The mutation strength adaptation properties of a multi-recombinative $(\mu/\mu_I, \lambda)$-ES are studied for isotropic mutations. To this end, standard implementations of cumulative step-size adaptation (CSA) and mutative self-adaptation ($\sigma$SA) are investigated experimentally and theoretically by assuming large population sizes ($\mu$) in relation to the search space dimensionality ($N$). The adaptation is characterized in terms of the scale-invariant mutation strength on the sphere in relation to its maximum achievable value for positive progress. %The results show how the different $\sigma$-adaptation variants behave as $\mu$ and $N$ are varied. Standard CSA-variants show notably different adaptation properties and progress rates on the sphere, becoming slower or faster as $\mu$ or $N$ are varied. This is shown by investigating common choices for the cumulation and damping parameters. Standard $\sigma$SA-variants (with default learning parameter settings) can achieve faster adaptation and larger progress rates compared to the CSA. However, it is shown how self-adaptation affects the progress rate levels negatively. Furthermore, differences regarding the adaptation and stability of $\sigma$SA with log-normal and normal mutation sampling are elaborated.
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The $k$-center problem is a classical clustering problem in which one is asked to find a partitioning of a point set $P$ into $k$ clusters such that the maximum radius of any cluster is minimized. It is well-studied. But what if we add up the radii of the clusters instead of only considering the cluster with maximum radius? This natural variant is called the $k$-min-sum-radii problem. It has become the subject of more and more interest in recent years, inspiring the development of approximation algorithms for the $k$-min-sum-radii problem in its plain version as well as in constrained settings. We study the problem for Euclidean spaces $\mathbb{R}^d$ of arbitrary dimension but assume the number $k$ of clusters to be constant. In this case, a PTAS for the problem is known (see Bandyapadhyay, Lochet and Saurabh, SoCG, 2023). Our aim is to extend the knowledge base for $k$-min-sum-radii to the domain of fair clustering. We study several group fairness constraints, such as the one introduced by Chierichetti et al. (NeurIPS, 2017). In this model, input points have an additional attribute (e.g., colors such as red and blue), and clusters have to preserve the ratio between different attribute values (e.g., have the same fraction of red and blue points as the ground set). Different variants of this general idea have been studied in the literature. To the best of our knowledge, no approximative results for the fair $k$-min-sum-radii problem are known, despite the immense amount of work on the related fair $k$-center problem. We propose a PTAS for the fair $k$-min-sum-radii problem in Euclidean spaces of arbitrary dimension for the case of constant $k$. To the best of our knowledge, this is the first PTAS for the problem. It works for different notions of group fairness.
We propose a randomized multiplicative weight update (MWU) algorithm for $\ell_{\infty}$ regression that runs in $\widetilde{O}\left(n^{2+1/22.5} \text{poly}(1/\epsilon)\right)$ time when $\omega = 2+o(1)$, improving upon the previous best $\widetilde{O}\left(n^{2+1/18} \text{poly} \log(1/\epsilon)\right)$ runtime in the low-accuracy regime. Our algorithm combines state-of-the-art inverse maintenance data structures with acceleration. In order to do so, we propose a novel acceleration scheme for MWU that exhibits {\it stabiliy} and {\it robustness}, which are required for the efficient implementations of the inverse maintenance data structures. We also design a faster {\it deterministic} MWU algorithm that runs in $\widetilde{O}\left(n^{2+1/12}\text{poly}(1/\epsilon)\right))$ time when $\omega = 2+o(1)$, improving upon the previous best $\widetilde{O}\left(n^{2+1/6} \text{poly} \log(1/\epsilon)\right)$ runtime in the low-accuracy regime. We achieve this by showing a novel stability result that goes beyond the previous known works based on interior point methods (IPMs). Our work is the first to use acceleration and inverse maintenance together efficiently, finally making the two most important building blocks of modern structured convex optimization compatible.
The Check tools automate formal memory consistency model and security verification of processors by analyzing abstract models of microarchitectures, called $\mu$SPEC models. Despite the efficacy of this approach, a verification gap between $\mu$SPEC models, which must be manually written, and RTL limits the Check tools' broad adoption. Our prior work, called RTL2$\mu$SPEC, narrows this gap by automatically synthesizing formally verified $\mu$SPEC models from SystemVerilog implementations of simple processors. But, RTL2$\mu$SPEC assumes input designs where an instruction (e.g., a load) cannot exhibit more than one microarchitectural execution path ($\mu$PATH, e.g., a cache hit or miss path) -- its single-execution-path assumption. In this paper, we first propose an automated approach and tool, called RTL2M$\mu$PATH, that resolves RTL2$\mu$SPEC's single-execution-path assumption. Given a SystemVerilog processor design, instruction encodings, and modest design metadata, RTL2M$\mu$PATH finds a complete set of formally verified $\mu$PATHs for each instruction. Next, we make an important observation: an instruction that can exhibit more than one $\mu$PATH strongly indicates the presence of a microarchitectural side channel in the input design. Based on this observation, we then propose an automated approach and tool, called SynthLC, that extends RTL2M$\mu$PATH with a symbolic information flow analysis to support synthesizing a variety of formally verified leakage contracts from SystemVerilog processor designs. Leakage contracts are foundational to state-of-the-art defenses against hardware side-channel attacks. SynthLC is the first automated methodology for formally verifying hardware adherence to them.
Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for a popular SDE-based sampler under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. To our knowledge, this improves upon existing convergence theory for both the SDE-based sampler and another ODE-based sampler, while imposing minimal assumptions on the target data distribution and score estimates. This is achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.
Hypothesis tests calibrated by (re)sampling methods (such as permutation, rank and bootstrap tests) are useful tools for statistical analysis, at the computational cost of requiring Monte-Carlo sampling for calibration. It is common and almost universal practice to execute such tests with predetermined and large number of Monte-Carlo samples, and disregard any randomness from this sampling at the time of drawing and reporting inference. At best, this approach leads to computational inefficiency, and at worst to invalid inference. That being said, a number of approaches in the literature have been proposed to adaptively guide analysts in choosing the number of Monte-Carlo samples, by sequentially deciding when to stop collecting samples and draw inference. These works introduce varying competing notions of what constitutes "valid" inference, complicating the landscape for analysts seeking suitable methodology. Furthermore, the majority of these approaches solely guarantee a meaningful estimate of the testing outcome, not the $p$-value itself $\unicode{x2014}$ which is insufficient for many practical applications. In this paper, we survey the relevant literature, and build bridges between the scattered validity notions, highlighting some of their complementary roles. We also introduce a new practical methodology that provides an estimate of the $p$-value of the Monte-Carlo test, endowed with practically relevant validity guarantees. Moreover, our methodology is sequential, updating the $p$-value estimate after each new Monte-Carlo sample has been drawn, while retaining important validity guarantees regardless of the selected stopping time. We conclude this paper with a set of recommendations for the practitioner, both in terms of selection of methodology and manner of reporting results.
We derive a robust error estimate for a recently proposed numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation, where $\alpha \in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ and $\beta \in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^{\beta}$ for all $h \in (0, 2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(\Delta x^{\frac{\beta}{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $\alpha=0$, then the rate improves to $O(\Delta x^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(\Delta x^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.
We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights $\gamma_1,\gamma_2,\ldots\in [0,1]$. Building upon the deterministic lattice algorithm by Kuo, Sloan, and Wo\'{z}niakowski (2006), we incorporate a randomized quadrature rule by Dick, Goda, and Suzuki (2022) to accelerate the convergence rate. This randomization involves drawing the number of points for function evaluations randomly, and selecting a good generating vector for rank-1 lattice points using the randomized component-by-component algorithm. We prove that our randomized algorithm achieves a worst-case root mean squared $L_2$-approximation error of order $M^{-\alpha/2 - 1/8 + \varepsilon}$ for an arbitrarily small $\varepsilon > 0$, where $M$ denotes the maximum number of function evaluations, and that the error bound is independent of the dimension $d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha} < \infty$. Our upper bound converges faster than a lower bound on the worst-case $L_2$-approximation error for deterministic rank-1 lattice-based approximation proved by Byrenheid, K\"{a}mmerer, Ullrich, and Volkmer (2017). We also show a lower error bound of order $M^{-\alpha/2-1/2}$ for our randomized algorithm, leaving a slight gap between the upper and lower bounds open for future research.
We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family. We propose certain conditions on the forward map $\mathcal{G}$, the variational set $\mathcal{Q}$ and the prior such that, as the number $N$ of measurements increases, the resulting variational posterior distributions contract to the ground truth $f_0$ generating the data, and derive a convergence rate with polynomial order or logarithmic order. As specific examples, we consider a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.
For any $\varepsilon > 0$, we prove that $k$-Dimensional Matching is hard to approximate within a factor of $k/(12 + \varepsilon)$ for large $k$ unless $\textsf{NP} \subseteq \textsf{BPP}$. Listed in Karp's 21 $\textsf{NP}$-complete problems, $k$-Dimensional Matching is a benchmark computational complexity problem which we find as a special case of many constrained optimization problems over independence systems including: $k$-Set Packing, $k$-Matroid Intersection, and Matroid $k$-Parity. For all the aforementioned problems, the best known lower bound was a $\Omega(k /\log(k))$-hardness by Hazan, Safra, and Schwartz. In contrast, state-of-the-art algorithms achieved an approximation of $O(k)$. Our result narrows down this gap to a constant and thus provides a rationale for the observed algorithmic difficulties. The crux of our result hinges on a novel approximation preserving gadget from $R$-degree bounded $k$-CSPs over alphabet size $R$ to $kR$-Dimensional Matching. Along the way, we prove that $R$-degree bounded $k$-CSPs over alphabet size $R$ are hard to approximate within a factor $\Omega_k(R)$ using known randomised sparsification methods for CSPs.
A posteriori error analysis of the virtual element method for second-order quasilinear elliptic PDEs
In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.
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