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In supervised learning using kernel methods, we often encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space (RKHS). Large-scale finite-sum problems can be solved using efficient variants of Newton method, where the Hessian is approximated via sub-samples of data. In RKHS, however, the dependence of the penalty function to kernel makes standard sub-sampling approaches inapplicable, since the gram matrix is not readily available in a low-rank form. In this paper, we observe that for this class of problems, one can naturally use kernel approximation to speed up the Newton method. Focusing on randomized features for kernel approximation, we provide a novel second-order algorithm that enjoys local superlinear convergence and global linear convergence (with high probability). We derive the theoretical lower bound for the number of random features required for the approximated Hessian to be close to the true Hessian in the norm sense. Our numerical experiments on real-world data verify the efficiency of our method compared to several benchmarks.

This paper investigates the problem of regret minimization in linear time-varying (LTV) dynamical systems. Due to the simultaneous presence of uncertainty and non-stationarity, designing online control algorithms for unknown LTV systems remains a challenging task. At a cost of NP-hard offline planning, prior works have introduced online convex optimization algorithms, although they suffer from nonparametric rate of regret. In this paper, we propose the first computationally tractable online algorithm with regret guarantees that avoids offline planning over the state linear feedback policies. Our algorithm is based on the optimism in the face of uncertainty (OFU) principle in which we optimistically select the best model in a high confidence region. Our algorithm is then more explorative when compared to previous approaches. To overcome non-stationarity, we propose either a restarting strategy (R-OFU) or a sliding window (SW-OFU) strategy. With proper configuration, our algorithm is attains sublinear regret $O(T^{2/3})$. These algorithms utilize data from the current phase for tracking variations on the system dynamics. We corroborate our theoretical findings with numerical experiments, which highlight the effectiveness of our methods. To the best of our knowledge, our study establishes the first model-based online algorithm with regret guarantees under LTV dynamical systems.

In this paper, we study the trace regression when a matrix of parameters B* is estimated via convex relaxation of a rank-penalized regression or via non-convex optimization. It is known that these estimators satisfy near-optimal error bounds under assumptions on rank, coherence, or spikiness of B*. We start by introducing a general notion of spikiness for B* that provides a generic recipe to prove restricted strong convexity for the sampling operator of the trace regression and obtain near-optimal and non-asymptotic error bounds for the estimation error. Similar to the existing literature, these results require the penalty parameter to be above a certain theory-inspired threshold that depends on the observation noise and the sampling operator which may be unknown in practice. Next, we extend the error bounds to the cases when the regularization parameter is chosen via cross-validation. This result is significant in that existing theoretical results on cross-validated estimators do not apply to our setting since the estimators we study are not known to satisfy their required notion of stability. Finally, using simulations on synthetic and real data, we show that the cross-validated estimator selects a nearly-optimal penalty parameter and outperforms the theory-inspired approach of selecting the parameter.

Evolutionary algorithms are bio-inspired algorithms that can easily adapt to changing environments. Recent results in the area of runtime analysis have pointed out that algorithms such as the (1+1)~EA and Global SEMO can efficiently reoptimize linear functions under a dynamic uniform constraint. Motivated by this study, we investigate single- and multi-objective baseline evolutionary algorithms for the classical knapsack problem where the capacity of the knapsack varies over time. We establish different benchmark scenarios where the capacity changes every $\tau$ iterations according to a uniform or normal distribution. Our experimental investigations analyze the behavior of our algorithms in terms of the magnitude of changes determined by parameters of the chosen distribution, the frequency determined by $\tau$, and the class of knapsack instance under consideration. Our results show that the multi-objective approaches using a population that caters for dynamic changes have a clear advantage on many benchmarks scenarios when the frequency of changes is not too high. Furthermore, we demonstrate that the diversity mechanisms used in popular evolutionary multi-objective algorithms such as NSGA-II and SPEA2 do not necessarily result in better performance and even lead to inferior results compared to our simple multi-objective approaches.

In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. For the sample size $n$, the step size $\eta$, and the batch size $B$, we derive uniform evaluations on the time with orders $n^{-1}$, $\eta^{1/2}$, and $\sqrt{(n - B) / B (n - 1)}$, respectively.

This paper proposes a new feature screening method for the multi-response ultrahigh dimensional linear model by empirical likelihood. Through a multivariate moment condition, the empirical likelihood induced ranking statistics can exploit the joint effect among responses, and thus result in a much better performance than the methods considering responses individually. More importantly, by the use of empirical likelihood, the new method adapts to the heterogeneity in the conditional variance of random error. The sure screening property of the newly proposed method is proved with the model size controlled within a reasonable scale. Additionally, the new screening method is also extended to a conditional version so that it can recover the hidden predictors which are easily missed by the unconditional method. The corresponding theoretical properties are also provided. Finally, both numerical studies and real data analysis are provided to illustrate the effectiveness of the proposed methods.

This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large P\'eclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the $L^2$-norm, the Galerkin projection onto this generalized finite element space even yields $\varepsilon$-independent error bounds, $\varepsilon$ being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate $\varepsilon$-independent convergence without preasymptotic effects even in the under-resolved regime of large mesh P\'eclet numbers.

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

Randomized smoothing is the dominant standard for provable defenses against adversarial examples. Nevertheless, this method has recently been proven to suffer from important information theoretic limitations. In this paper, we argue that these limitations are not intrinsic, but merely a byproduct of current certification methods. We first show that these certificates use too little information about the classifier, and are in particular blind to the local curvature of the decision boundary. This leads to severely sub-optimal robustness guarantees as the dimension of the problem increases. We then show that it is theoretically possible to bypass this issue by collecting more information about the classifier. More precisely, we show that it is possible to approximate the optimal certificate with arbitrary precision, by probing the decision boundary with several noise distributions. Since this process is executed at certification time rather than at test time, it entails no loss in natural accuracy while enhancing the quality of the certificates. This result fosters further research on classifier-specific certification and demonstrates that randomized smoothing is still worth investigating. Although classifier-specific certification may induce more computational cost, we also provide some theoretical insight on how to mitigate it.

In this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an $\varepsilon$-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose an first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for finding an $\varepsilon$-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are parameter-free or almost parameter-free except that the knowledge on convexity parameter is required. To the best of our knowledge, no prior studies were conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are entirely new.