane to risk-averse optimization and to data science in general.

We study the class of first-order locally-balanced Metropolis--Hastings algorithms introduced in Livingstone & Zanella (2021). To choose a specific algorithm within the class the user must select a balancing function $g:\mathbb{R} \to \mathbb{R}$ satisfying $g(t) = tg(1/t)$, and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a universal limiting optimal acceptance rate of 57% and scaling of $n^{-1/3}$ as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of the product form. In particular we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimise this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, optimal choice of balancing function under a Gaussian noise distribution, and optimal choice of first-order locally-balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings and in particular show that a bi-modal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.

Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.