Adaptive Bound Optimization for Online Convex Optimization

We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2-squared, and modify it only via a single time-dependent parameter. Our algorithm's regret bounds are worst-case optimal, and for certain realistic classes of loss functions they are much better than existing bounds. These bounds are problem-dependent, which means they can exploit the structure of the actual problem instance. Critically, however, our algorithm does not need to know this structure in advance. Rather, we prove competitive guarantees that show the algorithm provides a bound within a constant factor of the best possible bound (of a certain functional form) in hindsight.Comment: Updates to match final COLT versio

Ultrafast Dynamics of Relativistic Laser Plasma Interactions

This thesis documents the experimental and theoretical investigation of laser pulse evolution in relativistic laser-plasma interactions for plasma-wakefield acceleration and ion acceleration experiments. Power amplification of the Astra Gemini laser in a plasma was observed, with the compression of an initially 55 fs, 180 TW pulse down to 14 fs, with a peak power of 320 TW. This was achieved in a laser-driven plasma wakefield operating just below the self-injection threshold density for a propagation distance of 15 mm. Self-guiding of the laser pulse was observed, while pulse depletion was characterised as a function of density and propagation distance, showing that the pulse evolution scales equally with both. These measurements displayed good agreement with a depletion model based on pulse front etching. Particle-in-cell simulations were seen to closely reproduce the experimental results, which were concluded to be predominantly dependent on the longitudinal properties of the laser and wakefield. The simulations also revealed a new wakefield instability that is driven by the far red-shifted component of the laser pulse. In the case of high-contrast solid-density interactions, oscillations of the front surface of the plasma were seen to result in the generation of the second harmonic of the driving laser for a p-polarised interaction. Conversion efficiencies of 22% into the second harmonic were measured, while the total plasma reflectivity into the first and second harmonics remained relatively constant at 65% over the intensity range of 1E17 - 1E21 W/cm2. For normal incidence interactions with sub-micron thickness foils, the cycle-averaged surface motion was measured using a FROG diagnostic. Targets of a few nanometers in thickness underwent an acceleration away from the laser, but the measured surface velocities did not match the expected hole-boring velocities or the measured ion energies, due to the thermal expansion of the plasma. 2D simulations revealed that studying target motion in this way is affected by the scale length of the plasma and photon acceleration that can occur in the tenuous plasma in front of the laser-reflecting surface.Open Acces

Corn yield variability on the Des Moines Lobe of Iowa: Assessment of extent and soil-related causes

Precision agriculture techniques are an essential component to modern row crop agriculture in Iowa and can be used to create crop yield variability maps via geographic information systems. The first objective of this thesis was to explore the methodology that could be used to locate significant long-term corn yield variability on the Iowa Des Moines Lobe. A 158-ha site, consisting mainly of Clarion, Nicollet and Webster soil map units and containing multiple years of geo-referenced corn (Zea mays) yield data was selected. A cluster analysis tool was performed to locate spatially consistent corn yield variability (high, low and mean yield clusters). The mean yield of the entire site from 2005 to 2011 was 12052 kg ha-1 compared to the high and low clusters which were 13747 kg ha-1 and 10420 kg ha-1, respectively. It was observed that 12% of the site was consistently variable, of which 82%, occurred within the Clarion. This identification approach could be used as a basis for variable rate management which could increase yield, net profitability and sustainability. The second objective was to explore the relationships of specific soil characteristics associated with the identified variability clusters. High yielding clusters were deeper to maximum depth of mollic colors, higher in total carbon (TC) and total nitrogen (TN) content at 0 to 25 cm, higher in Mehlich 3 phosphorus (M3P) and Mehlich 3 potassium (M3K) content at all depths and higher in clay content at 0 to 25 cm and 26 to 100 cm compared to low yielding clusters. At the 0 to 25 cm depth, 60% of yield variability was associated with TN, M3P and M3K and clay content. These four soil characteristics were positively correlated with yield (r= 0.73, 0.47, 0.41, 0.72, respectively). However, 55% of yield variability was associated with TC and TN content at the 26 to 100 cm depth. At the 101 to 120 cm depth, 33% of yield variability was associated with pH and M3P content. This study illustrates the ability to associate yield variability within fields on the Des Moines Lobe of Iowa with specific soil characteristics

Automatically Bounding the Taylor Remainder Series: Tighter Bounds and New Applications

We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function $f: \mathbb{R} \to \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 1$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$ for all $x \in [a, b]$. As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known atomic functions. At a high level, our algorithm has two steps. First, for a variety of commonly-used elementary functions (e.g., $\exp$, $\log$), we use recently-developed theory to derive sharp polynomial upper and lower bounds on the Taylor remainder series. We then recursively combine the bounds for the elementary functions using an interval arithmetic variant of Taylor-mode automatic differentiation. Our algorithm can make efficient use of machine learning hardware accelerators, and we provide an open source implementation in JAX. We then turn our attention to applications. Most notably, in a companion paper we use our new machinery to create the first universal majorization-minimization optimization algorithms: algorithms that iteratively minimize an arbitrary loss using a majorizer that is derived automatically, rather than by hand. We also show that our automatically-derived bounds can be used for verified global optimization and numerical integration, and to prove sharper versions of Jensen's inequality.Comment: Previous version has been split into 3 articles: arXiv:2308.00679, arXiv:2308.00190, and this articl