Prospectus, June 24, 1992

https://spark.parkland.edu/prospectus_1992/1014/thumbnail.jp

The Optimal Double Bubble for Density rpr^p

In 1993 Foisy et al. proved that the optimal Euclidean planar double bubble---the least-perimeter way to enclose and separate two given areas---is three circular arcs meeting at 120 degrees. We consider the plane with density $r^p$, joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincar\'e Conjecture. Dahlberg et al. proved that the best single bubble in the plane with density $r^p$ is a circle through the origin. We conjecture that the best double bubble is the Euclidean solution with one of the vertices at the origin, for which we have verified equilibrium (first variation or "first derivative" zero). To prove the exterior of the minimizer connected, it would suffice to show that least perimeter is increasing as a function of the prescribed areas. We give the first direct proof of such monotonicity for the Euclidean case. Such arguments were important in the 2002 Annals proof of the double bubble in Euclidean 3-spaceComment: 15 pages, 10 figure

In-context Learning Generalizes, But Not Always Robustly: The Case of Syntax

In-context learning (ICL) is now a common method for supervising large language models (LLMs): given labeled examples in the input context, the LLM learns to perform the task without weight updates. Despite ICL's prevalence and utility, we understand little about whether models supervised in this manner represent the underlying structure of their tasks, rather than superficial heuristics that only generalize to identically distributed examples. In this study, we investigate the robustness of LLMs supervised via ICL using the test case of sensitivity to syntax, which is a prerequisite for robust language understanding. Our experiments are based on two simple and well-controlled syntactic transformations tasks, where correct out-of-distribution generalization requires an accurate syntactic analysis of the input. We further investigate whether out-of-distribution generalization can be improved via chain-of-thought prompting, where the model is provided with a sequence of intermediate computation steps that illustrate how the task ought to be performed. In experiments with models from the GPT, PaLM, and Llama 2 families, we find large variance across LMs on this fundamental linguistic phenomenon, and that the variance is explained more by the composition of the pre-training corpus and supervision methods than by model size. In particular, we find evidence that models pre-trained on code generalize better, and benefit to a greater extent from chain-of-thought prompting

Microwave Properties of Ice-Phase Hydrometeors for Radar and Radiometers: Sensitivity to Model Assumptions

A simplied framework is presented for assessing the qualitative sensitivities of computed microwave properties, satellite brightness temperatures, and radar reflectivities to assumptions concerning the physical properties of ice-phase hydrometeors. Properties considered included the shape parameter of a gamma size distribution andthe melted-equivalent mass median diameter D0, the particle density, dielectric mixing formula, and the choice of complex index of refraction for ice. We examine these properties at selected radiometer frequencies of 18.7, 36.5, 89.0, and 150.0 GHz; and radar frequencies at 2.8, 13.4, 35.6, and 94.0 GHz consistent with existing and planned remote sensing instruments. Passive and active microwave observables of ice particles arefound to be extremely sensitive to the melted-equivalent mass median diameter D0 ofthe size distribution. Similar large sensitivities are found for variations in the ice vol-ume fraction whenever the geometric mass median diameter exceeds approximately 1/8th of the wavelength. At 94 GHz the two-way path integrated attenuation is potentially large for dense compact particles. The distribution parameter mu has a relatively weak effect on any observable: less than 1-2 K in brightness temperature and up to 2.7 dB difference in the effective radar reflectivity. Reversal of the roles of ice and air in the MaxwellGarnett dielectric mixing formula leads to a signicant change in both microwave brightness temperature (10 K) and radar reflectivity (2 dB). The choice of Warren (1984) or Warren and Brandt (2008) for the complex index of refraction of ice can produce a 3%-4% change in the brightness temperature depression

The Impact of Depth and Width on Transformer Language Model Generalization

To process novel sentences, language models (LMs) must generalize compositionally -- combine familiar elements in new ways. What aspects of a model's structure promote compositional generalization? Focusing on transformers, we test the hypothesis, motivated by recent theoretical and empirical work, that transformers generalize more compositionally when they are deeper (have more layers). Because simply adding layers increases the total number of parameters, confounding depth and size, we construct three classes of models which trade off depth for width such that the total number of parameters is kept constant (41M, 134M and 374M parameters). We pretrain all models as LMs and fine-tune them on tasks that test for compositional generalization. We report three main conclusions: (1) after fine-tuning, deeper models generalize better out-of-distribution than shallower models do, but the relative benefit of additional layers diminishes rapidly; (2) within each family, deeper models show better language modeling performance, but returns are similarly diminishing; (3) the benefits of depth for compositional generalization cannot be attributed solely to better performance on language modeling or on in-distribution data

Multisensor Observation and Simulation of Snowfall During the 2003 Wakasa Bay Field Experiment

This research seeks to assess and improve the accuracy of microphysical assumptions used in satellite passive microwave radiative transfer models and retrieval algorithms by exploiting complementary observations from satellite radiometers, such as TRMM/AMSR-E/GPM, and coincident aircraft instruments, such as the next generation precipitation radar (PR-2). We focus in particular on aircraft data obtained during the Wakasa Bay field experiment, Japan 2003, pertaining to surface snowfall events. The observations of vertical profiles of reflectivity and Doppler-derived fall speeds are used in conjunction with the radiometric measurements to identify 1-D profiles of precipitation particle types, sizes, and concentrations that are consistent with the observations

Debate Helps Supervise Unreliable Experts

As AI systems are used to answer more difficult questions and potentially help create new knowledge, judging the truthfulness of their outputs becomes more difficult and more important. How can we supervise unreliable experts, which have access to the truth but may not accurately report it, to give answers that are systematically true and don't just superficially seem true, when the supervisor can't tell the difference between the two on their own? In this work, we show that debate between two unreliable experts can help a non-expert judge more reliably identify the truth. We collect a dataset of human-written debates on hard reading comprehension questions where the judge has not read the source passage, only ever seeing expert arguments and short quotes selectively revealed by 'expert' debaters who have access to the passage. In our debates, one expert argues for the correct answer, and the other for an incorrect answer. Comparing debate to a baseline we call consultancy, where a single expert argues for only one answer which is correct half of the time, we find that debate performs significantly better, with 84% judge accuracy compared to consultancy's 74%. Debates are also more efficient, being 68% of the length of consultancies. By comparing human to AI debaters, we find evidence that with more skilled (in this case, human) debaters, the performance of debate goes up but the performance of consultancy goes down. Our error analysis also supports this trend, with 46% of errors in human debate attributable to mistakes by the honest debater (which should go away with increased skill); whereas 52% of errors in human consultancy are due to debaters obfuscating the relevant evidence from the judge (which should become worse with increased skill). Overall, these results show that debate is a promising approach for supervising increasingly capable but potentially unreliable AI systems.Comment: 84 pages, 13 footnotes, 5 figures, 4 tables, 28 debate transcripts; data and code at https://github.com/julianmichael/debate/tree/2023-nyu-experiment

Prospectus, March 23, 1992

https://spark.parkland.edu/prospectus_1992/1008/thumbnail.jp