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Deep ensembles have been empirically shown to be a promising approach for improving accuracy, uncertainty and out-of-distribution robustness of deep learning models. While deep ensembles were theoretically motivated by the bootstrap, non-bootstrap ensembles trained with just random initialization also perform well in practice, which suggests that there could be other explanations for why deep ensembles work well. Bayesian neural networks, which learn distributions over the parameters of the network, are theoretically well-motivated by Bayesian principles, but do not perform as well as deep ensembles in practice, particularly under dataset shift. One possible explanation for this gap between theory and practice is that popular scalable variational Bayesian methods tend to focus on a single mode, whereas deep ensembles tend to explore diverse modes in function space. We investigate this hypothesis by building on recent work on understanding the loss landscape of neural networks and adding our own exploration to measure the similarity of functions in the space of predictions. Our results show that random initializations explore entirely different modes, while functions along an optimization trajectory or sampled from the subspace thereof cluster within a single mode predictions-wise, while often deviating significantly in the weight space. Developing the concept of the diversity--accuracy plane, we show that the decorrelation power of random initializations is unmatched by popular subspace sampling methods. Finally, we evaluate the relative effects of ensembling, subspace based methods and ensembles of subspace based methods, and the experimental results validate our hypothesis.

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Deep Ensembles: A Loss Landscape Perspective

We outline the experimental concept and key scientific capabilities of AION (Atom Interferometer Observatory and Network), a proposed UK-based experimental programme using cold strontium atoms to search for ultra-light dark matter, to explore gravitational waves in the mid-frequency range between the peak sensitivities of the LISA and LIGO/Virgo/ KAGRA/INDIGO/Einstein Telescope/Cosmic Explorer experiments, and to probe other frontiers in fundamental physics. AION would complement other planned searches for dark matter, as well as probe mergers involving intermediate mass black holes and explore early universe cosmology. AION would share many technical features with the MAGIS experimental programme in the US, and synergies would flow from operating AION in a network with this experiment, as well as with other atom interferometer experiments such as MIGA, ZAIGA and ELGAR. Operating AION in a network with other gravitational wave detectors such as LIGO, Virgo and LISA would also offer many synergies.

AION: An Atom Interferometer Observatory and Network

The axion is a promising dark matter candidate, which was originally proposed to solve the strong-$CP$ problem in particle physics. To date, the available parameter space for axion and axionlike particle dark matter is relatively unexplored, particularly at masses ${m}_{a}\ensuremath{\lesssim}1\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$. ABRACADABRA is a new experimental program to search for axion dark matter over a broad range of masses, ${10}^{\ensuremath{-}12}\ensuremath{\lesssim}{m}_{a}\ensuremath{\lesssim}{10}^{\ensuremath{-}6}\text{ }\text{ }\mathrm{eV}$. ABRACADABRA-10 cm is a small-scale prototype for a future detector that could be sensitive to the QCD axion. In this Letter, we present the first results from a 1 month search for axions with ABRACADABRA-10 cm. We find no evidence for axionlike cosmic dark matter and set 95% C.L. upper limits on the axion-photon coupling between ${g}_{a\ensuremath{\gamma}\ensuremath{\gamma}}l1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ and ${g}_{a\ensuremath{\gamma}\ensuremath{\gamma}}l3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}1}$ over the mass range $3.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}--8.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}\text{ }\text{ }\mathrm{eV}$. These results are competitive with the most stringent astrophysical constraints in this mass range.

First Results from ABRACADABRA-10 cm: A Search for Sub-μeV Axion Dark Matter.

The landscape of QCD axion models

For the minimal QCD axion model it is generally believed that overproduction of dark matter constrains the axion mass to be above a certain threshold, or at least that the initial misalignment angle must be tuned if the mass is below that threshold. We demonstrate that this is incorrect. During inflation the axion tends toward an equilibrium, assuming the Hubble scale is low and inflation lasts sufficiently long. This means the minimal QCD axion can naturally give the observed dark matter abundance in the entire lower part of the mass range, down to masses $\ensuremath{\sim}{10}^{\ensuremath{-}12}\text{ }\text{ }\mathrm{eV}$ (or ${f}_{a}$ up to almost the Planck scale). The axion abundance is generated by quantum fluctuations of the field during inflation. This mechanism generates cold dark matter with negligible isocurvature perturbations. In addition to the QCD axion, this mechanism can also generate a cosmological abundance of axionlike particles and other light fields.

Stochastic axion scenario

Recently, the problem of unitarity violation during the preheating stage of Higgs inflation with a large non-minimal coupling has been much discussed in the literature. We point out that this problem can be translated into a strong coupling problem for the dimensionless effective coupling, and that the existence of these problems is highly dependent on the choice of higher-dimensional operators because they can significantly change the background dynamics and the canonical normalization of the fluctuations around it. Correspondingly, the typical energy of particles produced during the first stage of preheating can remain comparable to or below the cutoff scale of the theory. As an example, we numerically calculate the particle production in the presence of a specific four-derivative operator of the Higgs field, and confirm the statement above. Our argument also applies to multi-field inflation with non-minimal couplings.

On preheating in Higgs inflation

Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster algebra structure In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions, at any weight, whose symbol alphabet consists of cluster coordinates on the $A_n$ cluster algebra Using such a basis we present a new expression for the 2-loop 6-particle NMHV amplitude which makes some of its cluster structure manifest

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Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes

We explore the loss landscape of fully-connected and convolutional neural networks using random, low-dimensional hyperplanes and hyperspheres. Evaluating the Hessian, H, of the loss function on these hypersurfaces, we observe 1) an unusual excess of the number of positive eigenvalues of H, and 2) a large value of Tr(H)/||H|| at a well defined range of configuration space radii, corresponding to a thick, hollow, spherical shell we refer to as the Goldilocks zone. We observe this effect for fully-connected neural networks over a range of network widths and depths on MNIST and CIFAR-10 datasets with the ReLU and tanh non-linearities, and a similar effect for convolutional networks. Using our observations, we demonstrate a close connection between the Goldilocks zone, measures of local convexity/prevalence of positive curvature, and the suitability of a network initialization. We show that the high and stable accuracy reached when optimizing on random, low-dimensional hypersurfaces is directly related to the overlap between the hypersurface and the Goldilocks zone, and as a corollary demonstrate that the notion of intrinsic dimension is initialization-dependent. We note that common initialization techniques initialize neural networks in this particular region of unusually high convexity/prevalence of positive curvature, and offer a geometric intuition for their success. Furthermore, we demonstrate that initializing a neural network at a number of points and selecting for high measures of local convexity such as Tr(H)/||H||, number of positive eigenvalues of H, or low initial loss, leads to statistically significantly faster training on MNIST. Based on our observations, we hypothesize that the Goldilocks zone contains an unusually high density of suitable initialization configurations.

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The Goldilocks Zone: Towards Better Understanding of Neural Network Loss Landscapes

Multi-loop scattering amplitudes in $$ \mathcal{N}=4 $$
 Yang-Mills theory possess cluster algebra structure. In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions, at any weight, whose symbol alphabet consists of cluster coordinates on the A
 n
 cluster algebra. Using such a basis we present a new expression for the 2-loop 6-particle NMHV amplitude which makes some of its cluster structure manifest.

/pdf/hedgehog-bases-for-a-n-cluster-polylogarithms-and-an-3agn0dgown.pdf

Adam Scherlis

Papers

Stochastic axion scenario

On preheating in Higgs inflation

Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes

The Goldilocks Zone: Towards Better Understanding of Neural Network Loss Landscapes

Hedgehog bases for A n cluster polylogarithms and an application to six-point amplitudes