Near an optimal learning point of a neural network, the learning performance of gradient descent dynamics is dictated by the Hessian matrix of the loss function with respect to the network parameters. We characterize the Hessian eigenspectrum for some classes of teacher-student problems, when the teacher and student networks have matching weights, showing that the smaller eigenvalues of the Hessian determine long-time learning performance. For linear networks, we analytically establish that for large networks the spectrum asymptotically follows a convolution of a scaled chi-square distribution with a scaled Marchenko-Pastur distribution. We numerically analyse the Hessian spectrum for polynomial and other non-linear networks. Furthermore, we show that the rank of the Hessian matrix can be seen as an effective number of parameters for networks using polynomial activation functions. For a generic non-linear activation function, such as the error function, we empirically observe that the Hessian matrix is always full rank.

We introduce an extension of the partitioned local depth (PaLD) algorithm that is adapted to online applications such as semi-supervised prediction. The new algorithm we present, online PaLD, is well-suited to situations where it is a possible to pre-compute a cohesion network from a reference dataset. After $O(n^3)$ steps to construct a queryable data structure, online PaLD can extend the cohesion network to a new data point in $O(n^2)$ time. Our approach complements previous speed up approaches based on approximation and parallelism. For illustrations, we present applications to online anomaly detection and semi-supervised classification for health-care datasets.

We consider the problem of ranking objects from noisy pairwise comparisons, for example, ranking tennis players from the outcomes of matches. We follow a standard approach to this problem and assume that each object has an unobserved strength and that the outcome of each comparison depends probabilistically on the strengths of the comparands. However, we do not assume to know a priori how skills affect outcomes. Instead, we present an efficient algorithm for simultaneously inferring both the unobserved strengths and the function that maps strengths to probabilities. Despite this problem being under-constrained, we present experimental evidence that the conclusions of our Bayesian approach are robust to different model specifications. We include several case studies to exemplify the method on real-world data sets.

Many travel decisions involve a degree of experience formation, where individuals learn their preferences over time. At the same time, there is extensive scope for heterogeneity across individual travellers, both in their underlying preferences and in how these evolve. The present paper puts forward a Latent Class Reinforcement Learning (LCRL) model that allows analysts to capture both of these phenomena. We apply the model to a driving simulator dataset and estimate the parameters through Variational Bayes. We identify three distinct classes of individuals that differ markedly in how they adapt their preferences: the first displays context-dependent preferences with context-specific exploitative tendencies; the second follows a persistent exploitative strategy regardless of context; and the third engages in an exploratory strategy combined with context-specific preferences.

Things to Do in L.A. Tap to enable a layout that focuses on the article. This is read by an automated voice. Please report any issues or inconsistencies here . The Trump administration is moving to dismantle the National Center for Atmospheric Research, a leading climate and weather research institution in Boulder, Colo. NCAR's weather forecasts, climate models and atmospheric data are vital to research, emergency planning and industries from aviation to insurance.

California regulators are threatening to suspend Tesla's licence to sell its electric cars in the state early next year unless the car maker tones down its marketing tactics for its self-driving features after a judge concluded that the Elon Musk-led company has been misleading consumers about the technology's capabilities. The potential 30-day blackout of Tesla's sales in California in the United States is the primary punishment being recommended to the state's Department of Motor Vehicles in a decision released late on Tuesday. After presiding over five days of hearings held in Oakland, California, in July, Cox also recommended suspending Tesla's licence to manufacture cars at its plant in Fremont, California. But California regulators will not impose that part of the judge's proposed penalty. Tesla will have a 90-day window to make changes that more clearly convey the limits of its self-driving technology to avoid having its California sales licence suspended.

Breakthroughs, discoveries, and DIY tips sent every weekday. It's been found in ancient human feces . The U.S. government stored 6.4 metric tons of it in mountains . And a big hunk of it played a major role in a presidential farewell party . While too much of the popular dairy product can spell tummy troubles and high cholesterol for some, new research suggests that eating more high-fat cheese and cream could be linked to a lower risk of developing dementia .

Season 2 of Prime Video's postapocalyptic megahit is a riot. Enter your email to receive alerts for this author. You can manage your newsletter subscriptions at any time. You're already subscribed to the aa_Rebecca_Onion newsletter. You can manage your newsletter subscriptions at any time.

Director James Cameron criticizes Trump administration's environmental rollbacks and warns humanity is going backwards in interview about "Avatar: Fire and Ash."

OpenAI launched GPT Image 1.5, delivering 4x faster image generation and improved editing accuracy for ChatGPT and API users worldwide. PCWorld reports the update includes a new creative studio mode with an image tab for enhanced editing capabilities and filter applications. The accelerated release aims to compete with Google's recent AI advancements while addressing consistency issues in consecutive image adjustments. OpenAI has launched its newest image model GPT Image 1.5, which offers up to 4 times faster image generation, more accurate image editing, and better compliance with user instructions. One of the more noticeable improvements is that the new version should provide more consistent results between consecutive edits. For example, when adjusting lighting, facial expressions, or color tone on a specific image, GPT Image should no longer make drastic unwanted changes--something many AI image tools have difficulty with.