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Magnetic fields play a crucial role in star formation, yet tracing them becomes particularly challenging, especially in the presence of outflow feedback in protostellar systems. We targeted the star-forming region L1551, notable for its apparent outflows, to investigate the magnetic fields. These fields were probed using polarimetry observations from the Planck satellite at 353 GHz/849 μm, the Stratospheric Observatory for Infrared Astronomy's (SOFIA) High-resolution Airborne Wide-band Camera (HAWC+ ) measurement at 214 μm, and the James Clerk Maxwell Telescope's (JCMT) Submillimetre Common-User POLarimeter (SCUPOL) 850 μm survey. Consistently, all three measurements show that the magnetic fields twist towards the protostar IRS 5. Additionally, we utilized the velocity gradient technique on the 12CO (J = 1–0) emission data to distinguish the magnetic fields directly associated with the protostellar outflows. These were then compared with the polarization results. Notably, in the outskirts of the region, these measurements generally align. However, as one approaches the centre of IRS 5, the measurements tend to yield mostly perpendicular relative orientations. This suggests that the outflows might be dynamically significant from a scale of ∼0.2 pc, causing the velocity gradient to change direction by 90°. Furthermore, we discovered that the polarization fraction p and the total intensity I correlate as p ∝ I−α. Specifically, α is approximately 1.044 ± 0.06 for SCUPOL and around 0.858 ± 0.15 for HAWC+. This indicates that the outflows could significantly impact the alignment of dust grains and magnetic fields in the L1551 region.

 

Traditional analyses in non-convex optimization typically rely on the smoothness assumption, namely requiring the gradients to be Lipschitz. However, recent evidence shows that this smoothness condition does not capture the properties of some deep learning objective functions, including the ones involving Recurrent Neural Networks and LSTMs. Instead, they satisfy a much more relaxed condition, with potentially unbounded smoothness. Under this relaxed assumption, it has been theoretically and empirically shown that the gradient-clipped SGD has an advantage over the vanilla one. In this paper, we show that clipping is not indispensable for Adam-type algorithms in tackling such scenarios: we theoretically prove that a generalized SignSGD algorithm can obtain similar convergence rates as SGD with clipping but does not need explicit clipping at all. This family of algorithms on one end recovers SignSGD and on the other end closely resembles the popular Adam algorithm. Our analysis underlines the critical role that momentum plays in analyzing SignSGD-type and Adam-type algorithms: it not only reduces the effects of noise, thus removing the need for large mini-batch in previous analyses of SignSGD-type algorithms, but it also substantially reduces the effects of unbounded smoothness and gradient norms. To the best of our knowledge, this work is the first one showing the benefit of Adam-type algorithms compared with non-adaptive gradient algorithms such as gradient descent in the unbounded smoothness setting. We also compare these algorithms with popular optimizers on a set of deep learning tasks, observing that we can match the performance of Adam while beating others. 

Convex-concave min-max problems are ubiquitous in machine learning, and people usually utilize first-order methods (e.g., gradient descent ascent) to find the optimal solution. One feature which separates convex-concave min-max problems from convex minimization problems is that the best known convergence rates for min-max problems have an explicit dependence on the size of the domain, rather than on the distance between initial point and the optimal solution. This means that the convergence speed does not have any improvement even if the algorithm starts from the optimal solution, and hence, is oblivious to the initialization. Here, we show that strict-convexity-strict-concavity is sufficient to get the convergence rate to depend on the initialization. We also show how different algorithms can asymptotically achieve initialization-dependent convergence rates on this class of functions. Furthermore, we show that the so-called “parameter-free” algorithms allow to achieve improved initialization-dependent asymptotic rates without any learning rate to tune. In addition, we utilize this particular parameter-free algorithm as a subroutine to design a new algorithm, which achieves a novel non-asymptotic fast rate for strictly-convex-strictly-concave min-max problems with a growth condition and Hölder continuous solution mapping. Experiments are conducted to verify our theoretical findings and demonstrate the effectiveness of the proposed algorithms. 

SGD with Momentum (SGDM) is a widely used family of algorithms for large-scale optimization of machine learning problems. Yet, when optimizing generic convex functions, no advantage is known for any SGDM algorithm over plain SGD. Moreover, even the most recent results require changes to the SGDM algorithms, like averaging of the iterates and a projection onto a bounded domain, which are rarely used in practice. In this paper, we focus on the convergence rate of the last iterate of SGDM. For the first time, we prove that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from a suboptimal convergence rate of $\Omega(\frac{\ln T}{\sqrt{T}})$ after $T$ iterations. Based on this fact, we study a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with \emph{increasing momentum} and \emph{shrinking updates}. For these algorithms, we show that the last iterate has optimal convergence $O(\frac{1}{\sqrt{T}})$ for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of $T$. Further, we show a variety of results for FTRL-based SGDM when used with adaptive stepsizes. Empirical results are shown as well. 

SGD with Momentum (SGDM) is a widely used family of algorithms for large-scale optimization of machine learning problems. Yet, when optimizing generic convex functions, no advantage is known for any SGDM algorithm over plain SGD. Moreover, even the most recent results require changes to the SGDM algorithms, like averaging of the iterates and a projection onto a bounded domain, which are rarely used in practice. In this paper, we focus on the convergence rate of the last iterate of SGDM. For the first time, we prove that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from a suboptimal convergence rate of $\Omega(\frac{\ln T}{\sqrt{T}})$ after $T$ iterations. Based on this fact, we study a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with increasing momentum and shrinking updates. For these algorithms, we show that the last iterate has optimal convergence $O(\frac{1}{\sqrt{T}})$ for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of $T$. Further, we show a variety of results for FTRL-based SGDM when used with adaptive stepsizes. Empirical results are shown as well. 

As a novel approach for tracing interstellar magnetic fields, the velocity gradient technique (VGT) has been proven to be effective for probing magnetic fields in the diffuse interstellar medium (ISM). In this work, we verify the VGT in a broader context by applying the technique to a molecular cloud interacting with the supernova remnant (SNR) W44. We probe the magnetic fields with the VGT using CO, $\rm HCO^+$ and H i emission lines and make a comparison with the Planck 353-GHZ dust polarization. We show that the VGT gives an accurate measurement that coheres with the Planck polarization especially in intense molecular gas emission regions. We further study the foreground’s contribution on the polarization that results in misalignment between the VGT and the Planck measurements in low-intensity molecular gas areas. We advance the VGT to achieve magnetic field tomography by decomposing the SNR W44 into various velocity components. We show that W44’s velocity component at v ∼ 45 km s−1 exhibits the largest coverage and gives best agreement with Planck polarization in terms of magnetic field orientation.