Deterministic Nonsmooth Nonconvex Optimization

We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(\delta,\epsilon)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $\tilde O(\delta^{-1}\epsilon^{-3})$ first-order oracle calls, independent of the dimension $d$. It has been an open problem as to whether a similar result can be obtained via a deterministic algorithm. We resolve this open problem, showing that randomization is necessary to obtain a dimension-free rate. In particular, we prove a lower bound of $\Omega(d)$ for any deterministic algorithm. Moreover, we show that unlike smooth or convex optimization, access to function values is required for any deterministic algorithm to halt within any finite time. On the other hand, we prove that if the function is even slightly smooth, then the dimension-free rate of $\tilde O(\delta^{-1}\epsilon^{-3})$ can be obtained by a deterministic algorithm with merely a logarithmic dependence on the smoothness parameter. Motivated by these findings, we turn to study the complexity of deterministically smoothing Lipschitz functions. Though there are efficient black-box randomized smoothings, we start by showing that no such deterministic procedure can smooth functions in a meaningful manner, resolving an open question. We then bypass this impossibility result for the structured case of ReLU neural networks. To that end, in a practical white-box setting in which the optimizer is granted access to the network's architecture, we propose a simple, dimension-free, deterministic smoothing that provably preserves $(\delta,\epsilon)$-stationary points. Our method applies to a variety of architectures of arbitrary depth, including ResNets and ConvNets. Combined with our algorithm, this yields the first deterministic dimension-free algorithm for optimizing ReLU networks, circumventing our lower bound.Comment: This work supersedes arxiv:2209.12463 and arxiv:2209.10346[Section 3], with major additional result

Meta-Analysis of Cyber Intimate Partner Violence Perpetration and Victimization: Different Types and their Associations with Face-to-Face IPV among Men and Women

Cyber intimate partner violence (C-IPV) is a technology-mediated form of violence. It has been examined only in the last 10 years as a form of violence that can cause psychological damage to its victims. How this phenomenon connects to and differs from face-to-face IPV (F2F-IPV) has been, as yet, little studied. Research has not made clear whether sex differences may impact its use, particularly in light of the fact that no physical coercion is used in C-IPV. Thus, the current research aimed to investigate through a meta-analysis: differences between the average levels of different types of C-IPV victimization and perpetration; the association between C-IPV and F2F-IPV victimization and perpetration; and whether the answers to these questions were dependent on sex. The current meta-analysis drew on 46 studies, within 44 papers, with a total sample of 27,491 participants. Findings from 22 of these studies showed no significant sex differences between the average levels of different types of C-IPV victimization and between different types of C-IPV perpetration. These 22 studies showed positive large effect sizes for the correlation between C-IPV and F2F-IPV perpetration and victimization. Moreover, in both perpetration and victimization, sex did not impact the level of association. The findings suggested that C-IPV and F2F-IPV are highly correlated, and though not the same, they may share similar characteristics. Additionally, the results suggested that sex differences do not impact non-physical aggression, such as C-IPV. The implications for preventive strategies include that IPV interventions should also focus on alleviating instances of C-IPV

Gemini Near Infrared Spectrograph - Distant Quasar Survey: Prescriptions for Calibrating UV-Based Estimates of Supermassive Black Hole Masses in High-Redshift Quasars

The most reliable single-epoch supermassive black hole mass ($M_{\rm BH}$) estimates in quasars are obtained by using the velocity widths of low-ionization emission lines, typically the H$\beta$ $\lambda4861$ line. Unfortunately, this line is redshifted out of the optical band at $z\approx1$, leaving $M_{\rm BH}$ estimates to rely on proxy rest-frame ultraviolet (UV) emission lines, such as C IV $\lambda1549$ or Mg II $\lambda2800$, which contain intrinsic challenges when measuring, resulting in uncertain $M_{\rm BH}$ estimates. In this work, we aim at correcting $M_{\rm BH}$ estimates derived from the C IV and Mg II emission lines based on estimates derived from the H$\beta$ emission line. We find that employing the equivalent width of C IV in deriving $M_{\rm BH}$ estimates based on Mg II and C IV provides values that are closest to those obtained from H$\beta$. We also provide prescriptions to estimate $M_{\rm BH}$ values when only C IV, only Mg II, and both C IV and Mg II are measurable. We find that utilizing both emission lines, where available, reduces the scatter of UV-based $M_{\rm BH}$ estimates by $\sim15\%$ when compared to previous studies. Lastly, we discuss the potential of our prescriptions to provide more accurate and precise estimates of $M_{\rm BH}$ given a much larger sample of quasars at $3.20 \lesssim z \lesssim 3.50$, where both Mg II and H$\beta$ can be measured in the same near-infrared spectrum.Comment: 19 pages (AASTeX 6.3.1), 9 figures, accepted for publication in Ap

Gemini Near Infrared Spectrograph -- Distant Quasar Survey: Augmented Spectroscopic Catalog and a Prescription for Correcting UV-Based Quasar Redshifts

Quasars at $z~{\gtrsim}~1$ most often have redshifts measured from rest-frame ultraviolet emission lines. One of the most common such lines, C IV ${\lambda}1549$, shows blueshifts up to ${\approx}~5000~\rm{km~s^{-1}}$, and in rare cases even higher. This blueshifting results in highly uncertain redshifts when compared to redshift determinations from rest-frame optical emission lines, e.g., from the narrow [O III] ${\lambda}5007$ feature. We present spectroscopic measurements for 260 sources at $1.55~{\lesssim}~z~{\lesssim}~3.50$ having $-28.0~{\lesssim}~M_i~{\lesssim}~-30.0$ mag from the Gemini Near Infrared Spectrograph - Distant Quasar Survey (GNIRS-DQS) catalog, augmenting the previous iteration which contained 226 of the 260 sources whose measurements are improved upon in this work. We obtain reliable systemic redshifts based on [O III] ${\lambda}5007$ for a subset of 121 sources which we use to calibrate prescriptions for correcting UV-based redshifts. These prescriptions are based on a regression analysis involving C IV full-width-at-half-maximum intensity and equivalent width, along with the UV continuum luminosity at a rest-frame wavelength of 1350 A. Applying these corrections can improve the accuracy and the precision in the C IV-based redshift by up to ${\sim}~850~\rm{km~s^{-1}}$ and ${\sim}~150~\rm{km~s^{-1}}$, respectively, which correspond to ${\sim}~8.5$ Mpc and ${\sim}~1.5$ Mpc in comoving distance at $z~=~2.5$. Our prescriptions also improve the accuracy of the best available multi-feature redshift determination algorithm by ${\sim}~100~\rm{km~s^{-1}}$, indicating that the spectroscopic properties of the C IV emission line can provide robust redshift estimates for high-redshift quasars. We discuss the prospects of our prescriptions for cosmological and quasar studies utilizing upcoming large spectroscopic surveys.Comment: 20 pages (AASTeX 6.3.1), 8 figures, accepted for publication in Ap