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This suppression has been quantified using a semi-analytic approach in {{cite:e6a8d553f2e1747047e59b92d3411e61bb0fc7ef}} for the case of protons, with the results being also valid in the case of nuclei at the energies for which the interactions are negligible. In this paper we will extend the study to the case of primary nuclei in the regime in which interactions are relevant, analysing in detail the effects on the secondary particles produced by the photo-disintegration during the propagation. To this scope we have extended the SimProp v2r4 code {{cite:63cbf9c5b7498c18c02ad1409bb1cc96b7e70ba5}} so as to not only follow the energy, mass and charge of the particles as a function of time, but also to follow the direction of propagation of the CRs and the distance from their sources, in the presence of a turbulent intergalactic magnetic field. In the simulations with the SimProp code we have used for definiteness the photonuclear cross sections from Puget, Stecker and Bredekamp {{cite:037338f1573f6744a62150d7810c70a7e747c56a}} and the extragalactic background light model from Stecker, Malkan and Scully {{cite:66c93a27734bc6413a0378b56d4e7ee1d772a1e3}}. We also study how a discrete source distribution affects the high energy cutoff due to the attenuation during propagation, as well as the possible recovery of the CR flux that may appear at the highest energies if the source spectra were to extend to extremely high energies.
| i | dca239f6973909132dfce92194bc5d02 |
Modalities. We use optical flow, released by {{cite:0b39851dc7d0cceb45cc58e698541e159bbbc7f7}} for Something-Something and Something-Else, and bounding boxes & categories released by {{cite:4c0c38302175b42629e81d51aa36ec429fe2cde9}} for Something-Something and {{cite:168f047c3d011db51c08138441279eb55e3b71c8}} for Something-Else.
| d | ffe6e4a168d4dfbc43b91e62f428393d |
Graph backbone and aggregation functions. Various GNN backbones {{cite:9de1b8fe43285fdab5fe6218ea62339c1cfe8621}}, {{cite:e615dc2679b0423bdd4aa6ca4e4684b69b922cf1}}, {{cite:633c36e51ff82bd09ddcc88d66fac82bfb10ebf7}} have been devised to capture graph structural information. They mainly differ in the specific aggregation functions. As a comprehensive study, GraphSAGE {{cite:633c36e51ff82bd09ddcc88d66fac82bfb10ebf7}} adopts four different aggregation methods, namely max, mean, GCN {{cite:9de1b8fe43285fdab5fe6218ea62339c1cfe8621}}, and LSTM {{cite:b9904a4a96451661baa094ab2bbc0f7dc02288b6}}. Instead, Graph Attention Networks {{cite:e615dc2679b0423bdd4aa6ca4e4684b69b922cf1}} proposes attention-based methods {{cite:953bef560ea9c269c411a5cda6ca55ebacd6822d}}, where the edge weights are learned by network adaptively. Graph Isomorphism Networks (GINs) {{cite:c380fdd3342ded97d4831d1986eaeb4b438003ce}} proves that GNN can satisfy the 1-Weisfeiler-Lehman (WL) condition only with sum pooling function as aggregation function. Recently, DeeperGCN {{cite:607c0862b6bb79a2f599ef33f8f622940856317e}} proposes a trainable softmax and power-mean aggregation function that generalizes basic operators.
| d | cc8ed6c5e7e40923bbf0cf9c507b66b2 |
Here, we characterize the nature of such inductive bias in the case of dense networks trained to classify natural images, and reveal that the winning lottery tickets of FCNs display the key features of CNNs. We apply iterative magnitude pruning (IMP) {{cite:9b9d1703735cb830fea291fa5f8f9ebc29bb8dfe}} on FCNs trained on a low resolution version of ImageNet {{cite:ee8746d2cfa118c0a46e7f3fd3637e591651cfff}}.
We show that the sub-network obtained by IMP is characterized by local connectivity, especially in the first hidden layer, and masks leading to local features with patterns very reminiscent of the ones of trained CNNs {{cite:d50ecf57897ec276182bbda45139ab6258e8a2e8}}. Deeper layers are made up of these local features with larger receptive fields hinting at the hierarchical structure found in CNNs.
We study how the data and the task affect these properties. We show a crossover from a large dataset regime (large signal-to-noise ratio), where the inductive CNN-like bias is present, to a small dataset regime in which pruning loses performance and concomitantly the properties described above disappear.
We show that a similar crossover takes place when going from a meaningful task to one with low semantic value.
| i | 1bc16a5462483b759a4b285571913784 |
The basic concept is the algorithmic (or Kolmogorov) complexity (AC)
which measures the complexity of an individual object
by the size of the smallest program that can reproduce it.
In fact, the AC of a sequence {{formula:bcd673e1-6db1-4d03-922d-60ee94a176ee}} ({{formula:bb24dee6-66d8-444a-b814-b9b38a38c212}} ) is the length of the shortest
program which generates as output the sequence and stops afterwards {{cite:7efd05055211f428cf21d04fae8b607abce44292}}, {{cite:a3af28c1a2df407fcf4253d91f279cd3e898cc8a}}.
For any probability distribution {{formula:0126989c-4925-43d5-a36f-0915dc7d761d}} that is computable using a Turing machine (a very general condition),
the expected value of AC equals Shannon's entropy, up to a constant term {{cite:9d2925e3ecba88e1134f9aa04a040e31b9297401}}.
From this result it follows that Shannon's entropy is asymptotically equal
to the expected complexity: {{formula:4cd718ce-a4bd-493c-ba94-226c01c3c0fe}} {{cite:9d2925e3ecba88e1134f9aa04a040e31b9297401}}.
| m | 4270e38db95024fae095365fb85a1eb2 |
In the past few decades, with the rapid development of laser cooling and trapping techniques, extraordinary advancements in optical atomic clock accuracy and stability have been demonstrated {{cite:0ba9e01e4e3c8682478f111395226ab257629b38}}, {{cite:a54277f08eab65ce3e4b4b34d60d25bf3ed8fcaa}}, {{cite:8b35ec8c5d1fe9c092f05c644eeb3637434a4b17}}, {{cite:b223f52ca835a79999d527272ab6866de648036b}}, {{cite:58b752fb9367815a9fb47325f0b30c99530dcc14}}, {{cite:9ce265478558f5a4be42586bc3b15e31cac9abe2}}. The high-accuracy optical clocks can be used for performing precision measurements of fundamental physical constants {{cite:b20569ecfabfa1193315bdc31b1212acc7c6b48d}}, {{cite:1045450036dd0af99f201bad5248383743ca8e05}}, testing the local Lorentz invariance {{cite:0168cb8d085d729dcac5193501ad6a0c919986a2}}, {{cite:5b79ac6d4aecafadf1cdf9d7e4461c8b8495a041}}, exploring variations of fine structure constant {{formula:0a207382-9336-4a94-a394-8b3ac6456d78}} with time {{cite:35cdc5f0970d6ad59615c9047a87a7488acd456a}}, {{cite:c0868ea0b8692e85e2d0cc9663af419f7ed627b7}}, probing dark matter and dark energy {{cite:4b2008853b94b2937d21cd6191efb284a9dc9353}}, {{cite:7f5fe6292e0decb64b46922df3c44c6adee0f111}}, detecting gravitational waves {{cite:ac3674763ca0555e3214e4403283f003058f98de}}, and detecting new forces beyond the standard model of particle physics {{cite:ac3674763ca0555e3214e4403283f003058f98de}}, {{cite:7f5fe6292e0decb64b46922df3c44c6adee0f111}}.
| i | 007e4579a9358ffc1ea03752ff86a834 |
A key step forward was taken in {{cite:e261116fc450d8e1484fa82ed6b8c8c78efa6b35}}, {{cite:d97a890b4c2889c9db1d8775c439a9f133b7dd00}} where it was realized that by applying a simple perturbation coupling the two sides of an eternal AdS black hole one may make the wormhole traversable. This, in principle, allows us to probe behind the horizon without dealing with issues of bulk locality - all we need to do is send an observer from one side to the other and ask them how they felt. Susskind has predicted that we will be able to perform experiments of this type `within the next decade or two' {{cite:d1e7d5da24d9e1e15c26241023eb4a40143aea7c}}. The eternal AdS black hole is dual to the thermofield double state of the two boundary CFTs {{cite:24bbb27e53769fa1962d4278a3fd2d55c35ce930}}, {{cite:c18ea10c6d15e6a96dac2358c16bbd5648a13e40}}. The thermofield double state is also of interest beyond the context of black holes, in the study of thermal field theories.
| i | caf940921ecd5b830c2c09c49511d6de |
The BP-PDF was evaluated for each TS of {{formula:7d3e6adb-2fce-448d-bb65-aeadcd812e87}} data, stochastic and
chaotic, following the lexicographic pattern-order proposed by
Lehmer {{cite:f6b9bbfdffa8c1fe3acc44b7957ac2baf6cbe71f}}, with pattern-lengths {{formula:b6d90e20-e64e-4ba5-a5e9-0b599e4223f1}} and time lag
{{formula:200f89f5-9abc-4cd0-8af7-74f9216e413c}} . Their corresponding localization in the causality
Fisher-Shannon plane are shown in Fig. REF . One
can use any of these TS for evaluating the dynamical system's
invariants (like correlation dimension, Lyapunov exponents, etc.),
by appealing to a time lag reconstruction {{cite:8c56556860ba6a9480d503305b12f17e3853857f}}. Here
we analyzed TS generated by each one of chaotic maps' coordinates
when the corresponding map is bi- or multi-dimensional.
Due to the fact that the BP-PDF is not a dynamical invariant
(neither are other quantifiers derived by Information Theory), some
variation could be expected in the quantifiers' values computed
with this PDF, whenever one or other of the TS generated by
these multidimensional coordinate systems.
| r | 491ba1ce90e13b1fda2eb5bbd19560e4 |
Recently, due to the success of CNNs in many fields {{cite:7debbb0faf9021ab9d0f0e203f43ed2e43944bd3}}, {{cite:0efda089e3778154b09d18acebc213e3fed0e096}}, {{cite:4a6deae183b42e7d316ac86770c1a91ba172817b}}, {{cite:25d357845b1f6eccbda80d6a47cdc42a3bc858cd}}, {{cite:cc60e828cbe72bbd1b192859d391595f8354520c}}, CNN-based methods are widely used in crowd counting. The density map generated by CNNs records the count and location information of the crowd. Zhang et al. {{cite:40765fc9a84eb531d30373ea21ff2cc6f69524dd}} proposed the MCNN to overcome scale variations. The MCNN leveraged three-branch CNNs with different convolution kernels to extract multi-scale features. Based on {{cite:40765fc9a84eb531d30373ea21ff2cc6f69524dd}}, Sam et al. {{cite:3145a2c6b95436ea9789f6e3777e8d6f6bcb2bd4}} designed the Switch-CNN with a classifier to select the optimal branch to encode a density map according to the variation of crowd counts. To employ the temporal correlation across frames in video sequences to assist crowd counting, Xiong et al. {{cite:13e2eed6350fc751e4fcb7d9b13eafb809d3fbff}} introduced the ConvLSTM that could extract bidirectional timing information. Sindagi et al. {{cite:4e3d6806c7631025d8d60e34ead25e7fe347d59c}} proposed the CP-CNN to incorporate global and local contextual features to encode a high-quality density map. Li et al. {{cite:c252db90c89dfb46e448adfbc35e39be95aaada4}} proposed the DecideNet model to adaptively leverage the estimations of detection and regression. Zhang et al. {{cite:12ab6a006b359b3fcc00347294ef09f091b70805}} proposed the MRA-CNN that could automatically focus on head regions by score maps. Hossain et al. {{cite:d0c6b5a9ea23332d69b295a55deb90470e9dddfd}} introduced a scale-aware attention mechanism to adapt the scale variation of crowds. Wang et al. {{cite:943c36630e4e1c43eb7d645449ec04b79bc19cc1}} designed the data collector and labeler to automatically generate and annotate the crowd data to reduce over-fitting caused by limited training data. In this paper, we propose a novel method to avoid misjudgements that can result in enormous errors of crowd counting.
| m | 33c6b4bb705f5ca085a9cefe9f40a27d |
Models of the X-ray emission from the ring {{cite:ed5114e454b3e69b903ee9450cbeea98ac2d2ee2}} suggest that X-ray emission comprises of a soft X-ray emitting component, generated by a {{formula:966d1c22-1db4-4d06-a468-fe222a09de4b}} km s{{formula:fef94175-1377-488c-9ebc-535c0a0307b8}} shock propagating through the dense clumps of the ring, and a hard X-ray emitting component, generated by the shock propagating through the lower density inter-clump region and the less dense medium in which the ring is embedded.
The soft X-ray component is characterised by a temperature of about 0.8 keV ({{formula:453cbcd7-f0eb-46ad-ba25-5b148e76c362}} K), which is consistent with that behind a transmitted 680 km s{{formula:d8490fae-0f5f-4c45-bc75-4703c88ba657}} shock through the clumps {{cite:b8115079350d72fa6ed8539082275755eec154e0}}.
Using optical spectra of the ring, {{cite:adcb9fbc2c734f34572cd942c8a96cfaed652cc7}} derived preshock densities between {{formula:21e2890b-05bb-43d4-894a-9c0b09a1295c}} (1–30){{formula:415aa288-0c51-427f-ab3e-22e4db943d9c}} cm{{formula:ee59506c-72ee-494a-935a-d999fb32dc6b}} by fitting emission line light curves with photoionisation models.
These plasma densities and temperature are capable of collisionally heating the silicate dust grains in the ring to the observed temperature of warm {{formula:0af5b2fe-bc4d-4dc4-a0f1-a13be6251399}} 180 K
{{cite:c9fcce5de68debc2be8304508065bf5817f642b5}}. However, densities above {{formula:3ba7f43b-3f88-4845-b49b-807e758926de}} cm{{formula:b8c33101-2187-4383-992e-a86e9fc998ed}} are required to collisionally heat the secondary hot dust component to temperatures of {{formula:cfda23c4-3d02-455b-988b-c0efd8ba2a43}} 400–500 K.
| d | 278d756d02cb02a7380985d6d9a8bdd7 |
We train and evaluate our models on the COCO val dataset. We use the detection results from a person detector with AP 50.2 on the COCO Val2017 set. Following standard practice {{cite:136df470834722dcac0e53a09d7a48475b9afd60}}, {{cite:a3f84e4b5684892224eeb4bd303d81a325345d0b}} we use flip test and average the results from the original and flipped images. Unless otherwise noted, the methods we compare against use the same evaluation techniques.
| m | 7f3f89ee6573a85072da91b560a5413e |
We used ResNet152 {{cite:e88cadab7bb25ef48884f8fbb8c6caa4648d982b}} as pretrained model for the encoder.
We tested several set of parameters to find the one that would allow a transfer learning from Western fashion data. As the latter are more abundant, we wanted an architecture leveraging Western style captioning to improve African style captioning.
| r | 5f5348929dca05b8e951fa2ce0b06ae9 |
Here we specify additional details on the baseline methods used in the learning experiment.
The ADF method, called AP2 in {{cite:f7dddc71bea8240903bd3da15b5f911977465917}}, propagates means and variances through the hidden layers fully analytically. This method is the equivalent of the PBNet method in {{cite:03dba83080cf9f620239e1dee80690fcc7d0be31}} when the weights are deterministic. We only sample the states of the last binary layer are sampled, as a general solution suitable with different head functions. The ADF family of methods includes expectation propagation {{cite:c88fe29ceb33fa1a076cd0f127833b382348b38e}} designed for approximate variational inference in graphical models. It computes an approximation to summation (REF ) by fitting and propagating a fully factorized approximation to marginal distributions {{formula:8b8bb1c0-bff9-42f8-affa-6ad5b9bd0509}} with forward KL divergence. The gradient of this approximation is then evaluated. The PSA method differs in that it approximates the gradient directly and does not make a strong factorization assumption. From the experiments we observe that ADF performs very well in the beginning, when all weights are initially random and then it over-fits to the relaxed objective.
| m | cc8d8415f9b3718da30300844fa1b401 |
Limitations.
Extensive discussions about social data biases are presented in {{cite:83a2eb5c3a5a86f318aa9609afa3574b44b86478}}. The biases can be introduced due to the choice of social platforms, data (un)availability, sampling methods, etc. Here we discuss three limitations in our data collection process.
| d | a507e956e9e0e3d6d78b6ff0466476f9 |
Self-supervised learning is a widely used paradigm to learn representations from input data without human annotated labels. In computer vision domain, it has shown superior performance in many tasks, such as classification, detection, segmentation, etc. A popular self-supervised learning framework is a Siamese Network with two branches. A similarity loss {{cite:55de504629da0748932bb4cbbf81c1bb8e55cbd0}}, {{cite:3bb8540a6163ed5b1e0617e75b450b18e5cee982}}, contrastive loss {{cite:a71ed0bd89945152ef1dca2cd95d0136a47e08f2}}, {{cite:5e75f198fec8057968ae161391fcb44c473f843f}}, {{cite:65f9d5bbb0ac295f1651fdc28552f4719b8ef235}} or distillation loss {{cite:7ac5ce78b6f84463f784fb06ed39f0cbba986cee}} is employed to calculate the distance of the two branches. Recently, masked image modeling (MIM) {{cite:fec6c55450abe8ebd3087883e1da9a351fa31520}}, {{cite:a5b31c330fcb69bb77ee36fbb0c0400d179a576d}}, {{cite:0699a618663f4f919f422eaf04e62505aa2703d7}} has emerged and proven to be an effective approach to learn useful representation. To fully leverage the advantages from both the masked design and siamese networks, Masked Siamese Networks {{cite:0699a618663f4f919f422eaf04e62505aa2703d7}}, {{cite:8118ad0095b37dd10b7eb3fd231a2d12d4bda5bb}} have been proposed.
| i | f861c37db2c95b896a8730d185bbffe3 |
Algebra of quantum fields, rather than their particular realisation in terms of creation and
annihilation operators, should be of underlying relevance {{cite:4cc859e917398e8f9e6a27075a53cd3b2e77b390}}. We have
proposed in {{cite:1000b9c5ba86e33b158c4eb403e3ffe84592f9e9}}, {{cite:043b6bde4e17955d908b9bd00572ef4b9b06ce57}}, {{cite:033df50448d6a5708f9c62e4e4193178ebe625b9}} to pick up
operators from the quantum-field algebra over curved spacetime, which locally reduce
to those which create free particles in Minkowski spacetime. These operators are employed in
quantum field theory in Minkowski space by deducing the Lehmann–Symanzik–Zimmermann
reduction formula {{cite:8698b1019573f2b997e7f6dcb32ecb851f49d847}}, {{cite:46586f7d4fc233c9dec6f8e4dd5d06572ea8fb70}}. The selection is achieved by means
of a bi-scalar which, in the weak-field limit, provides the wave-function description for quantum
particles. This basic idea has been applied so far to model spin-zero quantum particles in
gravity {{cite:1000b9c5ba86e33b158c4eb403e3ffe84592f9e9}}, {{cite:043b6bde4e17955d908b9bd00572ef4b9b06ce57}}, {{cite:033df50448d6a5708f9c62e4e4193178ebe625b9}}.
| d | e7e894a59ffdbf39ba39ba96c0ce3988 |
We make use of the following basic results related to matrix inverses (see e.g. {{cite:48ced209368465efc3e12b3d0b8035253a681b93}})
| r | b8daa6b01aec9c6e4e5980d217b08343 |
Even though the classical wavelet transform (WT) serves as a powerful tool in signal processing and analysis, its analyzing capability is limited to the time-frequency plane. Fractional Fourier transform (FrFT)({{cite:94a4ac14d728372b83972bcb50d273e00b8c4001}},{{cite:6704370185b8830f9e784eda671d8ea9a2413551}},{{cite:21eb5a23c0579939c9e3348a00fa84645e7c4004}}) gives the fractional Fourier domain (FrFD) frequency content of the signal, but it fails in giving the local information of the signal. Mendlovic et al. ({{cite:32135b75a7b8a4db06ae899819a393c0627bdf9c}}), first introduced the FrWT to deal with the optical signals. They first derive the fractional spectrum of the signal by using the FrFT and performed the WT of the fractional spectrum. But the transform defined in such a way, fails in giving the information about the local property of the signal, since the FrFT gives the fractional frequency of the signal during the entire duration of the signal rather than for a particular time, and the fractional spectrum of the signal cannot be ascertained when those fractional frequencies exist.
| i | f0e3d867b78ccd9cd7f6670f333b159c |
Gradient {{cite:d36d5d9056f9b2783ad7bfdd9986aa6fabda3cb3}} uses the gradient image,
which measures the sensitivity of the confidence score of a target label to changes in each pixel.
Integrated Gradients (IG) {{cite:8a39e84c5f9422470010b7a9e7b6e03e297c4b2e}} ameliorates the gradient saturation problem by linearly interpolating between the input image and a reference zero image and averaging the gradients over all the interpolation samples.
| m | 3bfc87692f32147e04a789c67acedd54 |
Moreover, the deep learning imputation methods leverage deep learning models {{cite:165d2fb684f502006f1504b9bb78a7f6b65769b4}}, {{cite:6aa268ae0c13cb968f1b1fa19f0640a3b68c950a}} to impute missing values with the mini-batch gradient descent.
This category consists of i) MLP-based ones like DataWig {{cite:623f1a95fa15ff616dd6e7b3405dc23d5ed41469}} and RRSI (round-robin Sinkhorn imputation) {{cite:74d5d0fc3c5a96e4c63cbb3702027c8cb8612b64}}, ii) AE-based ones, e.g., MIDAE (multiple imputation denoising autoencoder) {{cite:4232ae108fab990b479e19e5f35105a37c9d08fa}}, VAEI (variational autoencoder imputation) {{cite:9158e27c396175f319d7c5cf7ccfd3bc4bca9f28}}, EDDI (efficient dynamic discovery of highvalue information framework) {{cite:5d8a3645d870abd52ad2d15608011a15f51679f3}}, HIVAE (heterogeneous incomplete variational autoencoder) {{cite:e23cb35c4055ac6501d75547a155e3243939926c}}, MIWAE (missing data importance-weighted autoencoder) {{cite:1bb265f336377d840259573d63d48213cb51e0c8}}, and not-MIWAE (not-missing-at-random importance-weighted autoencoder) {{cite:9225bd7345d00cfa0c62add38a116c24d7c3e811}}, and iii) GAN-based ones, such as GINN (graph imputation neural network) {{cite:01c6240af231d9b9191768c22abfc6b230e86010}} and GAIN (generative adversarial imputation network) {{cite:3e27341a266274d1bfa60295dc4f06991659aa99}}.
The above methods calculate the model gradients with a series of random partitions of the dataset, to train the imputation models over large-scale incomplete data.
Nevertheless, both the iteration times and training cost of these methods are dramatically increasing with the rising volume of incomplete data.
| m | efef9d8e79cd3540c56a9b6a697208d7 |
The interest in kernel methods has recently surged due to their profound connections to deep neural networks {{cite:3b609d07876cf29ce6051bcbf41e026bf1283f80}}, {{cite:e9c4ff42f7506121d422c5aa1f587c0cc41cbd96}}, {{cite:ad90a8c5eb82951cb6ca2fad0c11926fe2407dea}}, {{cite:4e814706c0b8c7e14725e3c3f7d417dfd30c9f89}} and replica method of statistical physics proved useful in studying its generalization properties, inductive biases and non-monotonic learning curves {{cite:9155e0663134663741634d07526d452ca94b3a8c}}, {{cite:8b5ead0ea2718144145f59299ff5cf4cdeebe1af}}, {{cite:900c62853c27dc425fa4dd626bafabfba94f88a5}}, {{cite:1a3e44a8b34293b4c077d36b27d77f8826142cd0}}, {{cite:2eba85956fa6fcbce5777cbf700a966f9134707a}}, {{cite:90840aa2ca7dcc9a431792a78e28d5bc914b94e7}}, {{cite:12d1c3612d59c1e7301926a0320e913d3082ed9f}}, {{cite:9edf1742ee1bad493e71e96f76a8aa29f97b2da1}}. Along the lines of these research, we analyzed generalization performance of kernel regression under distribution mismatch and obtained analytical expressions which match experiments perfectly including those with wide neural networks. We considered several analytically solvable models to show how learning rate with training set size changes with training distribution and cases where choosing it different than test distribution helps. We demonstrated that our formula can be used to optimize over training distribution on MNIST which revealed that certain digits are better to sample more often than others. We applied our results to linear regression and demonstrated how dimensional reduction of the training distribution can be helpful in generalization, including shifting of a double-descent peak. Our theory brings many insights about how kernel regression generalizes to unseen distributions.
| d | bd065283f4260d891797354630fd2c99 |
In the experiments although we restrict the visibility of the robot to real world (lidar) sensor ranges and subject the robot to noisy state estimation, we do not account for occlusion and uncertainty in observing the environment. A tighter integration with a perception or mapping pipeline would be necessary here and we anticipate minimal degradation in performance by employing systems similar to past setups that have used trajectory optimization methods. Additionally, we currently precompute the SDFs and cache them in 3D environments and compute them from scratch online for 2D environments. Solutions to building SDFs incrementally online on GPUs {{cite:2751e2368510112e463586f26f32e98250c6421d}} can be utilized here to support the perception input for our approach and maintain real time requirements for the overall system.
| d | a10df6a1b0f2c9b474795608eefd8ba4 |
We used Steinhardt order parameters{{cite:d9d8d8b237a34704036f1cf98c8519a21286d652}} to classify particles as having AlB{{formula:d03d60e9-d6e5-4709-8d8f-3d84868d89ce}} , mixed FCC-AlB{{formula:13af9056-557e-41c3-b9b7-e043291bc434}} , or fluid environments.
We discuss our specific use in section S2 of the Supplementary Material.
The freud software library{{cite:b681aa41d117f22d48ed40f2dc39517aad05a672}} was used to calculate radial distribution functions and Steinhardt order parameters.
We used Ovito{{cite:17ce8772ba25c2590e2aae82a7cb6cea5f42d5ca}} to visualize particles throughout the work.
| m | c48db349e8f8fd40053a6d9a006c9a38 |
In this paper, we set the transformers as rational-quadratic splines. In fact, other splines could also be considered as long as the transforms are invertible, for instance linear and cubic splines {{cite:0a2cf2c4ccbda18d292fa3c2092ddd13bcf80e35}}, {{cite:06d95fd8891bd8b093f3d1cec069ccc50d5864d5}}. However, it is hard to compute the inverse path of high-degree spline based transforms. As suggested by {{cite:9f1b3021746413f96b2b4ce70b6daad70c54e428}}, calculating the inverse path of a cubic spline is prone to numerical instability. On the other hand, it is required that transforms are flexible enough, which translates into saying that there is a trade-off between complexity and flexibility. The adopted rational-quadratic spline based flows are more flexible than linear and quadratic spline based flows.
We use linear and cubic spline based transforms for a comparative study in Case 1 and Case 3. The CRPS values for linear and cubic spline based CNF models on wind farm 1 in Case 1 are respectively 9.34 and 9.11. The ES values for linear and cubic spline based CNF models in Case 3 are respectively 9.28 and 9.22. That is, the performance of the rational-quadratic based flow is superior to that of the linear spline based flow and comparable to the cubic spline based flow. Certainly, more advanced normalizing flow models could be used.
| d | f2abbfedd44115e75fe1c4d03657c371 |
Ensembles derive a prediction based on the predictions received from multiple so-called ensemble members.
They target at a better generalization by making use of synergy effects among the different models, arguing that a group of decision makers tend to make better decisions than a single decision maker {{cite:1e0f27ec7f8c190ad7f1b78018778c98afd32dca}}, {{cite:9443828948135ef0a0e5fec779c3f4fad03e603e}}. For an ensemble {{formula:21d891e8-07e8-47d3-8da8-06a9b4b775dc}} with members {{formula:28d812e3-b072-4a0c-a2c8-9251c243abe7}} for {{formula:56fc9519-6dbb-48c0-bc0d-ce7d698f01db}} , this could be for example implemented by simply averaging over the members' predictions,
{{formula:5e16b250-acf2-4554-92ae-2b776c44d1af}}
| m | 4360dab9de98874b1102b3f9585457da |
An elliptic Painlevé mapping comes with nine parameter points, whose configuration determines its dynamical properties {{cite:0a59ba24cafb2ee9322edca908192168f73cf0e3}}{{cite:224fca42806c6cb2bcfb9146cf604ce4ceb092c9}}. For some special configurations of the parameter points, one can find autonomous reductions of the elliptic Painlevé mapping. More precisely, if we start with nine parameter points which are the base points of a index {{formula:d33d3ae2-2b61-4e5d-b860-a2c0575cb89d}} Halphen pencil, i.e. they are the base points of a pencil of degree {{formula:f0de3d0c-5ddd-48c5-bf9d-dcd1979febef}} curves, each having multiplicity {{formula:31509907-ad45-473e-acfb-1a48e9611ed9}} , then the {{formula:f00a6296-decb-4415-9b33-3420a0b0bf1a}} -th iteration of the elliptic Painlevé mapping becomes autonomous and preserves the Halphen pencil {{cite:a49bbae6c9d7b8bc42bc753ef7d9d62a7745bf78}}.
| i | bf7b7a4e2549f22f3643cde6b681a69a |
For the PKU-Sketch dataset, we compare our method with Triplet SN {{cite:1bf5e5d46d709dbcbc62267eae44aeccdad25525}}, GN Siamese {{cite:96dd09bf40f5f9e6652c2d9c148489a5e71f1bb4}}, and CD-AFL {{cite:d9c406f9ece9d4b9d7979391bbe502d1f95dc140}}. Among them, GN Siamese and CD-AFL are pre-trained on the Market-1501 dataset. Note that CD-AFL only applied adversarial supervised DA on PKU-Sketch dataset. When considering Market-1501 dataset together, it simply applied pre-training and fine-tuning. Our model takes the Market-1501 dataset as the source dataset and uses the IHDA framework. The rank-1, rank-5, rank-10 and rank-20 re-identification accuracy of each method are reported in TABLE REF .
{{table:7acb442c-2422-41c6-8602-edc1786234f8}}{{figure:4865f671-88ab-47fd-a5f5-f35b6cd56ff5}} | m | c7502a69034bbc3c910185081889f30b |
Limitations of RSC .
First, to guarantee the model accuracy, we only replace the sparse operation in the backward pass.
Thus the upper bound of RSC 's speedup is limited.
However, we note that the backward pass usually is more time-consuming than the forward pass, which is also empirically shown in Table REF .
Second, some GNNs reply on the scatter-and-gather instead of {{formula:47a27da7-74a3-4943-9e44-7b7d4266c627}} (and its variant) to perform the aggregation, such as GAT {{cite:d6b2e9372c699edf73bb5218f4bc70c7bdc7eef8}}.
They are not covered in this paper.
However, scatter-and-gather based GNNs
can also be accelerated by RSC because the column-row sampling is also applicable to scatter and gather operation.
Similarly,
the caching and switching mechanisms are also applicable to them.
However, for the resource allocation Algorithm REF , the scatter and gather operations requires tailored error bound and the computation cost modeling in Equation (REF ).
We leave it as the future work.
| d | bb916ce7c9a293368402cbf0e5aa9132 |
In contrast, hamiltonians of PTQM models with broken {{formula:429cab0d-66d9-4d10-adef-230ec8e00fe0}} -symmetry are associated with indefinite metrics, and are referred to as pseudo-hermitian {{cite:5d89706bee6b2115c72a380130eee99bf680cf83}}, {{cite:07e80aa3dbbfba34b087635f3343da1fe26c26f5}}, {{cite:cce2bdf0eca1880db4ce846214fa0778f8ddcd4a}}, {{cite:d00817ae8e20e812b35985e03f364bc25761b206}}. They act on state vectors in Krein (or Pontryagin) space {{cite:0cdd6651e6a879dc430458532df98819bb519933}}, {{cite:f699850f1fb8548c26d70322b6afa657f96293d7}}. Their eigenvalues are either real or come in complex-conjugate pairs {{cite:e717030face06284969d4f3e87d19a36bb9cae1f}}, {{cite:e110bd2f1f22dcbd9964ce95e6f9c3545dcb12b5}}, {{cite:ffab74fc6918aa88254a59dd7bb04cf07c159088}}, {{cite:575040c6a2d1beaa7b0726db79539fdafee42b3a}}, {{cite:77b86647f9c39ded26413b6bce6a1fd774a21a11}}.
| i | 8ada7c897c37d94a1494b1b3b32c2428 |
where {{formula:1a1662fc-caef-42a1-ac35-c29ea8bc7bfc}} are the theoretical predictions of the N=156 observables used in the fit and {{formula:7f150be2-1801-4e03-8ef3-252d1b212ff2}} are the corresponding central values of the experimental measurements. The total {{formula:9ba302ad-303e-4069-8ee4-b8829706cd9d}} covariance matrix is obtained by adding the individual theoretical and experimental covariance matrices.
The theoretical predictions of N=156 observables along with the theoretical covariance matrix are evaluated using flavio {{cite:d7b9950560b4606bb2bdad600d881c1f1f817d6f}}. The experimental correlations, {{formula:6f385254-3f9d-4378-a8a8-6e3ff8f06c0c}} , are admitted for the angular observables in {{formula:a9c09896-c2ea-42eb-aaf1-9e61e3638588}} {{cite:aaaea59e712561a20a6c05c3c7cae77600b804be}}, {{formula:7fe1d32d-c206-4539-9ed7-12a63e0e9954}} {{cite:154d7a324a81e3773f780236c553dfe82305894c}} and {{formula:6b3f2660-6bb7-4324-b672-59f8ebdd9555}} {{cite:8698de102aaeeb3f23068c4b1d004ddbd24ac371}}. Further, for asymmetric errors, we use the larger error on both sides of the central value.
| m | 46146dde5bae88f73b8f6aef58196d28 |
If all the matrices {{formula:b6348cee-e26d-47e9-aebe-7246d4f9d5f7}} commute with each other,
the function {{formula:7a983fe9-3c42-4b11-81bf-69454db6fdb6}} is called a prepotential or a free energy (cf. {{cite:e3ede04b403e383462e54b51b2a2f0f63481f174}}).
Writing down the matrix entries of {{formula:7cb4a9f0-cd6b-4fbd-90e2-205cd10b1180}} ,
we obtain nonlinear differential equations for {{formula:65618bcf-0f2a-4b67-ac21-fb42ef9cbed9}} .
The collection of such differential equations is called the WDVV equation.
As a consequence, a prepotential is a solution of the WDVV equation.
Moreover in this case, {{formula:c53b9a26-ad84-4048-8485-979cd66a0cb9}} is called a flat coordinate.
| i | 057270888bacfeee8bac73ac787232b1 |
where {{formula:78baaa95-9e6f-4066-bbcf-e3bfe5f23abe}} is a class of explanation models; {{formula:b655f015-9a02-49e2-87f2-12e28f0a688c}} is a fidelity function which quantifies unfaithfulness of {{formula:092acbd1-70f7-4424-a494-82bdbb314c62}} in approximating {{formula:f00bf6d4-87cd-49fb-a053-d603bdbe8bde}} within the proximity of {{formula:2b9acd1c-74ff-4737-b251-e2505d4f54cd}} , defined by {{formula:5b56358a-cae9-4802-866a-89e30507299d}} ; and {{formula:a48b53c8-7a88-46f9-b9f1-2babd135e5aa}} is a complexity penalty. Essentially, {{formula:683209d4-72d6-420e-ae60-33aa1b59fe77}} is a locality-aware loss function and, in practice, is minimised in a model-agnostic manner. Usually {{formula:1ca17dcb-18d1-468c-a3a9-2fb5b6fad879}} is chosen to be a restricted class of models that are intrinsically interpretable (see Section REF ), e.g. linear models, GAMs etc. During training, instances are sampled around each data point {{formula:faf48f7e-0f98-47f4-a16c-3473caa34b97}} weighted by {{formula:1fd828bc-b314-4d53-abd5-cbf828ff5e01}} . In addition to local explanations produced by {{formula:ea64c5b6-91f8-4c79-8688-4d11a078275a}} , Ribeiro et al. {{cite:37447985b29b72001c0b9b1909993c373bb6cc9b}} introduce a procedure for obtaining a global understanding of the model {{formula:400e04b3-5312-4a67-8a15-6ec6f4ef9fe5}} . Given a limited budget, this procedure picks a number of explanations based on greedy submodular optimisation {{cite:2cd2d889e25fdde150725fc0c702b52af246b218}} and aggregates them into global variable importance statistics.
| m | 87eabb7be7ce81316ab37e64bf5086b2 |
Another area to explore involves the particulars of the DNC's external memory. It could be useful and interesting to analyzing the memory contents through the timesteps of simulations, to get a grasp on its inner workings as it moves through different parts (tasks) of an environment. It is also not certain what an adequate memory size is. Whereas we use the memory length, width and read heads settings used by {{cite:3200a319b7232717628c2a76d775aa6d6d42ca96}} universally, either smaller or larger memory sizes may be more optimal depending on the type of environment and tasks. This memory could also be exposed to the RL agent to provide it with potentially useful information the DNC is tracking while producing predictions.
| d | 1aaac1c29c0123065461baf02746b876 |
Understanding the physical process of thermalization within the framework of quantum mechanical principle has been a
long-standing problem. Thermodynamics and statistical mechanics are built with the hypothesis of equilibrium
{{cite:462d491a7d7dd3b29b29745c1844936bf742d36b}}, {{cite:1ba89c77436c8d3cc0a72e93c687a90318385865}}, that is, over a sufficiently long time, a macroscopic system which is very weakly coupled
with a thermal reservoir can always reach thermal equilibrium, and its equilibrium statistical distribution does not depend
on the initial state of the system. Over a century and a half, investigating the foundation of statistical mechanics and
thermodynamics has been focused on two basic questions {{cite:5bc51b03671b2ca29413262b2d90ec9965fbd188}}: (i) how does macroscopic irreversibility
emerge from microscopic reversibility? and (ii) how does the system relax to thermal equilibrium with its environment
from an arbitrary initial state? Rigorously solving these problems from the dynamical evolution of quantum systems,
namely, finding the underlie of disorder and fluctuations from the deterministic dynamical evolution, has
been a big challenge in physics {{cite:5bc51b03671b2ca29413262b2d90ec9965fbd188}}, {{cite:462d491a7d7dd3b29b29745c1844936bf742d36b}}, {{cite:1ba89c77436c8d3cc0a72e93c687a90318385865}}, {{cite:b3cbdc298a0ebd46544c02582abefbe508870948}}, {{cite:ef5b26604febe105d2dba6925c83b11598b3055d}}, {{cite:d4576d5fcbfb263f25b1058a300f80cdec5282c0}}, {{cite:1f7d52e97c9f825613b61019baeb9d34a56b7bd0}}, {{cite:1a414fecc11f53c78fa05c762f5e05222c731cdc}}, {{cite:d08a62039d8213379fbf00863f3e5b08182603a0}}, {{cite:53186173bf5552f79757721635a2b573e1ad7cf3}}, {{cite:a76cbf30875564b579091519e087aeacefe39551}}, {{cite:ae1595f1c5fd5308f2ec7cfce1827d2af0bb7c12}}, {{cite:e1074ae54085490a8182413e08290fc690ad75c7}}, {{cite:7f4d7c0d014465f01384eb4d087f0af1fb04de54}}, {{cite:9c94afd13845381df0c8f232fdb74480170e5185}}, {{cite:291428d930d5120730fe57a741c262ad6be7b5c9}}, {{cite:3977566ec90bc9924630d3403303ed905ec70105}}, {{cite:67af4d9b1ea4bb42430594b6c76cc867f3eb81a5}}, {{cite:d87c161e12363c39beee8dcf9acd8d7987da1dc4}}.
Obviously, the foundation of thermodynamics and statistical mechanics and the
answers to these questions rely on a deep understanding of the dynamics of systems interacting with their
environments, i.e., the nonequilibrium evolution of open quantum systems.
| i | 24e77f97543f5e8bc4a1e7f4847dbc51 |
Table REF gives the normalized signal and background cross-sections due to all selection cuts.
These are obtained by multiplying the production cross section given in the first row by acceptance efficiencies.
The production cross section are estimated by multiplying the leading order (LO)
cross section obtained from PYTHIA8 with the corresponding
{{formula:7a5a23cc-0c26-43bd-a6c0-2639457acaee}} factors The appropriate {{formula:99e8f96c-c085-4066-890f-74178db19888}} factors for {{formula:8be6e1a3-10f2-4af8-a782-a371f76396e1}} and
WZ processes is 1.6 {{cite:fa785e4d4bcb213240e97a7902c59baedb688472}} and
1.7 {{cite:f4ecd460a997033be9e09c35386287639d8a418c}} respectively while for the signal
it is 1.5 {{cite:846da60efe1dabd5f20ec0b482a736882503870a}}..
Corresponding to these signal and background cross sections, we also present the signal significance by computing
{{formula:9b26c006-df98-4ab0-a1c3-ac755a27bf46}} for integrated luminosity 100 fb{{formula:e778d017-7875-477d-b5ae-5d761a9cbb81}} as shown in the bottom of Table REF . Although case(b) corresponding to {{formula:b86450ca-0067-435b-9d73-69d94cd687af}} -like jet veto results in the largest cross section for all
signal parameter space, signal significance does not improve
due to comparatively less suppression of SM backgrounds.
With the increase of gaugino masses acceptance efficiencies goes up
as final state particles become comparatively harder, but {{formula:4010a360-35ba-487e-a3af-ed327aafbd72}} is
depleted due to drop in {{formula:8cccb2d8-1f74-41f8-b9ea-47569a422270}} pair production cross-section.
While estimating signal rates and significance, we assume
a maximal flavour violation i.e. {{formula:4e0eca60-8b35-488e-9cca-e017f574794d}} .
Obviously, a further suppression is expected by a factor {{formula:f1fa9ab6-70bf-4fb1-bea4-d3db49354929}}
which depletes the BR of {{formula:f9fea13e-0374-4e5a-ae25-4bf6caa6a714}} , (see Eq.REF ). For a given {{formula:ede01d44-034a-4b28-b926-3d97dd54fe39}} , {{formula:c1d60490-02da-44c3-9187-a424e525be8f}} is a function of the slepton mass as well as the mass splitting {{formula:34b2f0c3-fc1e-40a6-bc31-1c610c2966cc}} as shown in Fig. REF .
For instance {{formula:c236d1d0-4388-4cec-a187-05d657b0065e}} may suffer by an order of magnitude
for {{formula:f709d5af-d788-4692-b72a-320fb3411e25}} .
While the lower end of the spectrum can lead to a larger {{formula:c2c035a1-1646-4aa3-9e98-4b4ce8d78544}} ,
the corresponding {{formula:570f5939-d20f-4073-a05c-9804d74b5c00}} decreases as we move further
towards the IR part of the slepton spectrum. This can be attributed to
stronger bounds on {{formula:6511fafb-861b-44e7-8387-a7168aa701a0}} for lower slepton masses.
Though the lower mass is not yet ruled out, it is more economical to
consider relatively heavier slepton masses as the bounds from current
and future experiments will be relatively weaker.
| r | b7bca8b0c4f196eba514a4eab88e87b9 |
In quantum cosmology, the wave function of the universe is studied in problems of its quantum origin {{cite:6b7e82bcd4304b96c998f30d0991dd8ff326c34e}}.
Quantum tunnel transitions with a change in the signature of spacetime are described in the imaginary time
formalism {{cite:33e6aafa5d2015cdadbaaafac25157c70405cb0d}}. The description of the classically forbidden state using imaginary time means the complexification of the conformal superspace {{cite:23e03040d39a0e9437ee652624a204b74803124c}}.
This complexification makes the conformal time variable purely imaginary and transforms the Wheeler–DeWitt equation from hyperbolic to elliptic. This is analogous to the transition from the hyperbolic to the elliptic Klein–Gordon equation for Wick rotation.
| d | 4e9a8f4e24bd80517a14dc61b848a358 |
This statistical model implies many other strong
performance guarantees for a variety of tasks
such as covariance estimation, OLS, Lasso, PCA etc.,
which directly apply to data transformed by Gaussian embeddings, regardless of the
distribution of the original data matrix {{formula:0818f465-26fd-4aa6-9939-99762cfc466c}} . We will refer
to this phenomenon as Algorithmic Gaussianization. Unfortunately, Gaussian
embeddings carry a substantial preprocessing cost, compared to, say,
uniformly down-sampling the data, and so, many other sketching
techniques have been proposed, e.g., Sparse Johnson-Lindenstrauss
Transforms (SJLT), Subsampled Randomized
Hadamard Transforms (SRHT) and Leverage Score Sampling {{cite:a1a2fdc08d6a8a1f00938be155d6be239033026a}}, {{cite:77daf1823116f5157175dbb26622c83bc6df28e3}}, {{cite:9cb4fba873237c5796fe7685d670766be06bb7b2}} (see Section REF ), which can be viewed as
computationally efficient algorithms for transforming a large dataset into a
small data sample. Existing work has shown various approximation
guarantees for these algorithms, via arguments based on subspace
embeddings and the Johnson-Lindenstrauss property {{cite:98ce938ccbbeb22057feb0175a723f693bafb6b7}}, {{cite:9a97915af209fbf2e64982b901ec3b2c6013bc5b}}, {{cite:dcce7f926d15b47043ee9df05b3343c7da9b2587}}.
However, even though some level of gaussianization is implied by these
guarantees, exact parallels with Gaussian
random designs are not available.
| i | 5e2b7cb40647b974d147d886aefc56ad |
Experiments were carried out on a dataset acquired from 2012 - 2020 across the German federal state of Lower Saxony. The dataset contains a total of 19,995 aerial images, each of which consists of RGB and near infrared bands with a spatial resolution of 0.2m. The dataset was randomly divided into training (70%), validation (15%), and test (15%) sets. Dominant tree species within stand polygons which overlapped an image patch were used to attain the image-level labels. To evaluate the WSSS results, we rasterized the polygons corresponding to the test set in order to obtain pixel-wise labels. A total of five classes were extracted from the polygons: Pine (Pinus spp.), Spruce (Picea spp.), Beech (Fagus spp.), Oak (Quercus spp.), and Cleared. Cleared represents any open areas without tree crown cover (e.g. meadows, clear-cuts, water bodies, etc.).
For the experiments we used DeepLabv3+ {{cite:d8086df686fbb98033bf78c0fd2669b702a86735}}. Given that the architecture was designed for semantic segmentation, we replace the segmentation head with a multi-label classification head in order to obtain predictions at the image level.
For evaluation of the WSSS performance we utilized {{formula:276cc8df-e047-4ff2-a1ef-a97bd7fc91e3}} , and also examined the number of model parameters (in millions) and the segmentation time per image. All results were obtained using {{formula:c9260802-b4b1-4605-bdd5-f0cda33a9fc6}} . The number of seeds used in SEM was chosen empirically based on the performance across the validation image set, resulting in a seed value of {{formula:97a755ae-53fe-4852-8aa4-18fd4346b0de}} .
{{table:59d85451-9577-4a37-a3fb-3b5bba7892ec}} | r | 561cb42571f2eed9b7e78b4ade352788 |
To compute the conductance of such a wire, the Landauer-Büttiker formalism of coherent transport
is used {{cite:cfbedb4639be035bbd57cc82e4985c8a93fed87f}}. It allows to relate the dimensionless conductance {{formula:dec38adc-edb1-409c-95d7-d47a7548defd}} of a diffusive wire with the scattering matrix {{formula:ed349b95-1770-435f-9a9b-942676b8c161}} :
{{formula:050584fb-4461-4e72-8b14-7216aa62f308}}
| m | 47135911e629b0910425cf11ece3a5b0 |
As the 2.5-D slice classification network, we use a 2-D ResNet50 pre-trained on ImageNet {{cite:9f9b4dfc2780f80f0fe86f12ef321eb3d0ae148e}} as feature extractor with a consecutive LSTM layer. Furthermore, we also incrementally evaluate the benefits of segmentations on the input images to perform an informed cropping, as opposed to the previously shown centre cropping. Moreover, we incorporate the additional image channels from the segmentation task in the classification network. Lastly, we utilise the network weights of the segmentation encoder. As a segmentation network we used a DynU-Net from MonAI {{cite:032dfb78b9647d7883ef4bdabe13b43057fc1a12}}. Extracting the encoder from the DynU-Net and reusing weights allows us to reduce the network size and thus the computational power and time for training and inference. For classification without weight transfer from the segmentation network, we use a ResNet-50 architecture.
| r | c11a092c226f1c72e1f028a7d3ff21f1 |
Human study to evaluate concepts' explainability. We performed a Mechanical Turk study to evaluate the explainability of our extracted complex concepts to the end-users. The participants were asked to select from four different options (presented in random) of what they consider to be the best possible Explanation for the classification of a given video. The four options are: Concepts predicted by bottleneck models without attention, Concepts predicted by bottleneck models with attention, Word-level concepts and a Random set of 2-5 concepts from the set of the most frequent concepts. The methodology of this study was inspired by {{cite:e524a6ce050c91814d584da27489ceb38d52b096}}'s paper. Figure REF presents the aggregated results of the Mechanical Turk study. The complex concepts predicted by the concept bottleneck model with attention was considered as the preferred explanation by 68% and 57% of the responses in the MLB-V2E and MSR-V2E datasets respectively followed by the concepts bottleneck models without attention in 20% and 28% of the responses for the two datasets. The presented confidence intervals are calculated using the bootstrap method as described by {{cite:d41bf759ae4b460ec7dba2cd860211f0be076e5a}} for 95% confidence.
{{figure:ada59649-9720-417e-8d6b-a90888629c95}} | r | a6665d6486920d897abfd49a04b82e32 |
Sensor fusion algorithms are often developed with, and evaluated on, data sets with perfectly synchronized sensor measurements, e.g. the EuRoC data set {{cite:e44c40f242eef519788ec74027e8d7a3d3900661}}.
In reality, separate sensors are never completely synchronized.
The degree to which they are synchronized depends on the which synchronization primitives are used and how they are used.
There exists few commercial products for flexible synchronizing various sensor sources.
System designers are typically left to implement their own synchronization solutions {{cite:648827df9eacdde59a73ab8fed039f9af867f739}}.
Not only is this error-prone, but they lack a framework for understanding the precision requirements for such a solution.
For extreme precision, one could implement a custom timestamping circuit on an FPGA yielding deterministic timestamping and parsing of sensor data.
In the other end of the scale, there is software timestamping of sensor samples “on arrival” running under some non-real time operating system.
The first approach can achieve synchronization precision in the order of nanoseconds, while the second solution can suffer synchronization precision in the 100s of milliseconds range {{cite:7326b5dc93ba57b1a38297e2a61f4c46f2b6ce71}}.
| i | 551c4907de29c9f5e1575d057df86fb7 |
denotes the relative entropy of {{formula:94d0cb8a-fcd8-4cc4-ab9e-e7b76e69b536}} with respect to {{formula:8837d3c2-cee2-4fb2-8090-21081b3d316d}} . In the special case when {{formula:8aa59df3-dfbe-4f87-a3c9-0377c9b13251}} and {{formula:886e3bb4-67a8-407f-8c88-c6584207ea92}} are probability measures, (REF ) reduces to the relative entropy defined for the space of probability measures; see, for example, {{cite:87e3b3f474e2d92afbabca9681fb6c38a81e2705}}. In our setup, however, {{formula:f4a6f60a-b492-465e-98d2-f42a4ae7bf99}} and {{formula:8b5c89d3-2729-402f-ab9a-78a098e9c4c1}} are not necessarily probability measures, so we need to extend the definition of relative entropy as in (REF ).
| r | edc9aa0f1cc66bea212b617f30cce14e |
1. Replacing pronouns by entities using the neural coreference system {{cite:30c1fa1362f1c3ae3a11486d5f9b023b3bb2d1c7}},
| r | 2e21d3e61a89c7c02486df00465e98ed |
We consider a memristor Liénard system with external harmonic perturbation from Eqn. REF
where the parameters {{formula:f892246c-fc9f-44ae-8526-6fa36dbb4d3f}} are known. We generate our synthetic dataset by simulating data points from the nonlinear differential equations using the Runge-Kutta method {{cite:971ef68b3222e1cbb0f0395004b9c131acada890}}. We train an LSTM network on the simulated time series while enforcing the physical law
on the network as a regularization term. The network learns complex patterns from historical values and the physical distribution of the target values. In the standard PINN model, the time index is considered an input, and the output is the solution of the differential equation. Computing the regularization term amounts to differentiating the multilayered perceptron (MLP) network and computing the time derivatives using auto-differentiation {{cite:1cc9f198c4d23cd3cdbc6ce6e51bf9315ac8db83}}. However, time series are discrete observations and there is not a substantial mathematical equation governing the observed variables in time, and subsequently, it is difficult to compute the derivatives in time. To overcome this, we compute discrete derivatives of the time series. For an observed time series {{formula:fcf55467-c1a9-48c4-8485-9f12b97b5b9a}} indexed over time {{formula:8505815f-68ae-4e40-aa1a-ec389dbb66f8}} , the discrete-time derivative of order one can be written as:
{{formula:d1e9a0e6-daa2-44c7-9c77-107102f22720}}
| m | 4e74a2b91f97ecf2e246c4dd15a2afb7 |
The theoretical result is based on CSP, which is not computational feasible. As mentioned in {{cite:ef7215a502de6207c139c5394d87509436fa4bdf}}, finding the solution of the CSP optimization through exhaustive search over the codebook is infeasible, even for small values of the video size.
It is hard to verify the values of the parameters (such as {{formula:e9303ee6-cffc-4779-bfbe-8e9b79d5b205}} ) in real applications.
The mask is considered to be i.i.d. Gaussian, which is unrealistic in real SCI systems. Specifically, the mask or DMD response of light is always non-negative. Therefore, there is a gap, or at least a constant—probably a DC (direct current) term—between this theorem and real SCI systems. This gap arises from the `zero-mean' requirement in the proof of the theorem.
During the implementation of modulation, especially using a DMD, a binary mask {{formula:1d0b8480-51e4-43c3-8d6b-8b6d28e2e319}} is generally used.
In this case, an extended theory by considering the mask being the Rademacher distribution (i.e., random variables taking values 1 or -1 with equal probability) will be closer to real applications. An explicit bound (similar to Eq. (REF )) can be derived using the concentration of measure as used in {{cite:90c164e16fa8c0e2d762592a61fe57ce7a2c4192}} by considering the Rademacher distribution for the mask.
Another solution stems from the hardware side. By employing a beamsplitter into the optical path {{cite:07bcca135326f3c6987a714777ebcbf324d5f620}}, {{cite:609232e5e4f334ceaebdd215ae289a3b97e6aff0}} and imposing two conjugate modulation patterns (one with {{formula:efbce9c9-4a38-4b16-b4af-9b9217d2b659}} and the other one with {{formula:a4fc728d-02c5-4083-82ad-04b723f7c86e}} , when one cell of the first mask is 1, the corresponding cell of the second mask is 0) and then using the subtraction of these two measurements, we can get the measurement derived in the theorem.
This solution can also potentially increase the dynamic range of the measurement and thus improve the reconstruction quality as discussed in Sec. REF .
| r | 252c399096c6d406730bfa3f671e8b8e |
ALM was first proposed by Hestenes {{cite:ee34fc7bf5b038139d7a01f741ac6aa28bfa4b20}} and Powell {{cite:860f2e95fe087d7829af2afdd3269dac4d4f059c}} in the 1969s as an alternative to the penalty method for constrained optimization. For the penalty method, the Lagrangian multipliers are absent ({{formula:fd6291d5-7ff2-4c8b-98ef-89fe04457814}} ). The motivation of developing ALM is that penalty method generally requires to increase the penalty parameter {{formula:354d79e5-49f6-474a-ae92-0ef205ef4601}} to be very large (e.g., infinity), making the resulting relaxed problems ill-conditioning and very difficult to solve. Moreover, the method was found to be very sensitive to the round-off error caused by the means of analogy computing at that time. In such context, ALM proposes to add Lagrangian multipliers to the objective of penalty method. It was found that when the Lagrangian multipliers are close to its corresponding optima, one does not require quite large penalty parameters, alleviating the difficulty of ill-conditioning.
| m | 1cbd66a5c034d09d6eaf76abf7aa2850 |
[leftmargin=*]
I.
Active Ornstein-Uhlenbeck Process (AOUP): We first consider the scenario where the active force at each boundary undergoes an independent Ornstein-Uhlenbeck process {{cite:65486ce850aef28c97ca8c2da37dea4c3a29b1fa}}, {{cite:4d59b0a9c330207ace9ff07921110625b20f9610}},
{{formula:02a6a75b-8450-4af3-a030-0715e3834b62}}
where {{formula:db723894-76c9-49ba-88f2-861bf3421a0c}} is a Gaussian white noise with {{formula:4441e112-2a28-4211-b89e-52c9c476714a}} and {{formula:ae252bb0-9634-4164-b345-3bb472c35070}} ; the diffusion constant {{formula:94be83c3-3ffc-416e-8590-413f70eb4244}} denotes the strength of the noise. The linear nature of the process and the Gaussian nature of the noise leads to a Gaussian propagator for the active force {{formula:8fcd46d9-faae-4825-a264-bbdd2df637d0}} ,
{{formula:e11d6444-d6d2-43bd-a573-c887d33d9fb8}}
Evidently, the stationary distribution of {{formula:d2b12640-0acd-4096-8ca1-48f9736e366a}} is also Gaussian with {{formula:16c26a66-00c4-4bd2-8f1e-5a5fc3271fb1}} and {{formula:0a705953-6fb8-492f-a75c-8bda61502723}} . Equation (REF ) implies that the stationary two-point correlation of the active force {{formula:47ff1a8d-49b6-45a5-a16b-6f5e5bc8e58b}} is given by Eq. (REF ) with,
{{formula:9a4f6e82-d99d-4e1c-a3db-444bba956ed9}}
The linear nature of the system and the Gaussian nature of the active force {{formula:1a591761-17cc-4225-8679-88846ddaf794}} ensures that, for the AOUP drive, the joint probability distribution of {{formula:d978d901-ffb4-4abf-a0d0-39e755fd18af}} is also Gaussian,
{{formula:21ecad41-d89e-4282-b57a-80426fbc4f2d}}
where {{formula:387d84cb-921b-4f5a-a4cf-dff1f0890e74}} and {{formula:2d36a964-a4b3-4b25-9d95-1dbd1feca2d4}} is the corresponding {{formula:31c54cce-f410-4521-a79d-283fdcf84b9c}} dimensional positive-definite correlation matrix.
{{figure:3a2499a9-0615-4928-ab0d-0b7bd928375d}}II.
Run-and-tumble process (RTP): In this case we consider the active force {{formula:58489076-b29e-4153-a0fc-d15a0975208d}} to be a dichotomous noise similar to the famous Run-and-Tumble process {{cite:2179be23cf1f7b14088b13bc63aec2b61c46cac2}}, {{cite:e46bea10183dfeeabf0883336e0228de90f72720}},
{{formula:0990bc7f-9431-42b9-a1f7-826fdf583f15}}
where {{formula:2df6b83c-fbea-4251-bc6e-0c19ea5788f9}} alternates between 1 and {{formula:a8c4dc0a-f2eb-4a0a-a04f-d6bcbf03ed65}} with rate {{formula:a05c2c7a-f5df-4f51-9719-6c2fd4091d89}} . In this case {{formula:415f3cfa-3b6a-495e-9bc4-57661a642e69}} can take only two discrete values and the corresponding propagator is given by {{cite:07657d2ad6d61fec2b2da477f593344dad844343}},
{{formula:fe407bd4-802f-4686-80f5-b48d7bc6a971}}
Clearly, in the stationary state, the two values of {{formula:b0b7f532-3072-4d88-a7c9-b53fe1999253}} occur with equal probability {{formula:51778bdd-726c-4626-8128-9b3a32e38608}} . It is straightforward to see that this process leads to the two point auto-correlation of the form Eq. (REF ) with
{{formula:3225060a-6825-46f7-b43d-4b08f0789125}}
However, the higher order correlation of {{formula:3ed5da9a-aabf-4f3a-8d81-4fcd51745b6f}} , computed from Eq. (REF ), are quite different from that of the AOUP, and, in general, the stationary state distribution {{formula:e2bf2e77-334d-4cd5-a5c5-d6410a2bcfce}} is expected to be non-Gaussian.
III.
Active Brownian process (ABP): The third case refers to the scenario where the active force evolves according to the active Brownian dynamics {{cite:d9cb2a4ca2a75c5a7192911e17dc2b742d02508f}}, {{cite:d7a59421f3e126cd9774cdf23261f9aae621c7d2}},
{{formula:60304f09-ff9b-4bda-aa66-e6fd57e4fa28}}
where {{formula:90cb99f7-a68d-4a92-8849-8dc1f524266a}} refers to a Gaussian white noise with {{formula:d05fb7ec-5b77-4822-b832-59236873c1be}} and {{formula:d4bfa119-05af-48c2-be62-6484ff4e12fa}} . Clearly, {{formula:93025890-e7b3-45f5-9340-151abc8c3c1d}} undergoes a standard Brownian motion which leads to a Gaussian propagator,
{{formula:ea8491b8-0b84-4271-96f5-47aa424a0c84}}
Corresponding distribution for {{formula:bc2d8ab2-3f51-46e3-93a6-e8ea732e2d72}} eventually reaches a stationary state,
{{formula:082a2900-a702-4d15-814b-ef3bdf00f1f8}}
The auto-correlation {{formula:0bd1929f-2926-45b9-b30d-18a92a73a33f}} is given by Eq. (REF ) with
{{formula:512e498c-367b-4094-af82-d214e2bddaa6}}
However, the higher order correlation for {{formula:4b26b743-b619-4c66-bd62-2ac5807f9084}} for this case is different than that of both AOUP and RTP and the stationary state weight {{formula:4e38dee7-3dd2-4af9-a096-a805fe4d557f}} is expected to be non-Gaussian as well as different from that in the RTP driven case.
| r | be0d3cfe47ac1a4df769c5ec6800135e |
In this section, we discuss the power of conditional SI.
It is important to note that a conditional SI method is still valid (i.e., the type I error is properly controlled at the significance level {{formula:7ead0f78-e6a4-46d2-bfe6-70d70f6c1e1f}} ) in the case of over-conditioning {{cite:5245bd2829d5f6f1df614e1c0145efdc6a0676cb}}.
In fact, in the early papers on conditional SI (including the seminal paper by {{cite:431459b26f17fb1d2795ed7fe36a5777b68f02f4}}), the computations of selective {{formula:e72bfc23-9e17-493c-8d64-61fea1ecf685}} -values and CIs were made tractable by considering over-conditioning cases.
In the past few years, for improving the power of conditional SI, a few methods were proposed to resolve the over-conditioning issue {{cite:1eb34c55a8d594f8d1f0a70d3b626efcedfc39f2}}, {{cite:6dc3f35e051a23e877f9c528f2c9840c84a9428a}}.
In {{cite:1eb34c55a8d594f8d1f0a70d3b626efcedfc39f2}}, the authors pointed out that the power can be improved by conditioning only on individual features, rather than on the entire subset of features selected by Lasso.
In {{cite:6dc3f35e051a23e877f9c528f2c9840c84a9428a}}, the authors proposed how to avoid over-conditioning on Lasso conditional SI by interpreting the problem as a search problem on a line in the data space.
In this study, we introduce these two ideas into the task of quantifying the statistical reliability of the locations and components of multiple CPs.
Specifically, by only considering the locations of {{formula:10931ecf-6afc-454b-8a05-ff8c22c195db}} points before and after each CP, the statistical reliability of each location can be evaluated independently of other locations, which results in improved power.
Furthermore, the approach of searching for matching conditions on a straight line in the data space is inspired by the method proposed in {{cite:6dc3f35e051a23e877f9c528f2c9840c84a9428a}}.
In the experiments in §, the SI conditional on the over-condition {{formula:062c40ce-c04f-46b5-afa2-ef24aaa4b28c}} is denoted as “proposed-OC”, and it is compared with the minimum (i.e., optimal) condition case denoted as “proposed-MC”.
| d | 9d0a57673ddc538fb7d09c7d10935124 |
We employed the Gillespie algorithm (GA) {{cite:1f4906ffb434aedb06e458d00a36b0b302492338}} to simulate the protein networks.
Each protein network was made up of these reactions:
{{formula:42b5cd10-0388-43a0-8e35-db4d8d84e21c}}
| m | 74218246beb688c5388e56150bcd0e98 |
In fact,
we adapt the single-spin-flip variant of the Metropolis algorithm {{cite:b610fe6fdec4dab148615ee3a0eeb782137ed037}}
since we use the single-junction-flip dynamics to generate new states.
We refer to the literature to elaborate more sophisticated variants {{cite:b610fe6fdec4dab148615ee3a0eeb782137ed037}}.
| m | 256600544d0637bb81f2093848744e6c |
A black-bounce spacetime suggested by Lobo et al. {{cite:616648dbc641a8e78f7057c7c49e46def55179a9}}
forms two photon spheres which can be observed by an observer in the parameter region {{formula:c2bc382a-b5e0-4f0a-83fe-4d7dd6a90a5f}} .
We apply the formula in the strong deflection limits to a supper massive black hole candidate at the center of our galaxy.
As shown in Table I, the images slightly inside of the primary photon sphere is several dozen times brighter than
the ones slightly outside of the primary photon sphere.
The images slightly outside of the secondary photon sphere are quiet fainter than the other images.
{{table:950b911b-f68c-41d0-8c95-34783b996184}} | d | cb341bf348167ee6d4ea03206d5f9fdd |
Full data.
We follow the setup of {{cite:ae7afe55e38718b89ed44feed89af3bc4da0bb0d}}, except that we apply a dropout rate of 0.1 on the expert MLPs (as done in {{cite:5fc2ec709721950a4d29f013e584d1daccdd26a4}}),
and we halve the number of fine-tuning steps for all datasets other than ImageNet.
fig:imagenet2012testaccflopsdays shows the results on ImageNet (averaged over three runs).
Here, V-MoE also performs better than dense counterparts, though we suspect the fine-tuning protocol could be further improved and
tailored to the sparse models. See tab:modeltable for all details, including results on other datasets.
| r | 7dae2b8bbfbfff82d605c46879611a9e |
Federated learning supposes to support co-building models among multiple devices, while preserving privacy in-between.
Among these devices, each holds one replica of versioned model and its local data objects. Usually, one small set of
devices, called servers, are powerful that can publish a master model, and compute the alignment between all trained local models. Rest of the devices, called workers, subscribe models from servers, and repeat the training process in-sync, i.e., federation.
As such, raw data objects are not exchanged among parties, while the target model may converge after repeating the training process within a sufficient time.
In theory, the federated learning framework is anticipated to scale, working for millions to billions of devices {{cite:cb4060dc9fd270979f85c9646a1464ea2a5af069}}.
| i | c56cbd0d9572ebd2eb04f3b2f8d3b8a4 |
the inverse of the minimum value {{formula:49aff405-b882-43fe-ad70-c7cc9c16ae00}} .
Adopting PDFA techniques takes us into a different direction from
existing approaches based on a direct clustering of histories such as
{{cite:541868f80717e9f92340554f2badee26f6a4bff2}}.
There, histories are clustered according to the probability of the
following observation state. Since the algorithm compares single histories, the
accuracy of the algorithm depends on the probability of single histories, which
can be exponentially-low in their length.
In turn, the required length of histories can grow with
the number of transducer states, and hence the approach can require
exponentially-many episodes in order to achieve high accuracy.
For instance, in our running example, a history of length {{formula:bd15e5c7-165c-42d5-ac0b-89149cfd168b}} has probability at
most {{formula:b30d5818-544d-428a-a952-16176af48ee5}} under any policy, with {{formula:a4734949-8d0a-4014-b38a-2f51eb35f61b}} the maximum probability among
{{formula:c82bed4c-c526-456e-a094-2c7f69e3f626}} .
Histories of length {{formula:f847376e-1681-4fb0-a458-4081abad703e}} are necessary to determine the best action at the
{{formula:6c3e7047-7903-4fef-8a4c-3848c840862e}} -th step.
To address the issue, PDFA algorithms build
states incrementally while relying on their distinguishability.
This way,
each state gathers the probability of all the histories it represents.
In light of our results, we believe that these PDFA techniques will be
instrumental in developing the next generation of tools for RL in RDPs.
Acknowledgments
This work has been partially supported by the ERC Advanced Grant WhiteMech (No. 834228) and by the EU ICT-48 2020 project TAILOR (No. 952215).
Appendix
The appendix is organised as follows:
Appendix contains proofs of all our technical results.
Appendix contains a detailed description of
our running example.
Appendix contains a comparison of our approach with the
one of {{cite:541868f80717e9f92340554f2badee26f6a4bff2}}.
Proofs
Proof of Theorem
We prove Theorem 1 through a series of lemmas, with each lemma showing
the necessity of one of the parameters given in Equation (REF ),
which we restate here for convenience:
{{formula:b9ef71fe-a37e-4357-af7b-23a88f1084c6}}
The following lemma shows the necessity of the distinguishability
parameter {{formula:fbf5d514-237f-4286-909c-7eb40025f142}} .
The result is based on a construction from {{cite:cacee034299bb297a4d386ac0275d151469c3afa}} and it relies
on the conjecture that noisy parity functions are difficult to learn. Briefly,
it is difficult to learn a parity function from input-output pairs if the inputs
are chosen uniformly at random and the outputs are flipped with probability
{{formula:8fb2e0d4-23e7-427a-a647-aec1bcd379c4}} , called the noise rate. A clear
statement of the conjecture can be found in {{cite:cacee034299bb297a4d386ac0275d151469c3afa}}.
Lemma 1 Under the assumption that noisy parity functions cannot be PAC-learned in
polynomial time, for every algorithm there is an RDP {{formula:df490e8b-4b9c-4eca-81e4-cb3607a9d613}} such that the
algorithm does
not reach accuracy {{formula:126d4e67-8d0f-4f7b-9519-6c8fb0d1d4ce}} and confidence {{formula:c393f544-2264-417c-908e-c2c546eae97b}} in a number of steps
{{formula:a3628024-4fc9-468b-8708-fa0b2b35726d}} if
{{formula:eb53bd66-9068-4930-a94c-0c0cfa5c76eb}} is {{formula:b5cff863-41e5-4771-bf86-a6024a6d47c8}} after removing the distinguishability
parameter.
We show a class of RDPs such that the existence of an algorithm that reaches
accuracy {{formula:129c469a-c29d-4226-afeb-4d42b62b8c3a}} and confidence {{formula:f0f87668-3790-4903-8e2e-396e9b364e85}} in the claimed number of actions
steps in all such RDPs contradicts the conjecture about noisy parity
functions.
These RDPs are inspired by the construction given in Theorem 16 of
{{cite:c3ac5e591aec9638f07f400afda0f0bf7b5af4a8}}.
Consider a parity function {{formula:244d76d6-6324-40a4-a5b9-496c8684b987}} where
{{formula:ff6b626a-9cf6-4d46-ba2c-5fa6221de90c}} and
{{formula:e6ccfa07-3bb7-4858-bce6-6089b5f56e44}} iff the parity of {{formula:13efcf3c-d3b3-4042-a2d6-0723f8ed535c}} on
the set {{formula:54dda4ba-3bcf-4811-9c2c-d6df2cdf3573}} is 1.
We then build an RDP {{formula:c56d9ebc-8742-40e8-b55d-384439eb4340}} where learning a 1-optimal policy amounts to
learning {{formula:d8ebb4c0-0a0a-4fdd-9633-ceff18d09911}} from noisy samples of {{formula:647f838e-abe6-4cc3-abbd-a2d01effcdc4}} for a given noise rate
{{formula:2f66dcc0-ce27-4f16-aa89-8e843b321dd9}} , which contradicts the noisy parity conjecture.
The RDP {{formula:4735a6f7-87f1-4e15-962d-13588369fbf1}} has observation states {{formula:592b3735-47be-4a32-a9f5-104b54095955}} and actions {{formula:4ec66c81-317b-4d73-abeb-cd8b6dfe9deb}} .
The initial observation state is 0.
When the history of observation states has length at most {{formula:87598f5d-8ce5-4403-b6c2-f5d4bc210db6}} , then the
next observation state is chosen uniformly at random regardless of the action,
and no reward is issued.
When the history {{formula:6981e755-1c54-4c57-b5f0-d7bd4a8c3453}} of observation states has length {{formula:e1496289-6224-4ae7-9d50-47fb363bbfdd}} ,
the next observation state is {{formula:b7a58f82-de05-4fdf-bc0f-9fdac0db8008}} with probability {{formula:f07de344-ba09-4d2c-9d0e-7eed3b05d9f7}} , and the
reward is 1 if the chosen action is {{formula:8e3ab746-7235-42af-9e3f-ecbf05d25c3f}} with {{formula:428123f6-3d33-4596-ba36-bea38bf71cf4}} —i.e., the
agent has guessed the value of the function correctly.
From this point onwards, the observation state is always 0 and no reward is
issued.
Note that the transducer of the dynamics described above has states {{formula:d816c863-4a1e-49ab-8e5e-d1b52ddc75b0}} , it is initially in state
{{formula:8914b3ad-7b1d-4b63-87ee-369ebb90dfc9}} , it is in state {{formula:9f843ffa-0812-4f5a-bf42-3acc552ef5ab}} when the history has length {{formula:4a3b01c0-364a-4d04-a147-5d1a097baca7}} and
the parity of the bits specified by {{formula:c4e896aa-9850-41e4-a413-8de185ff9d20}} is {{formula:9687a6df-9b6e-4482-93a3-25cb46bfd68d}} , and it is in state
{{formula:98905615-cd09-4854-a7e7-41a7a4cf7eba}} when the history has length greater than {{formula:bdbae766-6bef-463f-9404-f958a404daf2}} .
The output in states {{formula:21a82931-0eb1-4c19-a7bb-44ea01beab25}}
specifies uniform probability over the observation states regardless the
action, and reward zero.
The output in state {{formula:0db31502-2ed7-44a8-af19-029abc6b96d8}} specifies probability {{formula:4fa42cbf-b11a-4adb-94d6-2fde14212798}} for observation
{{formula:017e5ee4-582a-4721-8a21-077528860e90}} , and reward 1 if the action is {{formula:fcaa1414-8e65-49ab-957b-d534be3b4990}} regardless of the observation.
The output in state {{formula:b2d0e730-aa01-485a-be1a-957d3d7657ba}} specifies probability 1 for observation 0
and reward zero.
If an algorithm could reach accuracy 1 and confidence {{formula:eca1b2af-bb48-4429-9956-2fa6814428cb}} in
{{formula:4346224e-bec8-469c-9b2b-1b42f067c3a2}} action
steps, then the learned policy would encode {{formula:2954e1df-5ec5-4814-aef5-d302c0104cf2}} with confidence {{formula:d59932a2-f7bb-49ca-9393-995177aea727}} ,
which contradicts the noisy parity conjecture because {{formula:19695a0e-872e-4091-b151-9a93405ff515}} is polynomial—the
only parameter in {{formula:a3933c58-aa0f-49a7-b0c4-13e394cf6c38}} to grow is the number of states, which grows
linearly with {{formula:00f06957-08f4-4e63-afa2-c1be5e2783bd}} .
The following lemma shows that the required number of action steps increases
with the inverse of the degree of determinism.
In other words, the less likely transitions are, the more the agent has to
explore.
This is due to the fact that an agent that has not experienced some
transition of the dynamics transducer will not be able to determine the
resulting transducer state when the transition occurs.
Lemma 2 For every algorithm there is an RDP {{formula:236f08e9-f590-4ed5-8959-6ae5f89bfaf7}} such that the algorithm does not
necessarily reach accuracy {{formula:33c5f62b-d3b8-42a5-b6ad-e7a6a170106f}} and confidence {{formula:605bf272-9e7a-4ae0-8f05-349ec2731bf0}} in a number of
steps
{{formula:621287f2-019a-49f6-9ca0-95134603d79e}} if
{{formula:b20084b3-a488-4d80-951e-8b6c1ef21f39}} is {{formula:ab77b4e2-7e48-4ebc-b83e-e12aefc8ea56}} after removing the parameter for the degree
of determinism.
Consider {{formula:150e483f-7569-468f-ad5a-36be100e0ef4}} , {{formula:b6a7b32b-a8ec-4fd0-9b9a-6d3f3eb8e7cf}} , and an RDP {{formula:f7ea1f2a-f783-4052-bb33-9ac61657db6b}} with
two observation states {{formula:74bf76d2-9365-48eb-8406-88583f340120}} , two actions
{{formula:5779575c-8096-4429-8be9-4cc5b72e9fc6}} , and dynamics as described next.
The initial observation state is {{formula:213ea175-01ef-47e1-b456-e25ea716be53}} .
When the last observation state is {{formula:0062b1eb-67d0-4f26-a16c-5ac34bf6a98f}} , both actions yield observation state
{{formula:734b1c41-dcb6-401c-93f2-bd3eb0ab98ae}} with probability {{formula:45e3a0ff-2752-44aa-b861-c97747480abe}} and observation state {{formula:24900591-521b-4cfb-bd3a-bb852bb5aa36}} with probability
{{formula:506b877a-4867-41f1-9d2a-33cc99a534e3}} .
When the last observation state is {{formula:00d4c943-899c-473a-ae60-412c54f4c520}} , both actions yield observation state
{{formula:98bc1b50-f3fa-4df9-bb49-57714bd352a5}} with probability {{formula:5811e5b2-3339-40ea-9b50-708b081609fd}} and observation state {{formula:b185944d-045d-40d9-82c4-9f6982f357c4}} with probability
{{formula:5a2295e7-c0b1-4068-a0e2-8dd13f6dd194}} .
When the last observation state is {{formula:b04f15ca-f23c-4946-b57d-e2fc808d03d9}} , both actions yield observation state
{{formula:05f338ea-52af-4c29-892d-bcb1dd81658f}} with probability {{formula:70f70253-4a5a-4bd5-9473-0317eb43e018}} and observation state {{formula:9fda86ef-1a04-4376-96cc-03d3b5e196ba}} with probability
{{formula:0c11cccd-d616-4bfc-b307-19b17352272f}} .
Note that the dynamics of both RDPs are represented by a transducer with two
states. The dynamics transducer of {{formula:b7b09258-72dd-4704-8f35-2586086b37e2}} has two states because when
the last observation state is {{formula:c68ab1ce-3cc4-4616-927c-c3c9026d6f64}} it is the same as when the last
observation state is {{formula:f1e1ba4c-ccb7-4815-af48-f0abfc4453a9}} .
Thus, in {{formula:58821a75-6a9b-4247-b41d-4f14fc7986ed}} , action {{formula:25136071-8f9a-48cf-8841-549269cfa042}} is optimal when the last observation
state is {{formula:a1468737-5ca7-44ba-ba76-6df187c5b5ca}} ; it is instead {{formula:01570310-6fc2-416f-bf97-7d2e3c108801}} in {{formula:83a5eda8-e1ca-41fd-bff2-bc8092530ba9}} .
This assessment is required in order to compute an {{formula:792a8cf9-8992-4742-a570-56c1345927f2}} -optimal policy
with {{formula:3655d683-f504-4965-8035-5b31305e2008}} when the underlying RDP can be {{formula:f80a0db8-7e42-46d8-9d6a-9ba9e968ff85}} or
{{formula:096964af-8995-4226-8c3e-9867e399a68e}} . Therefore, it is necessary to observe at least one transition
when the last observation state is {{formula:8d011c5d-1d62-40c1-83d2-5d1bb363eff1}} .
However, the probability to never see {{formula:baeab447-2b61-49cc-88d1-2244b1a637b1}} in {{formula:6f0388a7-1bb0-4829-bc78-1af5b678f0d6}} steps is {{formula:bbbd0690-0ce8-445d-8cf8-2b3a679dd1ce}} .
If {{formula:0e323447-894c-4a32-aeb5-e4d81f703688}} ,
then {{formula:e3ac8bb6-4186-4c0a-a945-eb5b4f6fefa6}} can be made arbitrarily high by choosing an arbitrarily
small value for {{formula:a3a3d90d-e7a6-43a7-8f2f-1c4592d07bfc}} , since none of the parameters in {{formula:1a0e5956-4105-48a7-845c-6de890fba45b}}
increases when {{formula:7f1130ba-179c-4434-91a3-e0f43ffe38c0}} decreases.
The following lemma shows that the length of episodes to consider can grow with
the number of transducer states {{formula:839f6302-5c4d-4a3b-9931-344051a79fb5}} .
Lemma 3 For every algorithm there is an RDP {{formula:38a14387-4c7e-4a42-a53e-ef1c2152fbad}} such that the algorithm does not
necessarily reach accuracy {{formula:b4e46ad1-5023-4be4-a7a5-c1cb623603bd}} and confidence {{formula:f0cf6909-a2a4-498a-9734-23fb9493a71b}} in a number of
steps {{formula:50350ac1-7848-42e7-bac1-0f80050d5ccd}} if
{{formula:5aaaae25-e617-4681-bd4a-4b2bf7671b73}} is {{formula:e6c142f5-f2a1-465a-990a-660d0afc2267}} after removing number of transducer states.
Consider {{formula:1fec5c99-4289-45fb-9573-0ef4e85cd987}} , {{formula:0ee82af7-9acc-40cf-ad2d-3a0e95fb4d62}} , states {{formula:701fd7c1-c777-4150-a72b-0dc07e41bb02}} , and
actions {{formula:782353a2-2455-4a60-8d74-4c3a22e1cb95}} .
We describe an RDP {{formula:32403f8f-a90b-4881-acfe-013c98d00039}} .
The initial observation state is {{formula:06194493-2ef7-4923-baa8-d7e22ed4391c}} .
When the last observation state is {{formula:b8e688ba-39fc-4d50-a4f2-e9af7a8ee50e}} with {{formula:84665191-9f5d-4616-b9b8-2bbbc7baef8d}} , all actions yield
observation state {{formula:bd4254b5-f8e1-4e59-bede-46ab38f5f866}} with probability 1. The reward in all the above
transitions is zero.
When the last observation state is {{formula:b6f438ee-c410-46fb-9f4a-065fee404140}} , action {{formula:0b399c70-3fa3-40bb-ac82-d516978ee964}} yields reward 1 and
observation state {{formula:3dc937ea-025a-49a6-bee6-6ae4eef8e309}} with probability 1, and the other actions yields
reward zero and observation state {{formula:ccab2c0e-d1e2-4a73-910b-4fd84ea080b2}} with probability 1.
The minimum dynamics transducer of {{formula:6cae8a15-4630-41f4-9530-84103f110b94}} has states
{{formula:a13b5473-7a9c-4135-a1cd-0017f1dbeb23}} , where {{formula:740eeb0e-d1c0-4ebe-abda-8f30b6fbcdd7}} is the initial state, and {{formula:f79739f8-c743-40b1-ba21-7612a35680bd}}
for {{formula:4fe97f8d-b2d1-446c-88df-ba174338a2ad}} is the state when the last observation is {{formula:e7580423-92a3-486a-9b95-3d8adde2a5d4}} .
To determine the best action when the last observation state is {{formula:7ee07829-02a2-4660-9a4b-1956442ddc38}} and
hence compute an {{formula:95ae93e2-f365-45e3-8b14-c2c17320bd0a}} -optimal policy with {{formula:60061626-a142-4a9d-9cae-a6d3cba50140}} , it is required
to see at least one transition when the last observation state is {{formula:c0a2304d-8128-490c-a911-7c0a68f10bca}} , to
assess whether the underlying RDP is {{formula:673c9941-354a-4e16-b9c9-52fd3daf311f}} or {{formula:65e18cb9-c02a-4ce7-9b8a-cd5ae347c86b}} .
This requires an episode of length {{formula:1d64b6c7-7102-4e69-beee-2a74976e62ce}} .
The value of {{formula:c28bcad6-8dc5-4786-8e69-5dc63b2f019f}} can be chosen such that
{{formula:8e103a00-08b6-472f-907a-f2a5b4bcb9b7}} , since
none of the parameters in {{formula:862e05f3-46ef-4a96-a414-c88136ef6230}} increases with {{formula:77004879-a4ee-4d04-a2a2-f4e4264028f6}} .
The following lemma shows that the required number of action steps can grow with
the reachability parameter {{formula:4332c33e-5d46-4407-968c-45d4fd4526e9}} .
Lemma 4 For every algorithm there is an RDP {{formula:6d51377d-149e-4225-a5f3-3e55e0bbf575}} such that the algorithm does not
necessarily reach accuracy {{formula:48c5e462-a193-42eb-8b6c-27afa227a72a}} and confidence {{formula:089f1b33-6a19-4ca1-b12f-5b31fe8cd92c}} in a number of
steps {{formula:cf45d7ec-df2c-46e0-98bb-1c3ff0e9d791}} if
{{formula:91bb3084-cd08-49f7-8de1-a9067b29d075}} is {{formula:c6c6e205-8fa9-4bd0-8b16-a93d7548e703}} after removing the reachability parameter.
Consider {{formula:e92e83bf-5221-4fb8-9286-c70efd9ce573}} , {{formula:69f5557c-c494-4c5b-82e2-5af495bcd87a}} , states {{formula:8769979e-51c8-42bd-a536-32096fc6b31a}} , and
actions {{formula:ec853d69-7820-4c05-bd79-c9d2c2bc7774}} .
We describe an RDP {{formula:3e7e0b49-2574-44b3-88a8-7cf0545e939b}} .
The initial observation state is {{formula:6aa760d7-85ac-40c2-9f64-d8587201bdaa}} .
When the last observation state is {{formula:fb4bee67-1538-4fce-b88d-18572eb98712}} with {{formula:b9cad6d4-6d96-40eb-8d16-2ea99b4c1eec}} , all actions yield
observation state {{formula:e77b7327-0b2d-46d2-8926-06c38707d075}} with probability {{formula:9cd866dc-18a5-4e9b-bf82-9515f03dc2d1}} , and observation state
{{formula:6f8b3a21-a39e-4116-8831-54cd31a97656}} with probability {{formula:40414a3e-2260-442e-8d90-5d0edef0d83d}} .
When the last observation state is {{formula:2285a270-ec75-4671-b1a7-5bfa0ac7636c}} , all actions yield
observation state {{formula:59ce6c69-5e1e-453f-943a-b70677a373a3}} with probability 1.
The reward in all the above transitions is zero.
When the last observation state is {{formula:8ed7a257-d6bf-4e0e-b923-47f99e173a4a}} , action {{formula:d26ec3ac-5426-4c18-a7ee-9dbf20418f88}} yields reward 1 and
observation state {{formula:bd0a7cb1-c41a-4930-98f3-3ecc0c3e0e87}} with probability 1, and the other actions yields
reward zero and observation state {{formula:35328434-fe5e-48d4-9fdd-58aa9c0dead7}} with probability 1.
The minimum dynamics transducer of {{formula:a38f6273-aeb8-4e34-b31b-f941af6f3da8}} has states
{{formula:d93a4c3a-1098-4611-aaa3-be2f8ccbb531}} , where {{formula:1e2c8455-0d04-4619-b8b4-eaa34b889e46}} is the initial state, and {{formula:a64c1764-10f0-43cc-8828-6428ec237c77}}
for {{formula:eccb5928-34a4-4c98-b977-2e76d08a43c1}} is the state when the last observation is {{formula:f4e92ce8-1be3-46d3-b0b0-702866880560}} .
To determine the best action when the last observation state is {{formula:9bc048e9-bba7-428a-903f-a4f3a39cf8a4}} and
hence compute an {{formula:4960e230-e1d0-4f25-b4da-860970f218a6}} -optimal policy with {{formula:d931e659-c916-46ed-918b-06a8209ffed5}} , it is required
to see at least one transition when the last observation state is {{formula:469b67bd-68bf-4ce9-abbf-cd56ce44ca4e}} , to
assess whether the underlying RDP is {{formula:ebb1f758-c34d-415c-ae2b-71fb3a1f9a7a}} or {{formula:9ff52570-8fb0-477c-8e16-83c4af55f7aa}} .
The probability of seeing observation state {{formula:cbee337c-e166-4e24-bbd2-078ee357d5de}} in an episode is {{formula:93bd5691-8064-4c33-874d-f44def117c8e}} .
The probability of seeing observation state {{formula:0f0d3cb8-5386-41d0-950c-920f7a30998a}} in {{formula:4dc5a382-456c-4935-8592-310194a35c3b}} episodes is at most
{{formula:87b9d41d-c46e-4294-88ec-942180ddc743}} , by a union bound.
If {{formula:a5c347b9-ca35-40ca-b4b4-e3c8ce1ec54c}} ,
we can choose {{formula:4d32aadc-3e02-47da-aa07-c42dae1c3f78}} such that {{formula:2772ff30-1d47-4e2a-b968-3c31e1f227d2}} is arbitrarily small.
The remaining lemmas are based on a core lower bound for the multi-armed
bandit problem given in {{cite:b078028eb5a0fb39d54a2bc9477123b467431f88}}.
Proposition 1 (Mannor and Tsitsiklis, 2004; Theorem 1)
There exist positive constants {{formula:4b73e73c-4c9e-4933-9d9f-0e3447547a85}} , {{formula:f932f063-53fe-4079-8343-679e45592042}} , {{formula:a8a830db-46d3-4051-99e8-e2ddee32a396}} , and {{formula:85925d59-dedd-43a4-975b-39b5ffc8c874}}
such that for every {{formula:1a2238d9-d739-4e11-9953-edddc66287c7}} , {{formula:c4d6a7db-7374-49f3-bd61-74cb01e45237}} , {{formula:bdaecac7-14d5-4fb3-bdb8-18ecc3d4515f}} , and for every algorithm, there is a {{formula:3d6f67f9-f949-4af6-809c-0b16052794a2}} -armed bandit problem
for which the number of action steps for the algorithm to reach
accuracy {{formula:cce31387-6271-4fc1-be21-5b7baa90450d}} and confidence {{formula:78ef1b8e-3687-4d26-b073-5733dfec1787}} in the worst case is at least
{{formula:1bf483fc-ce34-424d-8432-617e4213c591}}
In particular, {{formula:2b12346d-a703-41a2-a263-5792d1fd52c5}} and {{formula:f81ad9b2-6623-431f-9204-f91ea92200ee}} can be taken equal to {{formula:7a5092e7-368f-4661-8581-8689cebbbaac}} and
{{formula:24b53227-78dd-448c-83dc-eb89598eea5e}} , respectively.
Furthermore, the mentioned {{formula:26da1ba2-de99-47c0-9806-c9bbb58f5054}} -armed bandit problem has rewards in {{formula:65d25e24-8324-47f9-b96d-b15c5fa530bc}} ,
and its maximum reward probability is {{formula:93c03b56-44e5-4af7-935e-3bc5ca4863cc}} .
Note that the bound was originally stated for the expected number of
action steps, which is a lower bound for the actual number of action steps in
the worst-case—this was also pointed out in {{cite:f3cbb9f5c029a199a677e085bddbcf939d38fa3e}}.
Note also that the original theorem was for algorithms that, as a last step,
output a policy and then stop.
However, the proof of the theorem shows that, when the number of action steps is
lower than the claimed bound, the probability of selecting the wrong arm is
bigger than {{formula:6386df4a-8268-4dfb-bd69-e0082adc330f}} , regardless of whether the algorithm stops or not.
The following lemma is implied by the previous proposition since the
number of actions corresponds to the number of arms.
Lemma 5
For every algorithm there is an RDP {{formula:33816d89-b49f-4741-8d81-f5ab18eba14f}} such that the algorithm does not
necessarily reach accuracy {{formula:228302dc-6700-47e7-96bd-0b50a9a548e9}} and confidence {{formula:98285a19-6daf-46cb-89ad-61ab3ce3d2fb}} in a number of
steps {{formula:bf63f98a-bdb3-4560-bc87-07c62a45b272}} if
{{formula:0a0be852-efcb-4649-97c9-2965063b847b}} is {{formula:595f0727-23e1-434c-8f0a-b7c77aa41204}} after removing the number of actions.
We show RDPs that encode MAB problems and argue that if this lemma
were false then Proposition REF would be false.
Consider a {{formula:c69723c8-d58b-48cc-b032-92e983772ad4}} -armed bandit problem with arm probabilities {{formula:615fec99-9b0a-434a-ae11-31fb6a10b47d}} ,
rewards in {{formula:d6652209-3e8e-4068-8229-52aa4140f824}} , and maximum reward probability {{formula:dd69caa9-f0b8-4131-bcef-5b166b8ee093}} .
The corresponding RDP has observation states
{{formula:6e049f30-5689-48f0-9c7c-717a8bd5c979}} and actions {{formula:0a4f42e1-21ae-4cb8-93a6-81e570e2d766}} .
The initial observation state is {{formula:7a6a36b8-e30c-407d-ac21-e62bcfe5114d}} .
Regardless of the last observation state, when action {{formula:3d4c747d-0901-4a62-9e2b-744fcb44f793}} is
performed, the agent observes
{{formula:0c875673-347a-49ab-a0b5-5df06ef6edf4}} with probability {{formula:0b446732-54d7-4b76-919f-08afcbfdd16a}} receiving reward 1 and, it observes {{formula:7908037b-151f-41f0-b945-d37bb036d696}}
with probability {{formula:3d169acd-a4ec-4558-8eaf-568e549403b2}} and receiving reward zero.
Note that there is one set of parameters {{formula:0a241033-8af5-4f86-b153-59b977336ec5}} that describes
all such RDPs.
In particular, the discount factor is irrelevant and can be
taken equal to zero, the maximum reward is one, the reachability is one and
the distinguishability is one because the dynamics function is state-less,
and the degree of determinism is {{formula:3cfcce60-dad8-4984-9833-b314c4eb4a5a}} because it is the minimum
probability that an arm issues a reward.
Assume by contradiction that, in each of the above RDPs, an agent can reach
accuracy {{formula:b10dee2c-ee00-46fe-99fd-da52b5ea85cd}} and confidence {{formula:2940f04c-26ee-42d8-ace2-6e2dfe113ae8}} in
{{formula:9e82b8f7-0298-4505-8dfc-2d6928d9cae5}} action
steps. It amounts to solving the corresponding MAB problem with accuracy
{{formula:f40ac3a1-e5da-4b69-9d02-7a95d2e67b92}} and confidence {{formula:2321c404-60a4-4048-8e0c-2e5864d48cbe}} in
{{formula:c81d8834-4453-42de-85de-82efdde67098}} action
steps.
For any given {{formula:e5f44bef-f5ee-4452-8f48-913a68b8bd12}} and {{formula:ffb959cf-b4ea-4c72-8963-e67e28d57385}} , the former quantity is constant, and
hence will be smaller than the bound of Proposition REF for
a sufficiently big number of arms {{formula:158b737d-72ae-436c-baba-67886b3f6657}} . This contradicts
Proposition REF .
The following lemma is based on the observation that, if the number of action
steps did not depend on the maximum reward value {{formula:2f725c1f-c1da-4891-a8bf-fe5491f4c7f8}} , then we could
artificially increase all the rewards given to the agent by an arbitrary factor
in order to obtain a higher accuracy in the same number of steps.
Lemma 6
For every algorithm there is an RDP {{formula:b9f73fbb-a668-439d-8279-31feb6c42d31}} such that the algorithm does not
necessarily reach accuracy {{formula:b114e26e-2037-4d13-b3a2-a5db94526e57}} and confidence {{formula:3c8e4766-7c40-475c-9a71-dd42e9dd2826}} in a number of
steps {{formula:342b5214-14d3-4ca1-90e8-1f210967a63b}} if
{{formula:d344d653-af9b-44d2-bc3c-6ed06172752e}} is {{formula:3199f1a1-586e-4e0d-8a5c-a5c3bb62c279}} after removing the maximum reward.
Consider the encoding of the MAB problem introduced in the proof of
Lemma REF , with the difference that, each arms yields
reward {{formula:ba0f0b77-482a-4157-9b57-56bf2d92533b}} instead of one.
Note that the number of actions {{formula:1e8dbc0a-7b1e-4869-9d75-64a0b3d352b8}} is the only parameter that can change in
{{formula:4621a02e-b8f9-4137-a520-4ae060693c7f}} for different RDPs.
Assume by contradiction that, in each of the above RDPs, an agent can reach
accuracy {{formula:e6f18bdb-bee1-41d6-a13a-f525fc33821f}} and confidence {{formula:5f848d1c-16df-42f3-a38e-db0e642f437f}} in
{{formula:423baecd-29cb-498b-ab4e-bc63bc188ebd}} steps.
We can use this agent to solve the original MAB problem—the one with rewards
in {{formula:4eb010f7-83dd-4948-ac2d-4ccccf27dcd6}} —with accuracy {{formula:7f99b6ec-6204-4cb5-82cf-342adcae1c38}} and confidence
{{formula:5d4c1115-61ea-4ff0-8944-9552caa39e49}} in {{formula:4d07f5dc-b436-433e-942f-a13925aac7d9}} action steps.
It suffices to give to the agent reward {{formula:a9568380-ae92-4a8d-b62d-0f24824c5cfa}} whenever the original
MAB issues reward {{formula:66891031-c37e-4240-8184-ffc743fb4250}} —i.e., to give {{formula:7f283209-8e97-4a0f-b9e7-fbe471e1916c}} when reward one is issued by the
original MAB, and zero otherwise.
Then, if the computed {{formula:02fce8d3-4ab9-4367-9f64-b6e23bfa5cc4}} -optimal policy selects action
{{formula:58c0c30a-5a92-4fef-9e62-e74995637311}} , the {{formula:44a2487e-93d4-4d10-b3c8-edc70c219397}} -th arm in the original MAB is {{formula:c6220cd4-130c-48cc-9eb8-36686d1772e1}} -optimal, which solves
the original MAB.
This contradicts Proposition REF because the number {{formula:c01b04b3-3fdb-4c4a-9be2-f503b3b64cd3}} of
actions steps can be made arbitrarily small by increasing the value of
{{formula:7925fea9-5190-423a-a970-554ce2c3cc23}} .
The proof of the following lemma is a variation of the previous one.
It builds on the observation that, in the setting with discounted rewards, even
if {{formula:6f88b22e-9e16-484c-abe9-f3bf853a2c2c}} , there can be a state whose value is arbitrarily large, for
decreasing values of the discount factor.
Lemma 7 For every algorithm there is an MDP {{formula:94a1ca7a-2220-439e-bb6e-2b37320e0a1e}} such that the algorithm does not
necessarily reach accuracy {{formula:0f71c21e-d370-44f9-b542-2b49bf40d7bd}} and confidence {{formula:9dcbdf19-3212-4425-829d-0e1e81035631}} in a number of
steps {{formula:2551f726-4209-42c3-8f54-da7c0fb047df}} if
{{formula:c62b50d3-16fb-400e-9894-48ccc06e1d8d}} is {{formula:4f5810e2-6547-4f01-94f1-e77fb2d1efda}} after removing the discount parameter.
The proof is based on Proposition REF similarly to the proof of
Lemma REF . However, it considers a different encoding of
the MAB problem, to ensure that there is an action whose value increases with
the discount parameter—instead of the maximum reward parameter.
Consider a {{formula:c7283889-4d02-4b7c-be9f-fa3eac1777d0}} -armed bandit problem with arm probabilities {{formula:6185cab5-a2b2-45a0-9ec7-5c289bcfa504}} ,
rewards in {{formula:28d55638-4d80-4cb0-a87f-4008bf0297bf}} , and maximum reward probability {{formula:c1efe5b6-266f-409c-a08f-c7d410fb767e}} .
The corresponding RDP has observation states
{{formula:586935bf-6588-499a-bb32-8b3035dc8956}} and actions
{{formula:a73ba0a8-723a-445c-b759-00c9bc4360f9}} .
The initial observation state is {{formula:90d4b958-652f-42e1-a759-f43f9240616b}} .
When the last observation state is {{formula:97456d10-cfd2-4d29-806e-057fbd529260}} , action {{formula:05360948-aea7-4045-bbe9-86af95f9ec5d}} yields observation
state {{formula:40d4ab79-0e51-432e-99f7-ad4911528c16}} with probability {{formula:6c97e1e1-8725-4400-a893-baf2b3962f59}} and observation
state {{formula:1097fe75-91d6-466f-b723-cc9a4be87c20}} with probability {{formula:de2dc6c1-ef70-49df-931a-da9787dad669}} , and the reward is zero in both cases.
When the last observation state is {{formula:ee715cfa-dc3c-4516-a44c-1d9ee343838d}} , all actions yield {{formula:e0ce98c0-fc5f-4428-a0c7-83be65940b39}} with reward
one.
When the last observation state is {{formula:f2ddac54-b9fa-4145-8cff-be3706189fda}} , all actions yield {{formula:0fcf81c9-ba9f-4edc-a8d4-23067571c041}} with reward
zero.
Clearly, during exploration, the agent can always go back to state {{formula:afcce9f6-4b5a-4685-9232-92667a97f442}} by
performing a stop action—as assumed throughout the paper.
The key observation is that the value of every action in state {{formula:f0f4c32b-7d04-4a0f-b6ae-fede2c39608f}} is given
by its probability of
leading to state {{formula:02fafd1f-693c-48b3-a7a2-07df1ed2d96a}} times the value
{{formula:58210e0b-9777-4704-9523-6dd2c11da700}} , which is finite and it amounts to
{{formula:c228b18f-9361-470b-bb59-0029729a9c06}} . In particular, {{formula:2a28f392-459a-4223-8470-d898ea513a37}} can be made arbitrarily big by
making {{formula:c450cab4-6a0f-4b6a-8cec-8351ea3dc0f8}} arbitrarily close to 1.
Thus, {{formula:a355ebf1-5f95-4a8a-a53e-0a7efa28d026}} plays the same role as {{formula:f18e047f-f089-4510-9220-ff3699b2be2e}} in Lemma REF ,
and we omit the rest of the proof since it proceeds similarly to the one of
Lemma REF .
Proof of Lemma
The goal of this section is to prove Lemma .
Let the target RDP {{formula:d7031eed-59ed-40cd-b23f-7bcd8b944305}} and the ideal MDP {{formula:caa3bd17-fd1f-4d4a-a04f-d3187659031e}} be defined as in
Section .
First, the optimal value functions {{formula:e0323911-4840-4f92-8d16-fe1085bdb4d6}} and {{formula:12f47306-75cd-4e8e-8ab2-afd57051d467}} of
{{formula:f69923ac-3ae7-40e9-ad7e-1842be223c5f}} can be expressed in terms of the optimal value functions {{formula:13816058-6a71-4fa4-abb2-98d014b8d843}}
and {{formula:c9274b38-c68b-4686-9caa-a0ff6aa435e4}} of {{formula:d99e023f-617a-4b09-9113-64d32bcf0e0c}} .
Lemma 8
The value functions {{formula:3adca7d7-8fe7-403e-a8df-89a8aadb507b}} and {{formula:b6bc2a65-a0c6-402d-95b9-09ef4dc7bcb8}} of {{formula:385fb97d-a520-4a43-8696-d7efd2129f09}} can be expressed
as
{{formula:bdb53fd2-d018-4e5b-95c7-626c6935435f}} and
{{formula:a404b839-307b-4ff5-a043-59c02f98f7fb}} .
We first show the claim for the value function {{formula:22d5cff3-19f4-40c1-af5e-8613d983bf3d}} .
We rewrite {{formula:422e2008-5b47-40d3-97ee-58d9e35f9d66}} by expressing the dynamics {{formula:af1a6b37-f9ca-4a17-90df-0846c1d2f5eb}} of {{formula:65c3673f-a5f4-4bc4-8059-e426f85c79d1}} in terms
of the output function of the dynamics transducer {{formula:29931d8e-16b5-47db-99f5-4052c90a1913}} .
{{formula:3cf224b3-8314-40ef-a138-b4e7b8de29a3}}
The equation above can be rewritten in terms of a new function
{{formula:0bd79fe6-aa29-44cc-8335-90093d678b34}} over states {{formula:a4e64321-5238-47bc-a2cd-e4bc516a6d6c}} .
In particular, {{formula:4e0b99c9-c7e1-4db5-90b3-8e2e4d111833}} .
{{formula:392776d0-ee4c-427c-9d89-3e7ba00ed63b}}
The definition of {{formula:a2381efe-2f39-45b8-9bfa-92722a20cc1e}} can be given directly over {{formula:3bce830a-f777-4435-b047-adbd62f8ce98}} .
{{formula:599a981a-baa4-49ed-9d89-98580e041817}}
We apply the definition of {{formula:d47e7cc3-662a-4791-934d-dbe89ca7fea1}} .
{{formula:7250fdce-45da-47de-882d-caaab12440c2}}
From the last equation, we can see that
{{formula:f76876ff-c154-4a6f-953a-3a3edd7acc35}} , and hence {{formula:4f476721-faf7-4c4c-ae20-7322b7f60578}} as claimed.
For the action-value function {{formula:9b254e19-cc33-4261-a5bb-67a17e1953d7}} , the steps are similar.
One difference is that we make use of the result above for {{formula:7af1b79a-7db5-4860-a3ce-d98683f223b6}} .
{{formula:09e30ad0-24c8-4e3d-857d-76ea5597df88}}
From the last equation, we can see that
{{formula:33aaa8bf-51bd-4cd7-a354-9678c7fdcad8}} , and hence
{{formula:ad8df7b1-3246-4fce-bd26-fe1fb9bc4c80}} as claimed.
Then we show that, for every policy {{formula:cba40e91-209d-4386-8ffd-0a0f665aa506}} for {{formula:8aab46dc-9e9d-47a5-accb-495da96fce22}} that can be expressed in
terms of a stationary policy {{formula:318a2d10-f418-4ce7-babe-67610ff97959}} for {{formula:c4ae0754-b414-4ec2-a67a-fec100744d4f}} of a certain form, the value of
{{formula:66d7091a-8f0f-4d3f-8f79-f54df558e22a}} in {{formula:ecccb5d4-98ca-4a98-a7a2-2d00cc7e107d}} can be expressed as the value of {{formula:89ea4785-3a1e-46ac-bae9-ce6f6dee2de9}} in {{formula:a9856700-8ff2-4c70-b833-6773fb62c468}} .
Lemma 9 .
Let {{formula:6ccf4169-793d-4d63-abb3-b9eb0d3b2bd3}} be a policy for {{formula:3a8b534e-d30f-40c8-96d4-3bd46c1b2abe}} , and let {{formula:2b6677a4-0c74-40a4-9103-1d11af5943a1}} .
Then, {{formula:cf4a6b01-edfd-4284-ac81-96085c12a94d}} .
The proof technique is the one from Lemma REF .
{{formula:fcea0d93-27bf-41ab-aff0-4fd6f0872a9b}}
From the last equation we can see that {{formula:ead4345e-c47e-47f8-a9f8-88141c86ef77}} ,
and hence {{formula:1b6e50f0-71c6-4bdf-8862-927b11231be6}} as claimed.
The previous two lemmas imply the main lemma of this section.
*
Let {{formula:cdfdbf6c-7e8d-4de7-9994-31e3bf1ee0f5}} .
Assume {{formula:fa28bab6-f6b3-4907-b62a-7ffae9b5aa36}} .
The former is equivalent to
{{formula:af5c3bdb-5c08-4fa2-a7ee-39c817978a6c}} .
The former can be rewritten as
{{formula:c7068a8c-690e-4c8e-b8a3-0deaf6b19bdc}}
by Theorem REF .
The former can be rewritten as
{{formula:d92112af-f96f-4476-84da-8064691cae53}}
by Theorem REF .
Proof of Lemma
The goal of this section is to prove Lemma , by
applying known results for MDPs
{{cite:dc2fac724fa157eec7302e10371d188a053ace7d}}, {{cite:b7a95493f90e6629863d5f7a097839fe33e4a251}}, {{cite:f3cbb9f5c029a199a677e085bddbcf939d38fa3e}} similarly to
what is done in {{cite:f3cbb9f5c029a199a677e085bddbcf939d38fa3e}}.
First, similar MDPs have similar action-value functions.
Note that {{formula:0ef16394-3b69-4bb1-94d7-ee34197dd7c5}} is the {{formula:743843d4-4016-462d-b4cf-e5a32a7a2f77}} -norm.
Proposition 2 (Strehl and Littman, 2005; Lemma 4)
Let {{formula:e4776ea4-06fa-4ccc-83c8-0c516e651517}}
and {{formula:f90dcc15-6f88-4cac-9295-0931694a99eb}}
be two MDPs.
Let {{formula:0e583168-8061-42ab-b73c-530fa091f882}} be the maximum value in {{formula:f61ac769-92f4-4458-a449-b168e0805f28}} ,
let {{formula:cf661f88-20e7-4b02-87f1-a87f9c5792bb}} be the expected reward after doing action {{formula:b4d7bb2d-3845-438c-a9b1-ba93496f43af}} in
state {{formula:07a85802-104c-4ada-b738-ffbed762a740}} in the MDP {{formula:251cb5be-df03-4527-a919-05b27a536bbd}} , and
let {{formula:40dfcab7-983c-4b63-a8d7-aaad144b4f59}} and {{formula:c3db9ed7-f3bb-483f-a54e-df2d4b25542b}} be the action-value functions of
{{formula:ea169076-7336-4f6b-ba01-cb1eed82cfd8}} and {{formula:1ffa04c6-cd89-4cca-a4d9-c5c2ae259c46}} for a policy {{formula:41895b21-fb08-4b60-b814-4b6089253b61}} .
If {{formula:da43fd8c-7645-40b9-bf03-797598c4f48b}} and
{{formula:b0196f83-ef1a-4b3a-91e6-474b31b7779b}} for every state {{formula:4c2aa8b4-526f-4d5e-b535-398a9472b98e}} and action {{formula:cf0dcad1-1b8b-40fb-b23f-daa4a4cafbcd}} , then the following
condition holds for every state {{formula:8c513974-9ecd-43e0-a06d-38f437efef61}} and action {{formula:4333d64a-07c4-484b-a46b-1f3a29dbc041}} :
{{formula:8a7fff67-44b1-4090-bc8b-d97846a7249f}}
In our case we have a bound on the accuracy of the dynamics function, which
transfers as follows.
Proposition 3
Let {{formula:278124e4-2eb0-43fa-a029-70adf0946152}} , {{formula:bf370ff1-7911-412c-a296-4004aaafd432}} , {{formula:0ff9073d-ac36-4f90-a867-40aa44e7397e}} , {{formula:900b456d-0109-4fc5-a30b-b7507df22052}} , and {{formula:5aa3dc26-b89f-4361-bfe6-4c2ebfc3c04b}} be as in
Proposition REF , and let {{formula:09522c43-3b43-4d93-935e-e2b3629a9352}} be the
dynamics function of {{formula:11b56bdb-3b79-45a4-8a3c-ef4fa4eb6ebc}} .
The following holds, for every state {{formula:9b42137a-2602-4475-b7f1-0c87a67b9746}} and action {{formula:fc993f92-288b-4af0-b3ca-1fcd4ab822e2}} :
{{formula:ccbf08b8-185c-4bd8-80eb-0756a7d66478}}
We have {{formula:58de000d-2b85-400f-9ebf-ea23ee2e2d60}} .
Then,
{{formula:6e5315a0-82c6-497e-8b98-fc169685d50a}}
This concludes the proof.
Proposition 4
Let {{formula:c34eefce-62b5-4ce6-aa78-dd13451a2399}} and {{formula:bbcffaf7-b30f-4325-9f4d-d280f54a8259}} be as in Proposition REF , and
let {{formula:ccdef89b-ad0b-45bd-bf78-54e2d16ddf3e}} be the dynamics function of {{formula:a1998488-2708-4516-83e5-96ad20c1bb1d}} .
The following holds, for every state {{formula:2e8d5189-c417-4efe-9b5f-bcd510e773ea}} and action {{formula:b51d5917-6124-45d6-82e7-52dfacc3c14f}} :
{{formula:b5c1023e-9574-4c40-860d-e1cc0be3e10c}}
We have {{formula:f098e3ad-fd6d-4d10-8cd5-acf085f7bb15}} .
Then, for every state {{formula:9d042cbb-a370-49a2-aa76-371698678a2b}} and action {{formula:a3d38350-ae2b-4fb9-b8b3-e4773b4107ed}} ,
{{formula:4226ed89-698e-4aae-afb5-6a861d4b8d16}}
This concludes the proof.
We apply the previous proposition to bound the action-value function
of an MDP {{formula:dfe7fc85-ac6f-4697-8f17-764dc0965ae5}} that is an approximation of {{formula:4deccde6-7c8a-4035-b388-e572c8d8f579}} .
In the following, to simplify the notation, we identify the states of
{{formula:77ae094d-e9d0-4170-8991-71255992cecb}} with the ones of {{formula:be7f8f6e-1f6f-4cfd-aa80-eff3c4183dac}} ; this is w.l.o.g. since we could rename each
state {{formula:3abc9217-935f-4635-94d9-7feefcd7e58b}} of {{formula:61561ebc-1109-460d-a530-e05517132c27}} with {{formula:214cdc69-e398-49e5-9204-511fe80e917d}} where {{formula:18f90be7-a7c3-4cbb-a087-758f1e7f894c}} is the bijection mentioned in
Definition REF .
Lemma 10
If {{formula:c1bd5ef3-def0-475b-a794-5064ed98aa5a}} is an {{formula:66309218-a834-44f3-855f-945f2f04b9ef}} -approximation of {{formula:2a97a522-88e6-47bc-83fb-2e063b6b304f}} , then
the following holds for every state {{formula:d16c7635-4fd7-479b-8db7-d0535abbb991}} , action {{formula:bdd930c9-64ca-43be-bbc6-ef1ab842d421}} , and stationary policy
{{formula:6467f7ba-26bb-48e3-a34a-8b38bcd56531}} :
{{formula:c037ad7d-b080-4252-b4a5-12832320db87}}
Let {{formula:6632778c-e9b5-42a3-b849-2b6f245b3ba0}} be the maximum value in {{formula:9cd4b5f5-6151-4f57-b8bf-23b37dad5c2b}} , and
let {{formula:09d64f0a-42e4-4851-ad1e-6072b56005a8}} and {{formula:dcf50e84-1434-4d09-b4df-52c7bef3b871}} be the expected
reward after doing action {{formula:7825efd6-d5eb-4348-99ae-2b6a9ea5db57}} in state {{formula:ecbb3358-72ff-4d3a-8776-72a76b2819c5}} in the MDP {{formula:a6b19d00-34d5-4f6a-9a5e-22ef1523207a}} and {{formula:03eab504-49aa-4522-ba31-edbfdc6e13c8}} ,
respectively.
Let {{formula:38d8a17e-4314-4910-968c-0abef450d437}} and {{formula:9c3f4681-eaf4-4783-8685-912d51dc7326}} be such that
{{formula:f45f2db9-5611-4651-938b-e8a7597f8f0b}} and
{{formula:e18830d5-75a1-4140-a70e-10e5b7336089}} for every state {{formula:3af01194-96b8-4d24-8191-0283cf66eeaf}} and action {{formula:4a68f422-51f0-4243-b61b-afed592c9865}} .
Then,
{{formula:89448e5a-b69a-4dbf-9b23-47491e8e528d}}
The first inequality holds by
Proposition REF ,
the second one by Propositions REF
and REF , and
the last one because it is obtained by removing a positive additive term at
the denominator.
A near-optimal policy for {{formula:eea459b1-0ffe-42b6-877e-aa3bdeffd45b}} can be computed via the value iteration algorithm.
Specifically, value iteration allows us to compute a close approximation of the
action-value function of {{formula:34c7e8cc-3f27-4e8f-b084-e470b6a5a103}} , and then the corresponding greedy
policy—that picks actions with maximum value—is near-optimal.
The proposition is a straighforward generalisation of Proposition 4 from
{{cite:f3cbb9f5c029a199a677e085bddbcf939d38fa3e}} to the case where {{formula:c55b236c-a572-4c68-a919-f315f8642537}} is not necessarily 1.
We rewrite their proof to include {{formula:4780ae5b-ddb9-4f4c-a384-c564051765b8}} .
Proposition 5 (Generalisation of Proposition 4
of [Strehl et al., 2009])
Consider an MDP {{formula:ee533deb-2ac3-48af-868a-904258938f92}} , let {{formula:642d4525-4a3d-4375-a364-1837ac54bff4}} be its discount factor, let {{formula:005feee6-641a-4583-aa36-ea85a5623918}} be its
maximum reward value, and
let {{formula:a37ba77a-99d7-4479-82a5-e5b575ee27a4}} be its optimal action-value function.
Let {{formula:cb504fb7-c705-4ab0-937f-ddd0c668f74d}} be any real number satisfying {{formula:78a7eb05-27f9-4696-b860-464b6821acb0}} .
Suppose that value iteration is run on {{formula:c25d5e74-6e04-4524-86cb-170d8fbacc9f}} for
{{formula:9fcabeb8-be77-4297-aac5-3014b315da02}}
iterations where the action-value estimates are initialized
to some value between 0 and {{formula:b35c994e-daa9-4f9b-854d-56812c2613fd}} . Then, the resulting action value
estimates {{formula:0365ac0b-1ac0-4479-bef7-73c6b163c68a}} satisfy
{{formula:4e05babb-be5c-412c-acd3-b23cfffd63f2}} for
every state {{formula:14c8becc-21d9-417d-9dcb-c20f58f8d644}} and action {{formula:bcfc52c4-4ad9-4a03-9ef1-4116ee92e4c3}} .
Let {{formula:ad4c07f4-d2df-46bb-b54d-8da2c6c529a4}} denote the initial action-value estimates, and
let {{formula:93566ccc-7f27-44e0-9aee-a9c648283a13}} denote the action-value estimates after the {{formula:09ca234e-bfc8-4422-96d3-d29123ea8560}} -th iteration
of value iteration.
Then, let {{formula:a1018a9a-12eb-4c39-a92a-f65836efce0c}} .
We have that
{{formula:0a164455-7325-4db6-b91d-db9aa93a3026}}
Using this bound along with the fact that {{formula:fc31a609-daed-4d96-8957-d679d2daae91}}
shows that {{formula:5fc30804-b279-4c59-a752-158efce84ef8}} .
Thus, in order to have {{formula:ae6fee3b-13bf-4a43-98d0-5e21389c3f3b}} , it suffices that
{{formula:18682084-698b-4e18-96db-280905342177}} .
Then, the theorem follows since
{{formula:8ad8bf55-8cd2-433a-a4af-dd61a31c2e49}}
To see that the former inequality holds, it suffices to note that
the inequality {{formula:0e6a5597-a9c5-4031-9e53-3afb5c865708}} follows from the
inequality {{formula:c21d989d-b815-4be0-9fd1-6e8ea52ff9f7}} .
Proposition 6 (Singh and Yee, 1994; Corollary 2)
Consider an MDP {{formula:c29034cc-708f-45cb-a965-3f7407040ac3}} , let {{formula:7c9f4695-cd79-45c2-8f9c-04cac1003ee2}} be its discount factor, and let
{{formula:a2145017-c54e-4977-8ed8-d7699290d486}} be its optimal action-value function.
Furthermore, let
{{formula:931f3c5f-4111-4366-8aba-1a5d8e939fee}} be an estimate of {{formula:645f4dfc-023b-4c00-964e-e583cc6c17f0}} , and let
let {{formula:362bf9f3-d4f5-4eec-a4d3-2c52b0bcd3b7}} be the greedy policy with respect to {{formula:4969f01b-df58-4653-9cce-68fc288bdea6}} , i.e.,
{{formula:5c69ec3c-3dc9-4b42-9a76-21f5d7ab2d48}} .
For any {{formula:0cfdd02b-3ae6-4cb1-ab2c-2830dc3f10d6}} , if
{{formula:3647a933-d39a-4195-9b78-2db98ff33ada}} for every
state {{formula:72faf86c-fedb-4016-b1d4-68b329cdbb33}} and action {{formula:0f1ee687-a30b-4483-8d3f-c6a0f602e03d}} , then
{{formula:ea6e34f2-bb4f-437f-9864-7c2d47bb51ab}}
for every state {{formula:e40cd321-2c91-452e-8252-1a9b4192f602}} .
To prove the main result of this section, we apply
Lemma REF and the previous propositions above.
Note that {{formula:697f37e3-d981-4550-ad03-a4b860bf7d68}} as specified in Table REF .
*
Assume that {{formula:c352ef55-d4cb-494d-955c-b6a7a38e2bb2}} is an {{formula:c21b127c-148f-469e-a4a2-1efb32f7f9fc}} -approximation of {{formula:bb98d53f-28cd-49df-8d88-b03a89a44042}} .
By Lemma REF ,
{{formula:e9f303fe-e916-4f92-b962-6ac59e4dfa47}}
Consider to run value iteration on {{formula:21615177-2ba9-45da-a2c9-e2536ff257c2}} for
{{formula:b5a084fd-604c-4f92-8ffe-2101f84e9aec}} iterations, initialising all action-value estimates to
zero.
By Proposition REF ,
it yields {{formula:091ae75e-27eb-4279-8356-23202accee73}} such that
{{formula:0adeb849-eb4d-42da-9bf4-2c0a2605a901}} . Note that there are two
levels of approximation; namely, {{formula:7a541d91-a255-4e14-89be-3ef2beaf2250}} is an
approximation of the action-value function
{{formula:275b2447-8590-4144-8a34-83cfa5c25c2e}} of an approximation {{formula:12540593-ec4a-4f4c-aad8-7ac2c361549b}} of the MDP {{formula:d7f93fcb-1404-4a8e-9e7b-716f85d79bca}} .
The former bound together with (REF ) yields
the following:
{{formula:eec5ee90-3dec-4ef5-a3bc-e6719ac5b538}}
Let {{formula:ff389860-042f-4642-8dc6-08b97b2962d9}} be the greedy policy {{formula:8445c5c2-cfbd-4c4e-8dd4-96498fa28809}} .
Thus, by Proposition REF ,
{{formula:6b55966e-38cc-4363-96ba-f2073262e11c}} for every state {{formula:0e2bd7a2-c3d4-4397-9af9-1aeaa86d79b5}} , and
hence {{formula:d725cbc8-a03e-422e-b50f-fdd351d902c1}} is {{formula:9194cce0-47a6-4e04-8ba6-775ae46431dc}} -optimal for {{formula:ad59dad5-cd88-4e66-b354-18fe28701836}} .
Proof of Lemma
To simplify the notation, we identify the states of
{{formula:d404fee3-033a-4218-91a4-9867c61120bf}} with the ones of {{formula:dd6eb7dd-1e4e-4dc9-b1d0-6898c3262434}} ; this is w.l.o.g. since we
could rename each state {{formula:683a45e2-80ee-4ca2-821a-faca31ceab28}} of {{formula:e3d25920-9a00-4d58-9f7f-3f5545c67b66}} with {{formula:1cdc0e54-acd4-4b6b-8732-7ee9d3472347}} where {{formula:1126119b-4c88-46ed-9f32-d2f5022ea149}}
is the isomorphism mentioned in
Definition REF .
Note that {{formula:90871f3b-787d-42a9-a883-56124dc9f2a8}} as defined in Table REF .
*
For any state {{formula:afbf3efb-9cdc-45bd-b715-615808ef83d8}} and action {{formula:5e1a9c1e-faac-4952-975d-43035468f279}} ,
{{formula:b309e70b-740b-45dc-b26a-f364bd2c1497}}
The first equality holds by the definition of {{formula:8cdc2002-21f1-43c0-b23b-71256d100536}} -norm.
The second equality holds by the definition of {{formula:e4aeec34-becf-4713-88a3-061a9068fb6c}} and
{{formula:4afc9ffb-9e0f-4804-ae8a-6251833a0bf6}} .
The third equality holds by the definition of {{formula:4c0555de-9d39-4da0-b8d4-d40f2287dd85}} .
The fourth equality is simply a partitioning of the elements {{formula:8543cb2a-7f00-4388-9d7a-299a4c740b6f}} .
The first inequality holds because {{formula:cf6ba2e1-ae11-4717-8ace-4cd829b4cd7f}} is an
{{formula:f898e429-4e1d-46d2-8184-382a03303374}} -approximation of {{formula:22aedcd6-ce83-46aa-8e5e-c89da802bbf8}} ; in particular, for all elements
{{formula:8af78a4c-d0be-43e2-80da-26cd9318f58c}} such that {{formula:756919b6-67ed-4d32-93d0-74985e8785e3}} , {{formula:d26276a7-1090-44a5-acb8-f5370047b558}} and hence
{{formula:7211d325-8a18-47e4-822c-48bfe0b62a5f}} .
The second inequality holds because there are at most {{formula:75946fd8-2a6c-4afd-b0f7-4a8bbb13d03a}} elements {{formula:7f2dbba7-c754-4a9d-be97-f989e9684942}} and at
most {{formula:cd8203ec-2778-455d-b6e9-c9daf991c06b}} elements {{formula:b092cba0-2034-4ccb-90da-3935cc2d7f5c}} .
Proof of Lemma
We split the lemma in three separate ones.
Note that {{formula:29cdb949-ba14-4a97-aa59-e9024d5c2e2f}} is the distinguishability of {{formula:bc490bb0-f281-429a-a619-428f62fa1d2b}} , {{formula:fe7cacd9-4070-4c4a-a68c-d1ffef15aae6}} is the number of states
of the dynamics transducer {{formula:4b96b9e4-8bd5-4f52-ba2a-35b4dc8cf6c4}} , and {{formula:45768696-1603-4562-beba-75b1ee5c53ed}} is the stop probability.
Lemma .1.
The distinguishability of {{formula:2e3e6503-3905-480f-bf2e-22e05f7b7992}} is at least {{formula:f39ed06a-dac8-4a75-95ed-9147b57f4c7a}} .
Consider two distinct states {{formula:64041737-b435-4d15-bba5-86a5c7181fcf}} and {{formula:ae5bb2f4-55ee-4daf-b011-51b05b80d37e}} of {{formula:39277fe8-e548-4a66-a133-93754bcdb496}} .
It suffices to show a string {{formula:b67efdd4-d6c4-4082-9a80-fa395a166c27}} such that
{{formula:9248c0b9-a352-4011-8c46-e8d4bef4c695}} .
Since {{formula:e31c14aa-c6d8-4d9a-a936-f3cb91f667d2}} and {{formula:7b5dbf19-7f18-4999-b707-eee7766f4e9a}} are distinct and {{formula:1e7b1cdb-b54f-403a-88ad-52ac5fea2023}} is minimum, there are strings
{{formula:90746ab0-2d76-41f3-ac8f-465b0980afc4}} such that:
{{formula:e5952e81-dfc7-4f2d-8da7-938894a3bdbc}} or {{formula:9acd0cac-44ae-4eb8-8496-e5339423b98e}} have non-zero probability of being prefixes of strings
generated by
{{formula:80c04252-3b5d-4919-bde1-e0fcca80698b}} , i.e., {{formula:5e0d4570-d3cd-49df-9c75-f420af9e9e24}} or
{{formula:ca572f91-7679-4bbb-8e46-46b206d98b0e}} ;
{{formula:21b5ed0d-64f1-4654-a707-99aafb6c23b3}} and {{formula:40aeeebc-d213-4d2c-8600-500d76a26e04}} lead to {{formula:8e4df446-1d43-4189-af9f-21ea4e29942d}} and {{formula:da02045d-889b-4edb-a2bb-a23966ef2f50}} , respectively, starting from the
initial state {{formula:f338c736-c76c-4844-9c5d-e1b4aa604272}} , i.e., {{formula:cc167590-e444-467d-8f08-907b3b8c8cbf}} and {{formula:30fa6bce-6734-4d3e-af62-d3d76177dd94}} ;
{{formula:ac61dbd3-c6e7-47f6-b99d-783381330d2d}} has non-zero probability of being a prefix of a string generated by
{{formula:407a51ac-e3b2-47d2-aa2e-5da4a3c7cc0d}} from state {{formula:4ea568ac-fb1a-4e00-8a5e-cfc0718eb90c}} or {{formula:443482ed-d9c6-4dbd-acc9-5b938af2987e}} , i.e.,
{{formula:e0dc4174-5241-411c-806e-cb4956b0ecbe}} .
Let {{formula:89409781-9724-477e-a46c-e6029bed8339}} and {{formula:49ccaa6a-8844-4b81-8e04-9c32ee6fe233}} be the histories obtained from {{formula:3b5ece80-94df-474f-ae49-28ff1583bce0}} and {{formula:96c149aa-7cb6-4e85-b075-190fe078d200}} by
removing all action and reward symbols.
By the construction of {{formula:72100b9b-ec93-4982-9919-2a4470ea624e}} ,
we have {{formula:f8229375-3407-45ad-902d-88cae4c80b89}} and {{formula:073e7aec-3f85-4b69-afb3-f146fe2481f7}} ,
and hence
{{formula:4670a4d8-1675-4822-a16f-c65181d58b6f}} since the uniform
policy {{formula:393a6b6b-50aa-4ad2-b098-04e66710c838}} does not decrease the probability of a trace {{formula:26c02f14-2e09-4e04-aad3-dea8f054df1a}} with
respect to the policy {{formula:7fee56df-a97c-42dc-8c86-537b42565219}} —which selects actions uniformly at random like
the uniform policy and, additionally, considers to stop at every step.
Thus, {{formula:b8bad054-369e-4954-aaf8-fb57d44cd6da}} because
{{formula:d97146d3-6e5e-4b57-b473-fd58eece280a}} is {{formula:196c97c3-6093-43f7-ac61-a0f3ea0257c4}} -distinguishable.
Since {{formula:42987238-505b-4d59-9757-93cddc83b352}} is represented by a transducer with {{formula:63cd1f1b-bdcf-4b54-8a41-0ed85e3d6efd}} states, and
{{formula:e00f64fd-e4e2-46b9-af20-b10f709c0085}} is stateless, also {{formula:278a4ae4-7a02-4c67-ba3f-781755a8d946}} can be represented by a
transducer with {{formula:8559fa2f-1440-4bf7-9dc3-41dc8db43832}} states. Thus, a pumping lemma argument implies that
{{formula:9ffdbcb8-ab65-4b3e-b99d-d6c92e7441bd}} for some
prefix {{formula:48106d79-9182-493c-a5d6-9777538817c6}} of {{formula:952151bc-f1ba-406d-9d36-2933924c9ca9}} having length at most {{formula:f985965a-3417-46c0-980a-bb40a104e001}} .
We have that
{{formula:af57dbf0-f877-4b88-b26f-9648728c32fa}} is given by
{{formula:494a03a9-79dd-42f8-b6b4-e34706f54482}} multiplied by the
probability of not stopping for {{formula:e6b92f9d-e9b8-4301-9ccc-277feb42d400}} times in a row, which is {{formula:b7237bc8-4cf8-4418-998f-c29e7fc2fe60}} .
Therefore,
{{formula:43f5e068-fbeb-42f0-81a9-59b91eaa9d41}} , which shows the
lemma.
Lemma .2.
The probability that a state of {{formula:4d8a909e-d95c-4b47-9f95-dab36a514338}} is visited during a run is at
least {{formula:2eb85425-4601-47fa-b417-bb6f1a16f6ca}} .
A state of {{formula:45b60419-2f93-4b3a-8385-1335a694977f}} is visited if the corresponding state of {{formula:c404b878-7d37-423c-b2bf-304f9aa3392f}} is
visited under policy {{formula:e08694ac-7fbd-4e70-8a64-f2657e48c903}} .
The probability of visiting a state of {{formula:35eb528b-d949-40eb-80db-c9fe564112f5}} under policy {{formula:3e17f9c0-4ad2-4831-a223-a5abe493d02b}} is at least
the probability of visiting it in {{formula:a52aad86-08d5-4ac0-af02-ab5848552ca4}} steps given that we perform {{formula:84f11f10-7dd7-4c27-a78d-c2a358f45b46}} steps,
times the probability of performing {{formula:9584bf7e-77c6-40d2-917a-bb4099f526c4}} steps.
The former probability is at least {{formula:ce66e7b3-a5c7-4b49-8d6c-d7be2071634b}} by definition, and the latter
probability is {{formula:ae89dccf-c7bd-49cc-914e-52528fec1f01}} .
Lemma .3.
The minimum non-zero probability of a non-{{formula:49905ac5-ff5a-4393-81c3-dcf72cb5284a}} transition of
{{formula:4fe4016e-cad9-4867-99ad-17650e9334ff}} is at least {{formula:f408e367-a0e4-4192-9b49-2177e7bee660}} .
The minimum non-zero probability of a non-{{formula:b77497b4-18db-418e-bd96-4df3a8a4fd90}} transition of
{{formula:3fe98eda-1072-44c4-a09d-13c11e623244}} can be established from the definition of {{formula:bacdd3ff-328c-4d7d-a95a-ae806479adf5}} . It is
the quantity {{formula:53a1fc2a-7a29-4fdd-b001-0d369bd569f6}} multiplied by the minimum non-zero
probability of an observation-state in {{formula:7a068d98-93e0-4142-bca6-ad681a4cd6c7}} for a given history an action,
wich is the degree of determinism {{formula:8817b087-307e-463d-bd64-d5fd8a62fe06}} .
Proof of Lemma
{{table:8494dc68-ed74-4468-95ae-42a292f40d65}}The following proposition summarises the guarantees of the {{formula:b64c0618-6c1d-41c9-87dd-f430c942947e}} algorithm.
In particular, it summarises Definition 3 and Theorem 25 of
{{cite:e98867c0f12508f6744faddd1bec34a3fbd92f96}}.
Differently from the original statements, the following proposition takes
{{formula:cd8add5a-ff07-4acc-a27a-0ef3379bd7da}} , {{formula:de80a751-9b66-439d-9293-a5d6865c4f1b}} , and {{formula:f7fcdb06-a7ee-4447-a179-a48ddb53a7f1}} as primary
quantities.We have named quantities in a way that highlights the
correspondence with {{cite:e98867c0f12508f6744faddd1bec34a3fbd92f96}}. Quantities {{formula:56b13aaa-7a34-4f59-abf7-24b972444d53}} and
{{formula:6d204894-948a-49ae-82ea-6dc78fc18455}} are as in {{cite:e98867c0f12508f6744faddd1bec34a3fbd92f96}}. Quantity {{formula:8dd06187-5494-4280-925a-416a7ef75cce}}
corresponds to {{formula:9fca5855-5e05-4a41-9288-b8ec72e32f14}} , where {{formula:6047ee35-3a9d-4f47-880a-28b4b97393c2}} is the expected length of strings. We
have maintained names also for {{formula:46ebd1a4-cf8b-49ec-99ab-ae8890f17df7}} , {{formula:c811abd4-c86c-4c0c-a6ab-eb8cbc62c363}} , and {{formula:b04473c6-a338-4a92-9581-ef18e7c7aef4}} .
The reason is that {{cite:e98867c0f12508f6744faddd1bec34a3fbd92f96}} were interested in learning likely
parts of the automaton, whereas we want to learn all the parts of the automaton,
and hence we impose upfront that they are sufficiently likely.
Note also that our notion of distinguishability yields a lower bound on their
notion of distinguishability for PDFA, and hence can be used in place of theirs.
In particular, their distinguishability also takes into account the supremum
distance {{formula:7ad5d4c5-1851-40bb-b121-0a78a85b7c73}} , which is not relevant to our setting—see
Definition 2 of {{cite:e98867c0f12508f6744faddd1bec34a3fbd92f96}}.
Proposition 7 (Balle et al., 2013)
Let {{formula:8a4470eb-f1c5-4f43-acd1-a953d6b5d015}} ,
let {{formula:246ae5e8-1425-44ae-a49f-ce11e3f869b2}} , and
let {{formula:619c916a-09e8-41b9-8da5-2866ac08baa4}} .
Furthermore,
let {{formula:7a9d0bbf-1b27-4b0f-beed-e942803c4044}} be a non-negative integer, and
let {{formula:c4cb9df1-5225-4a0a-a876-1e2b5cb62ebb}} be a set of strings generated by {{formula:24dbdffc-787b-43af-8ce4-76852115376e}} .
Consider the following conditions:
{{formula:7ed2d945-b51b-4dbb-8493-8fe6ea6d0569}} is an upper bound on the number of states of {{formula:a9664444-4eee-48c9-abf2-0d698171d3fa}} ,
{{formula:1010fff8-5cae-443c-9d95-b4df93646290}} is {{formula:17d553c9-89c3-45f4-bb91-09b3374a35ae}} -distinguishable,
the minimum probability that a given state of {{formula:887d42f1-7504-4b21-b393-d1ea688a6bb9}} is visited in
a run is at least {{formula:14748153-8ac0-4bdd-b53e-af73421d0fcf}} ,
the minimum probability of a non-{{formula:15fb49a4-8134-4436-a649-29231b874154}} transition of {{formula:b40c3a85-2b49-4784-98d2-e37881031c8c}}
is at least {{formula:a79b6f6a-fa0f-4bfc-acdf-1c588959c24f}} ,
the size of {{formula:65d5f7e4-56b4-4534-a966-b917ad6d2cf2}} is greater than {{formula:bf4cce7f-3a7f-4660-be2e-1eef253b0993}} , where {{formula:ec9f4c53-05ae-42ec-8ab5-8d662014c34a}} and {{formula:094e6021-fe8e-485c-9692-10742273c722}} are
defined in Table REF .
If conditions REF –REF hold,
then the {{formula:27cd432f-caf8-4579-94a0-2ce110ae3a63}} algorithm on input {{formula:909a4fa7-8d64-4b5f-b57a-76800868ede7}}
returns a PDFA {{formula:3544c08e-07a6-49fe-9be1-d8647ed1e287}} that is an {{formula:57be4018-472f-4864-8ffb-4b1a3b3e49fa}} -approximation of
{{formula:5c232c46-aa76-4918-ae1b-2f0e86b7d705}} with probability {{formula:aa35ccfb-31d2-4935-be10-59f29a121684}} .
Furthermore, the algorithm runs in time
{{formula:b1d1fbab-c025-454e-8c14-0f4fc7dc4fed}} , even if the above conditions
do not hold.{{formula:bf04cd08-6ea6-467f-8187-bda17162b00e}} denotes the size of {{formula:3cba837c-8077-49d6-badc-fec5ffe7a044}} defined as the sum (with
repetitions) of the length of all its strings in {{formula:61bdc503-d8b7-4d0b-85db-4bf62f4a2896}} .
We apply the previous proposition, using the bounds from
Lemma to ensure
that conditions REF –REF hold.
*
It suffices to take {{formula:8a20056f-1e8e-4f08-8b3c-37adbf4fdff0}} and find values for
{{formula:1f0792bf-8800-4fbb-bbcd-e9d6cc8d3e90}} , {{formula:140cd109-5327-4a49-84f1-024ddb454bfa}} , {{formula:ced38f16-7363-41b5-9390-58ba161e85f3}} , {{formula:5c744803-a393-4719-afb5-d208f7a343c3}} , and {{formula:e4768fec-2028-4deb-89e3-59ca01a410f5}} such that
conditions REF –REF of
Proposition REF are satisfied.
Since
{{formula:97db32b0-22ad-4dcc-af7b-1e4c0c2e83fa}} implies {{formula:4f5fcacb-5172-4d52-a928-d0b51cd8b9a7}} ,
by Lemma .1 we can take
{{formula:4b550d48-c8d4-4589-9cf0-f90daa60acdb}} , and
by Lemma .2 we can take
{{formula:63bde1bc-cdcd-4849-a7b0-099acc89aec5}} .
Furthermore, by Lemma .3
we can take {{formula:624bd586-cc7a-4bff-bdd2-4c5fcbf0caba}} .
Finally, condition REF is satisfied since
{{formula:a036167c-26ae-4a6c-b8fa-cba0521e4596}} and {{formula:51db249d-75b6-444f-b499-2f4191179268}} are {{formula:79f1c8f0-a02d-4aa8-b0cc-8ff60a11eccc}} and {{formula:5d7d6419-ed70-49c2-a046-38be8dd4efb0}} , respectively, when we
instantiate the parameters specified above.
Note also that {{formula:4cf54333-78c0-4ae0-8704-845799062a33}} since the length of episodes has a geometric
distribution with parameter {{formula:a0484ab6-a082-4227-b2e0-f22108d56ab2}} .
Proof of Theorem
{{table:c199efe5-dda6-41d5-8afb-e3f5b6304655}}Algorithm REF computes correct estimates for the alphabet and the
maximum reward, when the number of collected episodes is sufficiently high.
Lemma 11
Consider the values {{formula:1df9d3ff-d838-4586-b3fb-9222148536b5}} and {{formula:c88bd0ba-701f-43ea-b5ca-7f047eaab4fa}} computed
by Algorithm REF at Line 11.
If {{formula:66eecdc6-b6c4-4874-949a-2f5250f4a0b9}} , then {{formula:08f9d29e-ddb3-44b8-a4ad-738cdc253987}} and
{{formula:9a0d4fa4-c040-4663-9910-f90ad17df40f}} .
If {{formula:a3b12055-49c7-4159-b2b5-9e95f7533639}} , the {{formula:6e1a9300-94da-4479-a094-69d040e47a5b}} algorithm correctly
finds all transitions of {{formula:2fd923eb-5b2e-4f9d-b220-6751a418fdeb}} , by Proposition REF . For
each such transition, the algorithm requires
to see at least an example in {{formula:79bd0066-8fcc-4940-b9a2-72eb59ebbe6d}} , and hence {{formula:92cb1b25-981d-43e1-8a8b-d0d916071277}} and
{{formula:bf176494-f90d-4053-8603-1ffffe456b97}} .
Algorithm REF computes an accurate automaton in polynomially-many
iterations.
Lemma 12
Let {{formula:d154474e-4f4e-4d92-a508-4032a7a691ab}} with {{formula:036faa90-135a-4025-ad80-f0a450950435}} , {{formula:c73753e0-db3f-42f3-aa12-239af416f788}} ,
{{formula:d06fcb12-def9-492b-9d76-e914c774d5c6}} , and {{formula:5735cc8d-b072-4864-8f22-eda185aa62e6}} defined as in
Table REF .
Consider the automaton {{formula:07304253-646e-4650-a581-e4d1574cb2e1}} computed in Line 12 of
Algorithm REF at the {{formula:d30854c9-8336-400e-b2fa-95b3efebe581}} -th iteration of the main loop.
Then, {{formula:9ce21c54-1abb-4737-98c7-e0806eb557c8}} is an {{formula:90946c44-5564-4410-b2ea-0c4607c6c1c8}} -approximation of {{formula:b3ac3963-1845-448f-8274-5d55a2388c41}}
with probability at least {{formula:3e708c59-2f40-437a-b3d2-c6c0ab7054ab}} .
Consider the {{formula:4ae241bf-c70a-4323-a4db-624e4763185c}} -th iteration.
Since {{formula:943a95f5-af49-4eeb-9d8a-eb164dc24580}} , we have that {{formula:2daddaf8-84c1-4c2f-9cec-bfaf642b22b4}} and {{formula:b9eb0e3f-6a3f-4374-9e46-f186be898f29}} is
instantiated with {{formula:aab64aa0-2131-453c-b2d7-f0ed0654fc03}} as a correct upper bound for
the number of states.
Thus, the first two conditions of Lemma
are satisfied, and it remains to show that {{formula:59ac8552-a2c7-4f63-a005-c09cc94ba825}} with
sufficient probability; note that probability {{formula:1c771f03-c55b-4514-b4eb-39f1e8cd02fc}} suffices, since
when it is multiplied by the probability of success {{formula:3e93d104-3292-41ef-9813-bb6abb4c3230}} of {{formula:9ad862ad-cefe-49dd-9d4f-353ae7788499}} ,
it is still above the required confidence {{formula:437e1806-6b22-4147-8e24-7fc6b034cab4}} .
The size of {{formula:041b16ff-386e-4ce5-8c0f-424cb2c48b9c}} mentioned above also guarantees that
{{formula:b183910b-72c8-4402-8cac-d1c25628cf6e}} (Line 11),
and hence that {{formula:5c457e2c-b7d3-4b87-a945-8d52462071fd}} is called with a correct value for the alphabet size.
We first argue that {{formula:fb926572-6fd2-4550-a16a-d361f9a4e261}} ensures that the
number of generated episodes {{formula:874f35d6-5571-4eca-bf9d-a65e75d8682b}} is at least
{{formula:6edefb0b-5c88-411d-8343-63325dec4824}} with probability at least {{formula:6c42b2cd-5a8a-4447-8e68-7a20cdae8ec7}} .
The algorithm performs
{{formula:979040c5-9348-4cf5-a66c-84cd46b78c11}} actions.
The number of episodes corresponds to the number of stop actions among {{formula:8032ac94-5525-445b-bfc2-1a648c369817}}
actions.
Thus, by a Chernoff bound,
the probability that the {{formula:17f1304c-0982-480d-82ae-d5d56fadf976}} -th iteration generates less than
{{formula:a07b7054-fa50-4705-87eb-7a1a861b6cdb}} episodes is at most
{{formula:35a89565-bc70-407a-b3ba-c5dd2480ee7f}} , which is at most {{formula:d3cd0c45-1abc-47d7-9e3b-5a016bd98da9}} since
{{formula:0fbff3a1-9885-4959-bfd4-604e5f66015f}} .
Then, it suffices to show
{{formula:565c14fd-0be8-44af-976a-a57f1bd25147}} .
The critical aspect is that also the values of {{formula:0a5f8eb0-3255-46e1-b65f-1b4c5d817dab}} and
{{formula:5e4e413a-83f4-4009-a1b5-b01bd1e9301b}} depend on {{formula:10099136-471c-4fe5-a005-9ab22b533ae8}} , because {{formula:83013695-a65f-4b8d-a80a-65a3ed02d54d}} and {{formula:436eeb58-0353-4eb0-87cf-c2a2f5c5ef76}} in the definition of
{{formula:1dc2bef1-c0d5-42eb-b9ef-0352275b9189}} and {{formula:082fb8ae-9240-4ee2-9dea-ba43dad9647e}} are two be instantiated as
{{formula:476e0b20-c125-4afd-a7d0-f597d54d6b4f}} and {{formula:5f64e78f-c709-4670-b501-7ced7762ab6d}} .
It is easy to verify that
{{formula:f8eb3761-774e-4932-b7c0-b5f319dd1f0e}} when
{{formula:7eb0c808-0e12-40df-b153-44136f0de3b0}} .
The size of alphabet {{formula:2acdbcb5-f4fc-49fd-b307-6fc77a24750b}} is bounded by the parameters in {{formula:10d8aabe-b610-4adb-8b8b-24821540b468}} .
Note that
{{formula:a13e22aa-0a42-43f5-a409-1d8cae6e1489}} as defined in
Section .
Lemma 13
{{formula:56fbd886-04e5-4f90-bc97-a5b8c1cbc404}} .
For every history {{formula:8919b03b-75d5-4fc2-897f-b5b34eaf599d}} and action {{formula:45b5ef53-12a4-43c6-aec9-461d6e658754}} , if
{{formula:526172aa-0ed4-414f-ae18-60e29c07991e}} assigns non-zero probability to an observation
state, then it assigns at least probability {{formula:d91a1a1d-1c8a-482b-860e-1a75ad4f2ee9}} by definition of {{formula:63780ee0-f766-4c17-ba14-eb2778f72fea}} ;
thus, {{formula:16033289-362c-490f-aeef-3c2da9b5203c}} assigns non-zero probability to at most
{{formula:ebda03da-a397-4205-9aea-8dbe10be4b1f}} .
We have that {{formula:137d9cce-908d-483e-8f09-d6d364fa2a2a}} is {{formula:e72ce8f5-79e7-475e-ac88-22d2075ec060}} if
{{formula:a6b1aa6d-16c3-4e97-8fb4-199996648b81}} and zero otherwise, by definition; hence,
there are at most {{formula:2c2b95c1-0ae0-4f31-bb82-0670a41a8c5e}} pairs {{formula:d2ccb962-98fc-45d4-aa26-d44a74f8facf}} that
are assigned non-zero probability by {{formula:6909de83-8818-42d1-ab1c-d04fff15186d}} .
Furthermore, the dynamics function can be expressed in terms of the transducer
output, i.e., {{formula:f809e1fd-1f8b-4368-b19b-d4f6e5d48c58}} , and hence the
number of pairs {{formula:5eeeed45-21b6-4369-af6d-36ae030f15f1}} that are assigned non-zero probability when the history
is {{formula:bb9c25a7-0a21-4856-bb50-06c62ae64c5f}} and the action is {{formula:c8c5995a-49c4-4e50-982b-3cd58ef5426f}} are the pairs that are assigned
non-zero probability when the transducer state {{formula:5fb74b2f-c6bd-4767-baa8-2f0ba5eddd77}} and the action is
{{formula:7febf1be-46e9-4600-a183-4180e75668d1}} .
Since there are {{formula:003aad4d-40d2-451d-a7be-9b8814e25499}} states and {{formula:6bce9c38-8a48-456f-bd85-387986fbdfa9}} actions,
we have {{formula:90d23d4f-f2eb-42e9-8392-576f1da33413}} as required.
The main theorem of this section follows from
Lemmas –,
REF , and REF .
*
By Lemma ,
Algorithm REF returns a policy that is {{formula:fe2aa453-2d66-48a7-a580-c1fdc15fdb46}} -optimal with
confidence at least {{formula:c44d8a0b-72e0-443f-862d-d81a1a49b371}} when the policy {{formula:7efa12b2-213d-4837-83e6-87f3d081c4ab}} computed in Line 15 is
{{formula:689c7229-6efb-4ae4-a550-b276f9c814a6}} -optimal with confidence at least {{formula:45eaf1f0-f1a6-48df-8415-a8d592c4da96}} .
By Lemma , this happens when
{{formula:3fd6f6f8-c73b-41a5-b077-e7cc746cf551}} is an {{formula:7619a092-c231-4d18-9955-4cc902ea70be}} -approximation of {{formula:c92170c4-94c1-4dc2-b507-bc9662ea839a}} , assuming that
{{formula:12ab064d-6a3b-4469-8161-6f185250da9b}} .
By Lemma ,
{{formula:271cb559-4c73-4c17-a8d0-96cfa3cf1332}} is an {{formula:d0135942-6aa6-4dee-ad21-1dcce94074f0}} -approximation of {{formula:253961a1-57f2-4666-9ab9-e861dff78bda}} if
{{formula:679030fd-bbe0-44d1-89a6-821b7b4ddc1e}} is an {{formula:3d369813-5af9-4797-8df8-20c49c3164be}} -approximation of {{formula:e1628cb7-2771-42c8-a8b9-ccec92ea5b0b}} .
By Lemma REF ,
the former condition is true in the {{formula:96bda480-f7f4-413f-bd41-345c964389e8}} -th iteration with
{{formula:76bbf3dc-27d1-426f-b2ad-a53bf96a4f06}} ; this condition also guarantees that
{{formula:58345fca-d783-4296-8c87-4741c56f43b8}} , by Lemma REF .
We have that {{formula:1cdc93b5-ca35-4cd2-b502-00facaa5882f}} is polynomial in {{formula:7b4d2a50-4a60-4bb1-a51c-0380d0f7b9a3}} as given in
(REF ), considering also the
bound for {{formula:3037c0f9-c3af-477a-a2ef-217725aab390}} given in Lemma REF .
Since each iteration performs {{formula:983a5dd0-a4bf-43aa-8731-9257c1e4b643}} action steps (see
Line 2), and the {{formula:51d3cce8-d41b-40a4-934a-bf33a725aee1}} and {{formula:2aee86fa-488f-4f27-989a-61643c70c636}} algorithms run in polynomial time,
we have that each iteration performs a polynomial number of steps.
Therefore, the end of the {{formula:fa97c44f-7141-4b29-8cfd-fe188264247f}} -th iteration is also reached in a
a polynomial number of steps, at which point the algorithm returns an
{{formula:b87052c2-99b1-480d-8a13-7b8074d6203a}} -optimal policy with probability at least {{formula:910167b0-0e8c-4857-888e-7d18434d8cb7}} .
Finally, since the above reasoning applies to the following iterations as
well, every policy returned at a later moment is also
{{formula:27246574-973d-4d0b-9b66-ed66ffd1782a}} -optimal with probability at least {{formula:9cda6052-7930-4a4b-b31d-ca0df7df84e5}} .
Therefore, Algorithm REF is PAC-RL.
Proof of Theorem
{{table:2ebe4d21-a1d0-4617-bbce-9dd793acaf40}}*
It suffices to show that the expected number of action steps is at most
{{formula:8d7075e4-fb10-45b3-8d52-4315f84438b1}} , where
{{formula:43d00f9d-e1c8-4116-b673-683edddf74b5}} and {{formula:1740bd37-00a3-45db-ac9b-2afb61a1fa88}} are given in Table REF .
By the same arguments used for Theorem , based
on Lemmas
,
,
, and
,
the algorithm outputs an {{formula:58159efe-77d2-4ecc-a6e3-84504fa20889}} -optimal policy with confidence
{{formula:d6efb81c-9911-4cca-bd17-570aef6a161a}} if it has read {{formula:f6259b84-08f5-499f-8795-8fa9b34e9fee}} episodes.
In particular, the values {{formula:d205c9f1-bb90-4bbc-8f38-1cee5d11d875}} and {{formula:4e9f303a-2d7e-46b4-a0b6-3bcfa59d621b}} can be derived from the values
{{formula:2d442b22-a2c0-431a-906e-b4da65ef4779}} and {{formula:4f6a48b1-e90c-4ef5-bc5b-ab1fa16e8dcd}} of Table REF , respectively.
The expected length of an episode is {{formula:cb3d16b7-5242-42e2-8ae2-5fdbc1141d96}} , since the length of an
episode is a geometric random variable with parameter {{formula:01a8a62f-6723-4127-af0f-771540926282}} .
Therefore, the expected number of action steps is
{{formula:0f560357-4a4c-42bf-a202-3bbe01ad113d}} .
Additional Material for The Running Example
Formal description and transducer.
In Example REF we have described a family of RDPs
{{formula:4ff19279-1272-4ac1-bbf0-0e63c8e801c5}}
where
{{formula:9aaabf91-42c6-4d47-a9dd-5b77b717c5ef}} ,
{{formula:6877a9a2-a81b-4144-8605-755ef4919440}} , {{formula:c2c66bce-646f-49bf-99bf-eb84d7414ea9}} , {{formula:259038e9-b559-4d9e-8adf-786e2578acbb}} (the discount
factor is irrelevant in this case), and the dynamics function {{formula:eda2de50-63b2-4278-9894-9f3ba8132cba}}
is represented by the transducer
{{formula:8f777e79-aad8-495b-962e-08f762757a53}} where:
the states are
{{formula:bbeda388-1cf2-4138-a197-234a98e00e19}} where the first component denotes the current
agent's column and the second component is a bit denoting which of the
two sets of probabilities are being used by the enemies,
the initial state is
{{formula:060e6c6e-ef5e-42ca-974b-218d10082fcc}} ,
the transition function is:
{{formula:a57c60b6-9d6e-46a8-8bbd-b6ba3c9ba6d9}} ,
{{formula:f48eddf2-4f34-4a22-b0e7-3604bad14544}} ,
the output function is:
{{formula:0b37fd88-0acc-4244-94be-72bd8f77318a}} where {{formula:cb435c6b-d43d-4699-9df9-2a6bab86f986}} ,
{{formula:a3e8525c-23b5-4f30-b28d-018996cfaae7}} where {{formula:5e1cddca-9794-4c90-b715-97710b63bace}} .
Value of the parameters.
The reachability {{formula:a968cbe3-96f5-4bf7-8bbb-5ab62576cea8}} of the RDP {{formula:543de7bc-48d1-4375-b5b6-5d94cc77d7e4}} above is {{formula:ada349b8-4891-4fe6-b2d1-02c42d3a3462}} .
It suffices to show, by induction, that every state reachable in {{formula:8175f629-da91-4c44-ae06-a68587944e15}} steps has
probability at least {{formula:4dec068c-fd9d-4956-9f61-a050dc016d61}} to be visited at step {{formula:15f6fed9-ae58-48ee-a302-74fb7c6e8442}} , under the uniform policy.
For the base case we have that the initial state {{formula:6cbfeb1a-42df-41a3-85e6-50c610c799ef}} has
probability 1 to be visited at step 0.
By the inductive hypothesis, states {{formula:bacdaf96-23d1-4d2f-bb2f-b821758b3c6a}} and
{{formula:8deb822c-2cab-4d8f-8178-1ef814fd9c3c}} have probability at least {{formula:8869e14c-e5b0-4e5d-b1c8-a8b7cd941b08}} to be visited at step
{{formula:6edfe5bd-068d-4a30-965f-103dabee9ca2}} .
Thus, the probability of visiting
{{formula:3b51f99e-7c81-4c57-a2d3-e12521be507c}} or {{formula:612af53a-9ccd-4776-82c1-180f0b255fe1}} at step {{formula:36fde089-0ea2-4e96-be82-d868d15e77bc}} is at least
{{formula:3e03ee38-eee1-4e3c-9cdc-063c35fc2cb5}}
The degree of determinism {{formula:4de82436-269f-42cf-a7a7-514f72708ad6}} of {{formula:c31d31ae-e2fa-46af-adb2-790e31fb4fce}} is the minimum value returned by
{{formula:c9353eeb-679f-4a1a-9dd1-3b61806cf506}} , which is the minimum, for any {{formula:337a1303-d544-474e-bd62-5d70c2785f02}} , among {{formula:9b447077-a98d-4d16-8035-2425a2ae272b}} , {{formula:1e1ea0fd-6ac4-4c91-bbb2-90e74abb6d5a}} ,
{{formula:f551fe8f-7af6-4e75-980e-0aa456e8aee2}} , and {{formula:cdca1d0b-0bc5-492b-9112-29a944eb00ab}} .
The distinguishability {{formula:f85edc29-1744-4185-b980-d854d1432173}} of {{formula:4902a8c4-fa80-46a5-bd9a-adf265b43be6}} is determined by the minimum probability
difference of an observation in two different states.
For every two states {{formula:86c44a65-fb1a-4e15-bf1c-866a30434448}} and {{formula:c7651ce2-b354-4008-a1dd-d6524f5403c9}} with
{{formula:f8696a3d-6f99-4272-beeb-0b110d3499c0}} , we have that there is an observation
{{formula:1628186e-d722-4fb7-803c-ca383d9a5fe8}} that has probability at least {{formula:3d050868-2ece-4270-9556-22290c583b0f}} in
the former state and probability zero in the latter state.
For every two states {{formula:f76638ff-c9cc-429a-b79f-0cd91f894ce0}} and {{formula:1d680b1a-5016-4bb8-a06c-c96da42094c6}} with
{{formula:fdffc5f0-f295-41a5-ac8f-d22c0c5ae431}} , we have that the difference in the probability of
{{formula:0220b73e-7cab-41a1-a2e3-5244bc12c1fc}} is at least {{formula:9c1c6835-bbf6-4625-89a7-8ce2ee980fb9}} .
Taking into account the probability of choosing an action uniformly at random,
which is {{formula:d201c2cf-bb0a-494d-86ad-d7e6ac2b17cd}} , we have that the distinguishability is at least
{{formula:52ea042b-1300-453e-8382-fff999576ebd}} .
Comparison with [Abadi and Brafman, 2020]
We compare our approach with the {{formula:16082994-13cd-4c9f-8e63-4235e43c7540}} algorithm from
{{cite:541868f80717e9f92340554f2badee26f6a4bff2}} on the family of RDPs {{formula:ac867dce-962b-4dc2-b4df-f6915126e81e}} introduced in
Example REF and formally described in
Appendix .
Our algorithms output a near-optimal policy within a number of
steps that is polynomial in grid length {{formula:1c8cb1eb-6229-4fc3-ad14-f3ded7e110df}} , whereas {{formula:97f33f72-2b3a-4851-94e8-c76629126cb3}} requires a number of
steps that is exponential in {{formula:ae2be82b-9c48-4be3-9894-41de2cb3f294}} .
The performance of our algorithms follows from Theorems
and , and from the fact that the parameters describing
{{formula:76e70c35-52e2-493f-a745-2df7401d0f81}} grow polynomially with {{formula:ba26f249-da5b-4ae8-bc9d-0dffe1f87918}} —please refer again to
Appendix for a discussion of the parameters.
For the {{formula:c008fd57-225f-4754-b8fb-dcc7a6fae112}} algorithm,
we argue that it cannot achieve arbitrary precision and confidence in polynomial
time, since the accuracy of its estimates depends on the probability of
single histories of length {{formula:91209e35-06d1-404f-b24a-470306d6e861}} , which decreases exponentially with {{formula:5f534de9-31cc-4f5a-a31f-286aca9d2338}} .
Our argument to show a lower bound for the {{formula:fab9a06a-c93f-4e5d-b81e-0f5cf36636ec}} algorithm is based on the
fact that the probability of generating two identical histories up to the
{{formula:60c4fc9b-e934-481a-8808-93c4cb06624b}} -th step in {{formula:97d686d8-2051-4ae8-8c53-738eecb13760}} episodes can decrease exponentially with {{formula:e7f783be-8319-488f-9f5b-c49b513b0197}} —proof below.
Proposition 8
Let {{formula:2d0bfde4-cef0-4b3b-8906-c27861aa00d9}} .
For every history {{formula:1a938bee-82ef-486e-94d3-f43121c438e7}} and every policy {{formula:e6f49ea3-00b4-4a3f-9af8-08407daf1146}} , the
probability of observing {{formula:3e552046-b8c5-47d8-a7fc-c0a9eaa65055}} at least twice in {{formula:bd8efe42-65dd-417a-827c-867263cb77a0}} episodes is at most
{{formula:2e644c74-0edb-45d9-90b4-cf9d473f38d3}} .
If {{formula:952da5c8-5ed0-4ae0-a59a-c040a4f54b42}} , and {{formula:4cda2e16-738b-4e1f-91ed-6817f94b671a}} is polynomial in {{formula:3cda3d8b-f7a4-46cc-a99a-5f3e66f60001}} , the former bound goes to zero
as {{formula:d50db74e-ff34-4e51-84f3-dee9c2cec449}} increases, for {{formula:003ca4fc-73df-444f-a057-8c12d5a936da}} sufficiently large.
Thus, for large values of {{formula:d2fab828-6bcd-42e8-b9c9-f9383c4cd303}} , there is a high probability that every history of
length {{formula:686f841d-9079-4266-9884-cad0a785f276}} occurs at most once.
We analyse the behaviour of the {{formula:c57df963-94e3-46af-8abe-0bd0e5c5a98a}} algorithm when every history of length
{{formula:716c6a17-074e-4d26-bf2e-2dfd5643d2da}} occurs at most once. These histories are important to determine the action
to take in column {{formula:f856a175-6148-49bb-a765-ac6411c6ebcc}} of the grid.
The algorithm compares pairs {{formula:6bfa948e-9d4a-4a99-8661-5000e6a0d3e3}} of a history {{formula:fcd98839-a1fb-49d2-938b-11d740b9dfd2}} of length
{{formula:852eb77f-a3f1-49de-8002-a9d0c6295123}} and an action {{formula:9b5007de-aaae-4fa7-8523-04b5854ee46b}} based on the empirical distribution on the next
observation; the only possible observations are of the form
{{formula:0fa6014b-45b0-49de-8566-8cb541718570}} .
Thus,
the empirical distribution for {{formula:925980cd-9800-4d48-b828-e18b094e4cd7}} is either the
distribution {{formula:41b3be1b-81c9-4782-abac-1f3be2633e9f}} that assigns probability one to {{formula:87067d36-f030-4506-be42-719769a2e1ef}} or the distribution {{formula:1febb396-62d5-40b1-95f0-2d3c9faf7928}} that assigns probability one
to {{formula:f97914af-cafc-47ff-adff-8a148c9e02d7}} .
Let {{formula:ec09e39a-b9a3-4ce2-ab12-9632f262a5fb}} .
Regardless of wether {{formula:b377a3a8-6208-4fd4-a833-33dd54003954}} is assigned to {{formula:00d9a958-2810-4d08-968f-8fc5cf8c59b7}} or {{formula:0bb6f690-13c7-4c53-b7d8-ab8184c1f2f4}} ,
there is probability at least {{formula:32367120-8754-47b5-b833-80d53e608d03}} that the assigned distribution induces a
higher value-estimate for the worse action—e.g., if {{formula:03e05095-2dfe-4be7-a198-e32f9e99c2c0}} and enemy
{{formula:1aa1c0c5-ed75-4515-a1b2-8531a9cd5324}} is in cell {{formula:7f79fc91-5987-4717-b291-333fad0bedff}} with probability {{formula:15c873f3-e94a-49a6-a67d-637708cdc429}} , but we do not hit
enemy {{formula:1cdacc50-3406-4d65-b03a-f00c89dc451c}} .
In such a case, the error in the estimate is at least {{formula:e768109a-4122-4062-8f12-3c0f99dc9353}} .
Therefore, the {{formula:886ae373-5c1d-46f4-aaa0-d6dbae6df93d}} algorithm introduces an error of at least {{formula:f985e73f-dbf2-46c2-b68a-4940fa0f7009}} with
probability at least {{formula:8661f24e-42e2-4f31-95f9-2a8fb464c4c2}} .
For instance, if we take every {{formula:17835f27-7c34-4eec-bdf0-7ed9319e5f51}} and every {{formula:4ceda3cb-7462-442c-986b-b38d6043cc34}} ,
we have that {{formula:c6c2cc05-bf12-43bf-9565-5030dc3fddd8}} as required by the proposition, and the
error of the {{formula:3ab7815b-4d37-4e4c-b2d3-47502bd666fc}} algorithm is at least {{formula:6fc7a66d-b44d-4998-a2b8-34dfc332db51}} with probability at least {{formula:9519c3b8-6610-4027-a229-4e2132cd8c0d}} .
[Proof of Proposition REF ]
First, note that the probability of a given observation
{{formula:7558e1f5-d0ee-461d-ab46-208c927ec424}} upon performing action {{formula:0908c7c3-3f6e-4dbb-b950-f5cf46029e99}} is at most {{formula:6db327b9-d333-46af-bd62-3b24311a6f83}} if {{formula:bc6f7289-b9a4-4240-a722-12a0b7d0c32f}}
and zero otherwise.
We use a union bound over all histories.
There are at most {{formula:c7114e68-8c6b-4145-bf26-0aa8985f0909}} histories of length {{formula:2d762fee-acf0-436b-8e3b-d8c068819259}} having non-zero probability.
However, since the probability of actions {{formula:260b3b1b-e8ce-4e40-84f5-24d9f297534d}} and {{formula:f47705d8-547d-4867-bb30-4ddcb7a0e912}} sums to one under
every policy, and an observation has probability at most {{formula:2fb0ec7d-fcb7-48c2-b283-f231715bcf6c}} if the
action index matches its second component (and probability zero otherwise), we
can consider only one value for the second component of an observation and
bound the probability of the observation by {{formula:aff969b1-68d6-493d-b04f-3509b19e2a62}} .
Thus, the number of histories to consider in the union bound is {{formula:96152d04-7c76-415a-8ebf-c003e90b1e60}} , with
each history having probability at most {{formula:a2fbcc12-0ddf-437c-9177-772023d20ecd}} .
Overall, the probability that any of the histories occurs in any two episodes,
out of {{formula:ca37a698-2026-4945-9e71-bd0cd67a3f69}} episodes, is at most:
{{formula:4bd410c2-9901-4f37-8a35-69143e70c8e5}}
where the binomial coefficient takes into account the possible orders of the
episodes in which a history {{formula:a0074e9b-a07e-4f7b-8729-ebd6c592ee4e}} can occur. The inequality above makes use of
the fact that {{formula:3f6425c9-0f37-41e7-87fb-fdf601cfee83}} .
| d | 0fe77760411b34ad5f416a94578d7bde |
Hardy and Littlewood {{cite:2f979e7ff9b145aa320316123fe8dc8ba1077a9b}} showed that {{formula:0774a97e-66b5-446e-b7db-c741f01c7cf6}} , and Ingham {{cite:9d931bf671e1a54b26c425d978d1ed07e09f0c3a}} showed that {{formula:c01230d7-111d-4e7a-9daa-af27589f9c13}} . It is conjectured that
{{formula:53f22062-bb1e-4fac-af8e-fa0d6b8e4d14}}
| i | f267241071e3cb882f7cd54b5c78b8bd |
In real scenarios, we cannot obtain a solution directly from Eq. (REF ). Therefore, {{cite:cc6262a0dff277cd8a2610bffb8282c92fe3a901}} proposed a solution by splitting the {{formula:68f474c8-cc77-4c4d-9162-ba47b4947826}} axis into bins, computing the local effects inside each bin with a Monte Carlo approximation, and, finally, averaging the bin effects. As we discuss extensively in Sections REF and REF , this approximation does not scale well to high-dimensional datasets and is vulnerable to OOD sampling.
| m | c1ada2d993e7452b9dc78f4476bd973b |
By the same token the higher spin extensions of SDYM and SDGR {{cite:a8e3608b34ca569e76913b322381cc8548498936}}, which were previously discovered as contractions of Chiral Theory in {{cite:087fc6232adaf817c929284a75ebf5c3b30ebaa8}}, must be consistent contractions of the present FDA as well. We note that in the latter two cases the FDA of this paper should provide a complete solution of the problem. Indeed, the actions of these two theories are schematically
{{formula:cea71f9e-c50d-495f-95d7-4992da61cace}}
| d | 5309fb76d5457ca782b59617a1a1aff8 |
In contrast to active tracking, passive tracking collects information about people without the need for active participation. Under the category of passive tracking, some currently used methods include cellular activity tracking {{cite:2b1f5a52f807c6dcfcd47d283a85ff7f51b32bbe}}, {{cite:f1dcfca73f90ebe80c57160a82c68579c5ced6ec}}, {{cite:c897e15b9a2a63573d9d5814d821654b2410cda4}}, {{cite:a9b1f870d5a24bad477ccc8910adeb45c09bbb84}}, and the use of cameras{{cite:597516f5044f766f6f39e99bef6e775ae1026117}}, {{cite:7aac4393317d0f8cacf5622a4448c12d0c7de05b}}, {{cite:d1d17cfe3eade8d5986afcf906a28ab378d4fbdc}}. However, these methods may have certain drawbacks under certain conditions. For instance, the resolution of cellular methods depends on the density of cell towers in a region, which is usually in the scale of kilometers and would be too coarse to measure micro-mobility of humans. Camera-based tracking is usually limited to sparse crowds{{cite:597516f5044f766f6f39e99bef6e775ae1026117}} and a small coverage area {{cite:7aac4393317d0f8cacf5622a4448c12d0c7de05b}}. As limitations based on line-of-sight will be inherent in all camera-based tracking methods, we will look into using a different method of tracking human micro-mobility to avoid this. Alternative methods of tracking passively include the passive sniffing of Bluetooth and Wi-Fi signals from mobile phones.
| i | 0dc395395624e0ac684e2c85d0b80679 |
To alleviate this problem, we focus on the Deep Convolutional Neural Network (DCNN) which is a modified version of Xception {{cite:a303d1870690126d3438800c206bf91fae713028}}. We leverage multi-task learning to enforce the encoder to learn weather and time specific features. We add two simple identical models Weather-Aware-Supervisor (WAS) and Time-Aware-Supervisor (TAS). Each model is composed of two {{formula:b5087193-16f5-40e5-b289-a3661229a368}} atrous 2D convolutions with a rate of 2 and padding of 6. Each convolution is followed by a batch normalization and a rectified linear unit (ReLU). After this, the feature map is flattened and fed to 3 fully connected layers. The last layer predicts the weather for WAS and the daytime-nighttime for TAS. It is worth noting that WAS and TAS are only activated in the training process to guide the feature extraction learning process.
| m | 66eca8d45238a9494c68f8dc14e645d6 |
The mixture distribution facilitates the closed-from Markov-Bayesian recursion greatly in two means: First, a mixture of conjugate priors is also conjugate and can approximate any kind of prior {{cite:0835e051225623fa923362f242309751599a5328}}, {{cite:f92d97fad8c2b43dc3807b419ba74a4d01ac9e63}}. Second, the linear fusion of a finite number of mixtures of the same parametric family remains a mixture of the same family. These properties play a key role in the mixture filters such as GM filter{{cite:2efe4ded0b22643bafa07334b0d45d58b34dd727}}, {{cite:cd75219b049006c066825323c0240e6c3f3d3e29}}, Student's-{{formula:486f454b-261a-4c59-8fbf-dc0553ab8bd7}} mixture filter {{cite:23984102a2f209c6d3b19a049fd17d3b99a1c4d4}} and multi-Bernoulli mixture filters of various forms {{cite:8a3a22e45dbe005c28a4756ed6c438fddc27ee72}}, {{cite:1abc57d0b26570986506dcb4558d2205bea2a1ee}}, {{cite:92b5527cea9fd3ef8c70c64573c07a0d59f2bc3b}}. The AA fusion has demonstrated outstanding performances in many challenging scenarios {{cite:694860a83b1bddf8246c2565841e97db58908e23}}, {{cite:ce67b339cc1561b185cbfa10e64b8fbb8457b074}}, {{cite:fe2201a772d26c205544e42d13a442fda3a16529}}, {{cite:2cf6da445ef341ee419579d82e8b3fa6864898e7}}, {{cite:1baa19582eeb48b36508af34726311146bf9d23b}}, {{cite:c7d3ffd2ecde42ab4c3afaa329b59e0b00bb0418}}, {{cite:9b1c74e2fe5652afb1cc4dfc70520b12e0f3df6a}}, {{cite:ae72557e262d223b35cbc8d76a7083991ebaec65}}, {{cite:6b95999fd6e836243702d72210b7cc6090d9d967}}, {{cite:4dda2d193fe1e2e60b950cc63188ead182d879ca}}.
Nevertheless, statistical and information-theoretical study on the mixture/AA fusion of probability distributions (of the same family or not) seems still missing in two aspects, which motivate this paper.
| i | a4097529f4edcf4c72298eea7c601e43 |
With this multi-disease segmentation in mind, one of the main future directions for the BraTS challenge would be to expand beyond its current focus on glial tumors towards general brain abnormalities. Furthermore, the extension from solely pre-operative baseline scans to post-operative scans, and the inclusion of an additional label for the resection cavity would be a very interesting and clinically appealing direction, as it would speak directly to the assessment of treatment response and disease progression. To ensure robustness and generalizability of the computational algorithms, ample patient data from multiple sites, capturing diverse patient populations are desired. A major hindrance for accessing these datasets is data siloing due to tedious bureaucratic process, data ownership concerns, and legal considerations reflected in patient privacy regulations, such as the American HIPAA {{cite:cf6702ff01d96a058c830f9af0150df9a6e7c5f3}} and the European GDPR{{cite:533d23c2012e629b22a34ce53aad09d6bec26ecf}}. In future, we aim at moving from the current centralised data approach to a federated approach, which would enable researchers to access potentially unprecedented size of data and hence design more robust and generalizable algorithms {{cite:d0071dd34e4fa0efea5b667b179abc847fa811c9}}, {{cite:6864ca561bd30b68655decbe37fc83a2947b5172}}, {{cite:7ad8c19f1f6143e0a24d5482f4303b1bed194ffc}}.
| d | 519f0065ca6d5d000fe8c5e9d30cb1d1 |
All spectral fitting is done with XSPEC 12.11.1 {{cite:bf861bfc4e7b14bfe2b709c077051ea2a2bafe35}}. We use the PG-statistics, for Poisson data with a Gaussian background. We use the wilm set of abundances {{cite:5f61bf41bd67eb94961c7d8a8d76d84e1b1f66f8}}, and vern photoelectric cross sections {{cite:fec8f1f9767509b10614ca7d3bf497ff0b3fd61e}}, to account for the galactic absorption with the model tbabs.
{{figure:283f4149-27e3-4196-ac33-dc13cd6bd233}} | m | 82ea1980e433ea4b424a9ec5392f65a7 |
are generically produced. They come with completely fixed coefficients as a precise prediction of the model {{cite:7456c834203b0c03ced19ce07129e3231eb9b9b2}}, {{cite:e703d2285e0857b2aab45cc2c1834cebba0c894c}}, {{cite:562289e9ea109ccefb590e4e9f5e6a32777ddef2}}. The presence of such terms, where the {{formula:866eff9d-7728-400b-ab58-03cb4fe2918f}} indices {{formula:3e1e32ac-3cae-4292-944d-208a7991c315}} label the different fermionic families, has been proposed as a possible explanation for left-handed neutrino oscillations without a need to assume nonvanishing neutrino masses or sterile features for the right-handed states {{cite:211dc3b033e1b8b44066ed4eedf881943bf0ffa8}}, {{cite:9a0a03dfb3f047c580d454c228ddac0dfcdc1c27}}. Hence, u-SUSY could perhaps provide a mechanism for the missing neutrino puzzle, which could be an indication for an underlying supersymmetry in nature.
| d | 7bf6d94b444e16c9f5fef65dc2d5a69f |
The Fermilab Muon g-2 experiment {{cite:6c78deaec9a114869c6e255183911da1a210209c}}
reported recently their first measurement of the muon anomalous
magnetic moment, {{formula:8a05ebd7-5a4a-4053-85f6-0d550dd17c2d}} , confirming
the old BNL result {{cite:10b50323b7213e557e499dacc8b429efc3e48546}}.
The combined value of
{{formula:0e12af0a-ae0b-4fe7-977b-93d9a102ca66}}
disagrees with the SM prediction,
{{formula:1ff0ee47-66b2-47c4-a1e2-c20c7c3455f5}} {{cite:30eaa2095505f9d72bcde5732e918b4c351e7f91}},
by more than {{formula:4b29e438-1359-47ac-9536-61170565bcdd}} ,
{{formula:d682e137-d2ef-4f44-8692-651ddd53dd46}}
| d | bc6f17a105e8ee8ddd9b88881eab635b |
One can see this result in {{cite:096193f4de577fdffe415af5729a300240907c96}}.
{{formula:ab9757ad-eb6b-4d70-b483-a7080e738d1e}}
Next, we state the functional equation for the Dirichlet {{formula:d20f4504-2a1f-4cb1-a258-49863497cc61}} -function.
Lemma 4.5
Let {{formula:45524ee0-786d-4232-9b37-37e5cdb81cf9}} be a Dirichlet character of Modulo {{formula:109245e4-df01-43b1-9718-eb1866495535}} . Then the Dirichlet {{formula:7e1094d2-573c-4571-bf33-a111f9d19162}} -function {{formula:09d8d8ab-195c-4b20-9cce-1017bf5919c3}} analytically extends to the whole complex plane and satisfies the functional equation:
{{formula:66b61edf-17fb-4182-b557-a480078d0e06}}
| r | 3c512fbbc460bc3eaf85f9af4f0b7d3d |
Distantly-supervised Methods. (1) KB-Matching reports the distant supervision quality. (2) Distant BiLSTM-CRF, Distant DistilRoBERTa, and Distant RoBERTa fine-tune the corresponding models on distantly-labeled data as if they are ground truth with the standard supervised learning. (3) AutoNER {{cite:c93730997de1b3f6ad02e78b24788b1be3f7fdcb}} trains the model by assigning
ambiguous tokens with all possible labels and then maximizing the overall likelihood using a fuzzy CRF model. LRNT {{cite:6020f254618c8d8d311316d7ee771c7cf09cd63c}} applies partial-CRFs on high-quality data with non-entity sampling. Co-teaching+ {{cite:ce4a6ffcea4ec773429167010d88634670ece811}} is a classic de-nosing method in computer vision. NegSampling {{cite:1a6400cb3dc92185173bd25fb446a0878ad92313}} only handles incomplete annotations by negative sampling. BOND and SCDL both adopt self-training strategies that are straightforward competitors to ATSEN.
| m | cb11f9be629a5a63833b76440b77989e |
A pending goal in this research would be to obtain the effective electromagnetic action in our approximation and also as a non-perturbative result in the tilting parameter {{formula:d3a6a480-8f1a-4830-9f6e-c371ee508124}} . This would be especially important in the context of Weyl semimetals, since the value of {{formula:0ac06f1b-6973-4fe0-9613-41f2847475c6}} will not necessarily as small as it must be in the case of LIV in the SME. Furthermore, the case {{formula:fcff965b-5b02-476f-8cc8-bc6a57185d41}} is of special physical importance, as it corresponds to cones tilted parallel to the Fermi energy plane. This value is also relevant since it distinguishes the so-called type-I ({{formula:c868421e-35f5-4797-a80f-c98835fcf7f1}} ) from type-II Weyl semimetals ({{formula:29b502ea-569c-4ab1-88c6-44a2a66eb8c6}} ), and it is the point at which the density of states diverges, requiring an additional regularization {{cite:d4b6a44de6dc8b865cc79b79dc664b902ef0f609}}. Only a non-perturbative approach in {{formula:046bbc76-143b-4788-833d-f6bd05ed1ba1}} could probe strongly tilted WSMs. A more complete calculation of the effective action would also necessarily involve the incorporation of the separation of the nodes given by {{formula:24390644-b5ef-4509-a6f2-b45534d1d089}} in momentum space.
| d | 63db008e756cb5271277f6cc236daeb4 |
The proof of Theorem REF is quite long and technical, relying on the machinery of stochastic estimate sequences {{cite:0628faed751167604c5d2bfdbfc2b3b5f2a5ad31}}. Therefore, we have placed it in Appendix . The high level idea of the argument is as follows. Existing arguments (in the unbiased setting) based on estimate sequences {{formula:3cbd4176-6936-4f89-8372-f6ae5900eba3}} rely on lower bounding the error {{formula:c2a8b32c-629a-49d9-ae5f-ca306b6142ad}} ; see e.g. {{cite:184de57811cf13aa55b9e5e3a4cd9ff99c358b87}}, {{cite:0628faed751167604c5d2bfdbfc2b3b5f2a5ad31}}. The usual path is through a sequence of clever algebraic manipulations. During these manipulations, there is a term {{formula:cb8b6a11-bf91-458b-9c04-951494e41dc5}} that appears, which is lower-bounded by zero and ignored. In contrast, we show that this term balances the incurred bias of the stochastic gradients.
| m | b4bb8ab1795a8dcd249396953f76f18e |
The question arises (of interest from a theoretical point of view, though irrelevant for applications as in {{cite:5446486f3af5a1a843be0ed1110d4abe9bf2ab55}} or {{cite:d6cc13cfaa9b7fe31d306b605f07f1eceaec7936}}) what precisely happens at the thresholds. The first part of the following is concerned with this question. It turns out that here a monotonicity property of the quotient {{formula:d574917e-fe7e-42de-916e-7c6ed7da8adf}} as a function of {{formula:c22aba52-789d-4ddd-8b67-bdd74ca13dc7}} is useful, and this will be proved in Theorem REF . This monotonicity may be of independent interest. Here it allows us to give an elementary proof that the limit in (REF ) is equal to 1 also for {{formula:dd2d1679-2734-48b8-b8df-9a70783f5d05}} . Curiously, none of the proofs given in {{cite:2a1950cb47dd476ddd0e4c4c566635e74fbade05}} for {{formula:bc0234f4-6ff0-49a2-a87a-91c6aa08ccae}} or {{formula:ffe7b74d-8c49-48f3-a4f2-035640dd02fe}} extends to include the case {{formula:addbba62-1708-46c7-972c-d940aa6fd673}} , so a different argument is required. A proof using deeper deviation results from probability theory was given by Godland, Kabluchko and Thäle {{cite:d914224d23c3e5fd0b9fae309e41fea6224afb61}}.
| i | 8a05259a63e801a1c7fcee7c7c56c48f |
In this section, we demonstrate the performance advantage of the sample-based PMD-PD algorithm (Algorithm ) in the same tabular CMDP described in Section and in a more complex environment Acrobot-v1 {{cite:a16a8573a56cdfcc226c3cf161ae432e9285e3e2}}.
| r | b45c2204be0ee8675837943504a63262 |
The systematic study of stochastic optimization problems under changes to probability distributions extends at least back to {{cite:185bf8bf9d44d7c4c6078576bacb43b30cdf5e51}}, {{cite:0e53985e8e55b42af06ad4d1f6c5bd480909e1a7}}, {{cite:47e5ac2f90cef44373b3751200f59236577975e7}}, {{cite:3d60ffaed791d671014d6539387b30e8575f8d65}}; see {{cite:0fbc1a973b1bb4d5b003126e5f70df45c9438de5}} and {{cite:d19d54d90edead5662b14cdaab50308dff5fa814}} for more recent efforts in the context of convex problems and stochastic dominance, respectively. There is an extensive literature on how to address parameter uncertainty in a conservative manner; see {{cite:5076c8362d3bb81123deb23d39dd337126ef8e01}}, {{cite:df8b04b297d454358db58b04cced8ba64488b632}}, {{cite:c1d98bf9d5b162964c617e4a0bb4db78d13dc1cb}} for results from semi-infinite programming and {{cite:fb97c8fc0087a9b6874e18250a6324555c3acd1c}}, {{cite:40ed7459e3841512ab70a000882bf248f7c2c0bc}}, {{cite:9a6f13bfd431c402f72d7697ed93f2fd8d07c75c}} for robust convex optimization. Under distributional ambiguity, the focus tends to be on achieving a solution that is “good” across a set of candidate distributions, for example centered at a presently available empirical distribution {{cite:8f47192560714fb0c7668f337988d97c9de7864e}}, {{cite:283946a88fecbd4d5b01e967b6c899c9a6196c1f}}, {{cite:d0467645354a9922a7e2ba2103b8f8cb4b569fa9}}, {{cite:63edd4ea00405837239f52ac1de67c19f26c9922}}, {{cite:c2851eea11c9e7a21c74ffa2acb5f85d450db539}}, {{cite:e1179ecc7d0b927146cedaf6dc91a9f1de3d8f3b}}. Related risk-based approaches have similar effect, at least for monotone and positively homogeneous regular measures of risk {{cite:cac9fbb27bb4d8507142ccf47e7b5febeb6458b0}}, {{cite:544c0432f898b8c609df3189cf29fd21defaac6c}}, {{cite:0247e73e364856bb36b95a774dc233e59359d3f9}}. Adversarial statistical learning addresses ambiguity about training data through conservative perturbations of the support of the corresponding (empirical) distribution {{cite:bd0dc03094c9a579c47f13f28d5cce0ba928ab5b}}.
Robust regularization {{cite:2cd37ca9e085f94182ffadf970574f6d999ad419}} achieves conservativeness through perturbation of a decision vector, with applications to manufacturing {{cite:690f2cd203d7dff54907f1e237f9fc1e4bc770b4}} and statistical learning {{cite:83fdecbf495c4888d34747134929f18758f54962}}, {{cite:8d3ae2d48f8a1c335373bb1e2dbd3e908762dbdf}}, {{cite:86252665308e346c45c7134f70602adc49f3c767}} in the form of diametrical risk minimization. When approximations stem from empirical distributions, these approaches are well-motivated by the downward bias in sample average approximations {{cite:86252665308e346c45c7134f70602adc49f3c767}}.
We step away from conservative approaches, which correspond to being “pessimistic” about values of ambiguous parameters and distributions, and adopt an “optimistic” approach based on problem relaxation.
| i | 3a4b91f758faecee4e6cf638b1a7d2cb |
The output of the two approaches described above were compared using pixel accuracy, Intersection over Union (IoU), and mean IoU; three metrics that are commonly used in semantic segmentation analysis {{cite:8b9097beafd26ec34b7bb9fe5322aa8f5ed1c273}}.
| m | 9abe2313ecad232d8d39abfe26bf21ec |
Over the last several years, substantial interest has been
generated in the problem of solving underdetermined linear systems
subject to a sparsity constraint. The field, known as
compressed sensing or sparse recovery, has applications
to a wide variety of fields that includes data stream
algorithms {{cite:3d681be00daf97660e188b3e2acacef937b6192c}}, medical or geological
imaging {{cite:3ee6b7bbe0e376c6c5dd2a9813861bc78fe655d6}}, {{cite:fe529ac266df3d36a92523385fa08e45e1bf9f28}}, and genetics testing {{cite:01164b1fcb3880947bea93fb9598e15c8e2986b1}}. The approach uses the power of a sparsity constraint: a vector
{{formula:f1b903de-e1bb-4ef6-a687-d5d54d563330}} is {{formula:005ab9f4-9fb1-4e45-8a85-0e355539f517}} -sparse if at most {{formula:85a8f539-1a4f-450d-8c5a-3e7a09185bae}} coefficients are non-zero. A
standard formulation for the problem is that of stable sparse
recovery: we want a distribution {{formula:d1accb56-3434-40cd-b02a-e4ea478a3320}} of matrices {{formula:ee89d5bd-5361-4eda-b9da-014225577c33}} such that, for any {{formula:d5318d55-23fa-4082-a475-947838ae2ada}} and with probability
{{formula:068f33fa-7d6e-4887-9ef1-a4791ede167f}} over {{formula:525964c8-9878-42dd-bc22-dd9dd8671967}} , there is an algorithm to
recover {{formula:c245e552-8839-4291-9ef2-37e58a808c74}} from {{formula:1ef571b8-07a3-4785-aa6d-2c8c2406d31d}} with
{{formula:49c7ed1e-03a3-4747-bc33-fc138f0ed20c}}
| i | e1d84bcf2297dad05604188a7dd5be8c |
In general, an agent can experience a quadratic speed-up in its learning time if it can perform an arbitrary number of coherent Grover iterations {{cite:5a417ab9c233d19388202b44af99df12defb3b76}} even if the number of actual rewarded sequences is unknown {{cite:aa7531b1bcd29f851a9526cbf200107115a7a888}}.
| r | 40a4a0d51c9ee3a7200b82d0eea7fe9e |
In Table REF of the main paper, we compare to published works on the validation set of EPIC-KITCHENS-100.
Unfortunately, most works do not report on the leaderboard test set.
In Table REF , we provide results on the test set comparing our model to baselines from {{cite:81b3ca571824e14686b55e79e46f15d8a54185be}}, as well as Ego-Exo {{cite:4a336693606d89dd877bf39215ac5e0af5861a0d}} that distills knowledge from a much larger training set. MTCN outperforms all other methods, including the competitive method of {{cite:4a336693606d89dd877bf39215ac5e0af5861a0d}}, showcasing that multimodal temporal context from consecutive actions is more beneficial than pretraining large models (ResNet101) using egocentric signals from third-person datasets.
| r | 59fa7d93f794d82e284e1079540f4681 |
Label correction-based methods. In the early learning stage, the model predictions are accurate on a subset of the mislabeled examples. This suggests that label correction-based methods are potential for correcting the corrupted labels during the robust training process. For example, {{cite:ae923d9fb1844ec58b4c3adcc77de2c5146d77bd}} propose to adaptively correct noisy labels with new labels that are consistent with the probabilities estimated by the model. {{cite:1a25272c66b72048b0a3261af653bd294d756519}} learn a set of extra hyperparameters to correct noisy examples. Usually, label correction is associated with some iterative sample selection procedure or is with additional regularization terms. SELFIE {{cite:49f78365becb8ab0e1d6f92e8e3adb0aa53ad28d}} focuses on examples that have consistent model predictions, so as to minimize the falseness of label correction. {{cite:0c835c3b73cd6e06b1f8a440db71cc5730dd1afe}} fits a two-component mixture model to perform sample selection, and then corrects labels via a convex combination. DivideMix {{cite:4179be1210794385e5b5315c3d6678ba1abe6635}} uses two peer networks to perform sample selection via a two-component mixture model, and applies the semi-supervised learning technique MixMatch {{cite:8e8367c56dd0a668497363aef37b65e650b57b06}} to further correct the soft labels of the training data.
| m | 8923e7db1f62d54ce3578cba58562e1e |
See {{cite:81a6101a6ce8f00281c2f93c6e76c53efa63f8c8}}
| r | 781b6e458df9c69584f1a2e20458e431 |
Despite efforts to make the lectures self contained, some familiarity
with basic ideas of statistical data analysis is assumed.
Introductions to the subject can be found, for example, in the reviews
of the Particle Data Group {{cite:b7f42fa66abb764b9a102967d07c40d59dbc951a}} or in the texts {{cite:41f07f9cbe6bef89a8eb357660e0271cd784a0bd}}, {{cite:604a7723dcb5da7b7ff50671aa883e9ca8804136}}, {{cite:7dea24d4f6cfc869db93abb5be0b7498b1378c1c}}, {{cite:e0aea357b9c43b6bbfa37b0d9ed131f89209ada7}}, {{cite:e6527e3592e4d64dd2f97f57d4b3656850045906}}.
| i | bf49c51f4c6b12620adbe9e4923e67b7 |
Training time becomes an important matter for tasks that have high degrees of complexity, like the block manipulation tasks in OpenAI's Gym environment {{cite:8463f373591d530549388481ff93f53882670c78}}. The most popular method for learning in this kind of goal-based environment is Hindsight Experience Replay (HER) {{cite:976387daf8ad912e4b8e2995f08785ed24d5c85f}} trained in conjunction with Deep Deterministic Policy Gradients (DDPG) {{cite:8679896ab6f4ce0bc855e94c5bd13308be4e8027}}. HER is capable of successfully solving most tasks implemented in this environment, but the amount of simulated experience it requires for training is immense, exceeding {{formula:191bf940-eb6c-41e5-bbc8-62a88f0002a2}} time-steps in the most complex tasks, and even then it is not able to discover an optimal policy that is able to solve all object rotation goals. Furthermore, 19 CPU cores are required to generate simulated experience in parallel {{cite:fb3b7a83c54bf19ec743854d0e2d74e9445917bc}}.
| i | 0df0c29431787296dcd86503c54f4875 |
The recent discovery of billion solar mass SMBHs in luminous {{formula:79ab7dba-6b46-452b-bb57-f70e6760cab4}} quasars at the epoch of reionisation poses the most stringent constraints on the masses of the seed black holes and the early accreting mode {{cite:74dc8f755d8bf1353ce128165bcac1fdd28ff83b}}, {{cite:31c521bf3333a56200296d58a65d5eee15f27fea}}. Jets are closely related to accretion and AGN feedback of SMBHs, but no radio jets have been found so far in {{formula:7bb20447-15d6-4f23-a490-eb65ef98b74a}} quasars, and continued exploration is still needed.
High-redshift jets are difficult to grow to large sizes {{cite:0c153e03aa9a81d88eff23720a7d3b6de81978a0}}, so observing the jets of HRQs requires very high resolution, and VLBI is typically used to probe such radio jet structures.
Among the more than 200 quasars at {{formula:73e17e70-55ff-4319-bbe2-ab7dbfac3f60}} that have been discovered so far, only five others have been observed by VLBI:
J0309{{formula:0e2105ce-dc13-4523-990c-d1559028c664}} 2717 ({{formula:342a2fc0-a01f-40a2-96d3-2e8003642193}} , {{cite:1717821438dbf50850f93bad13d1c7fb5197f0ea}}),
J1427{{formula:9c555270-a8ec-4dda-849c-b7dc93cca803}} 3312 ({{formula:897920b5-78d7-4412-be92-85b80927501b}} , {{cite:bf0e6bf222e76680872458ab9aad4fb021fad9b7}}, {{cite:bba62b5f3e8ca2ebe1c134a43e71ef07692afbd7}}),
J1129{{formula:4bfa9fd2-5f72-4e3e-9506-4ea02d622a40}} 1846 ({{formula:f58c7cff-fb25-4382-8eb2-c6126e4ec76b}} , {{cite:0b3b186a7702955e1746522b20c51a6f8a6812e7}}),
J1429{{formula:231d0c8b-4c47-449a-a09d-32ce49def017}} 5447 ({{formula:b3aee42c-acf6-4be0-ae76-e4f29bb09215}} , {{cite:1971d563aaa558b298631a340d929c3635091113}}),
and J0100{{formula:9b957171-9758-46ac-907f-7b09719ef490}} 2802 ({{formula:d1af1634-bfd4-45ce-ae00-77fa03123d5a}} , {{cite:fc8c67b61a5738b8f52720f79292663848ffbbd2}}).
Among the VLBI detected HRQs, J0100{{formula:eb219c91-0440-454d-bad3-ad4e8f5f4d4a}} 2802 is the only radio-quiet source with {{formula:49db3d9f-7f3f-4e26-ba51-007d3008a37b}} . J0309{{formula:49d0984d-c663-40a5-abfe-ac0f324025e4}} 2717 (PSO J030947.49+271757.31) is the only blazar with {{formula:aeed5717-d3ed-47b7-acbb-8de256e7a57e}} mJy flux density at 1.5 GHz {{cite:1717821438dbf50850f93bad13d1c7fb5197f0ea}}. The flux density of the others is at mJy and sub-mJy levels. VIK J2318{{formula:f834d32f-949c-4e66-aefa-f99fe3e7a412}} 3113 presented in this paper and J2331+1129 (S. Frey et al., in prep.) add two {{formula:99bb7713-39d4-4124-9016-03447d9f0477}} jetted quasars which also have mJy flux densities in VLBI images. The general picture emerging from their VLBI imaging observations is that the radio-loud HRQs usually have compact but somewhat resolved structures, including one (J1427{{formula:f969330a-b1fc-4eb4-b1c4-74e66630d55e}} 3312) which is a double {{cite:bf0e6bf222e76680872458ab9aad4fb021fad9b7}}, {{cite:bba62b5f3e8ca2ebe1c134a43e71ef07692afbd7}}. Except J0309{{formula:1dab559f-b4b7-44de-830e-0a133cb72c66}} 2717, these {{formula:468f9430-9bab-4d20-ab95-2339eed2ae48}} quasars showed steep spectra in the (observed) gigahertz frequency range, and are identified as nascent radio quasars or a compact symmetric object, which could results from the GPS (Gigahertz-Peaked Spectrum) or MPS (Megahertz-Peaked Spectrum) source nature {{cite:cc775452314127ddb4b3639204fecbe63c6fba18}}.
| r | 5fbf38289f221ae1528f1a075a217cbe |
Numerical simulations reveal that cosmic structures develop universal properties, for which the radial density profiles of gravitationally-bound objects are an important example {{cite:7daa8f27df6db839f3318efadc77e5f916a2b00b}}, {{cite:1a1e482a486c239fb23073a1207d4d82aaf5cbf4}}, {{cite:a300e4310e4cdcd2dc4f5673afcb775b17a91882}}, {{cite:6eba7b5630ccf879d07d81eb7c6d0ab8512f506d}}, {{cite:f77b27055f3d391b8af7f98aa4f1d4c3baf902f1}}. Where does this universality originate from? This question can be addressed in the framework of an analytic theory of cosmic structure formation. We use previous results obtained with the kinetic field theory of cosmic structure formation (KFT) {{cite:4a000cfbaa7c14ffa4a8c7e868d63010bf6186ba}}, {{cite:541de0ac1c5a75b385464191cf19d548c374eb68}}, {{cite:33cf9648c171bd282743aa6e6173f5326b39badf}} to study the asymptotic behaviour of the power spectrum of cosmic density fluctuations in the limit of small scales, or larger wave numbers, {{formula:06d0d8d5-f785-4b30-8728-8fece8f891fc}} .
| i | b5a3dabe923185a90130b20977b144bd |
It is well known {{cite:75ea482bb6dfe5b77a57770c9696fecf190f9561}} that models using a balanced static batch can have higher performances respect to those that use an imbalanced dataset. It can be interesting to know if this aspect is valid for streaming methods, too. Our proposal is a meta strategy, as Swt, called RebalanceStream able to rebalance a stream and train a model with it. It is represented as pseudo-code in Alg. . The full code is available online on a GitHub repositoryhttps://github.com/alessiobernardo/RebalanceDataStream.
| m | 7afd780d189150fd61d2cee99f882a66 |
The consistency of the PLS methods was proved by Delaigle and Hall {{cite:396da1e7e35e33f927ab0adf130fbf123a2554fc}} where they found an equivalent space with explicit expressed basis functions
to the PLS basis space. For PQR and PCQR methods, it is difficult to find such equivalent space and therefore their consistency may not be easy to show. The difficulty
of the problem lies on the iterative nature of PQR and PCQR methods where basis is sequentially extracted. One way to overcome that is to find preselected number of basis
simultaneously {{cite:743c1db4ebc29e3fdb78880927e4ffb7006ee8d2}}. Another direction is to impose certain structure
on the selected basis, for example, sparsity and smoothness in PLS methods {{cite:75c7d149fda2101a9472ca3c3089b467f14dc540}}. This can be done for simultaneous basis selection as well {{cite:6f849d51d72a201853ecc334e44f8bc510fc7f84}}.
| d | 53f40dc2b788bc5363768d9480b4689a |
Many defense strategies have been proposed to counter adversarial attacks {{cite:3906b352c65da9f0270a0a816ecbf8380fab88cc}}, {{cite:99786d397b7b6db0c1d7a9883f92efccf4fe995b}}, {{cite:75aebbac18b4906d9bd9f350adf0460ea091120f}}.
However, it happened many times that a new defense strategy is broken down by a newer attacking strategy only a few months after its proposal.
Thus, the adversarial defense problem has not been solved even on the toy MNIST data set,
although defense is considered much harder for larger data sets.
| d | 9016e1d1d2bb106de55d3cc19eef1520 |
We show more qualitative results of the proposed method on the PROX {{cite:749a416f69fbfc03a1101080896ecda8d186d4c7}} and Matterport3D {{cite:c8c8f0325058e18a56f8bee82312df755f6a013a}} in Figure REF .
We show all three aspects of the synthesized scene-aware motions, including the human-scene interaction anchors, diverse planned paths, and completed motions.
These results demonstrate that our framework can synthesize diverse human motions in the specific scene contexts for the given target action sequence.
More qualitative results are included in the supplemental material.
| r | b3abe5a7ca89f6dde20938e6c865ea71 |
where we used the Jensen inequality. For the logistic model, the PAC-Bayesian bound Eq.(REF ) holds {{cite:ef4ab73d2981ef24e68fc8ff42e6e89ab62c1f37}}, we have
{{formula:9729716e-b575-4b43-9c98-c1d5ccaaa25c}}
| d | 65ed35ed2f83a47fc9e2aadbd027a439 |
The first set of diagrams, depicted in the top panel of Fig. REF , represents the correction to the quark-connected contribution in the electro-quenched approximation, where QED corrections for the sea-quarks are not taken into account. The right most diagram is the strong isospin correction while the other two diagrams are QED corrections. Those diagrams have been computed by the ETM {{cite:f8d6b88c151be83b2cd431f07f3b313c5c30ae1a}}, the RBC/UKQCD {{cite:e8f019e75657e80df24f3e575accb011a1cd8c0a}} and the BMW {{cite:2d469cc488a0c823d73fbbb211b1b6fe8626cb0d}} collaborations.
| r | b54857056cceb5feedf4c0532a0220d0 |
The FL {{formula:3a4a9eb1-1c72-49a1-8098-3f3860f364aa}} resistivity mechanism discussed here explains the behavior of resistivity seen near band edges in magic-angle TBG {{cite:f85c247dcdfceb05becd19495498984ba3b97ca5}}. In addition, while the “strange-metal” linear-{{formula:d857e2e4-66a8-4543-a07c-44c2499c8dc2}} resistivity in TBG conceivably arises from mechanisms that are not part of our FL-based picture, these mechanisms may originate from the strong el-el interactions enabled by compact Wannier orbitals.
Furthermore, in TBG with non-magic twist angles, the enhanced el-el scattering persists so long as Wannier orbitals remain compact.
In that regard, the key aspect of the spatial structure of Wannier orbitals that boosts umklapp scattering is a compact core, whereas the behavior in the tails – exponential vs. power-law – is of lesser importance.
As a result, strong umklapp scattering and {{formula:f951df57-8fb8-49bc-8e37-2ea883f21d35}} resistivity is expected for both the conventional narrow bands and the bands with topological obstructions for truly localized Wannier orbitals such as those discussed in Refs. {{cite:1281db11a59ad76dc07bd23bee05e3ce18c4efb3}}, {{cite:076b0cf99a1358e7654b2340263cdaab0e0bea8f}}.
These predictions can be directly tested in slightly non-magic TBG.
| i | 26a099d6624dc0a74aa82f36691cbefd |
x1Rn, y1Rm.
In problem (), {{formula:4879a99b-09be-48ef-965d-5c3e01f9ae4f}} are decision variables and {{formula:4591f724-48b5-4464-99e6-89db5ffae08c}} are fixed parameters. To handle problem (), we employ performance estimation method introduced in {{cite:7e69f0c13efba13f5307e4e4a5d142e5938476fa}}.
| m | e4c90b33aef971d4bcb4339a4c2d04ca |
Inspired by the recent success of the text-to-image generative approaches, such as DALL-E {{cite:a1153d7d36bf8398fae937ef84f1decbe458b25a}} and stable diffusion {{cite:8d15fcfe8b4ad25f68d59aa3112147bddd42167c}}, we propose DiffAlign - an approach to data augmentation where synthetic images are generated using the textual description of the novel classes and aligned with real images for improving few-shot performance. Specifically, we consider the stable diffusion approach {{cite:8d15fcfe8b4ad25f68d59aa3112147bddd42167c}} that generates realistic images given a text description. We consider class names as text descriptions to generate a large set of synthetic images corresponding to the novel classes. However, directly combining synthetic images with real images may not be effective due to the domain gap between these images. Thus, we propose a maximum mean discrepancy (MMD) loss {{cite:bfa451f6ec92c6776e94ee0361222539ad879246}} to align the features of the synthetic images to real images. Along with the aligned synthetic images, we make use of the pseudo-labeling to associate novel-class labels to the base class images {{cite:84e989f96a7fab929b03656039b29dc4a0719019}}. Our experiments indicate that DiffAlign outperforms state-of-the-art approaches on the benchmark datasets in both 5-shot and 1-shot settings. DiffAlign is particularly effective in the 1-shot setting where only one real image is available for each novel class. DiffAlign is also effective in the zero-shot setup where we rely on only the synthetic images.
| i | 49985ff7dabc1362ee6cf031d3fef4a3 |
More specifically, We analyze our per-class accuracy on CIFAR-10 {{cite:757bf4e3bd86c9abad6e14ae19fdf2d616aabe88}} with 40 labels, compared to FixMatch {{cite:59b6839e68565961cee1c67aa8f55144dbcd3590}} in Table REF . total accuracy of FixMatch {{cite:59b6839e68565961cee1c67aa8f55144dbcd3590}} is 87.11%, but it can be seen that there is a significant difference in accuracy between pairs of semantic labels, which can share the abstract feature space due to the similar appearance between the two classes. Fixmatch {{cite:59b6839e68565961cee1c67aa8f55144dbcd3590}} shows that the accuracy between pairs of semantic labels are largely biased on one side of label. For example, the cat label and the bird label have lower accuracy by 66.0% and 21.7% compared to their counterparts, the dog label and the
airplane label, respectively. On the contrary, Aggmatch shows that the accuracy between pairs of semantic labels is quite equivalent to the other. From this observation, we prove that samples, having ambiguous features and class probabilities, can be corrected by aggregating numerous discriminative within-class and across-class samples in the confident-aware queue. And it also proves that the confidence estimation can appropriately measure the confidence of the pseudo-labeling using multiple hypotheses with subsets of the queue.
| r | 02260a24e3da9288a16141106787e55e |
Some GRBs with hard spectra in the GeV energy band can be explained by the IC/SSC mechanism, while it is
hard to explain some GRBs with soft spectra by the IC/SSC mechanism. It is possible to solve the GRB spectral
diversity by the jitter radiation with the turbulent cascade scenario {{cite:998da0d23c36c9a5bffb15ff523c34c76a52243d}}. The IC/SSC mechanism
can produce the emission that has a time delay of less than 1 s to the emission produced by synchrotron
radiation, but it is hard to explain the time delay longer than 1 s. Because turbulent cascade requires a
time interval to reach a full development, we suggest that the jitter radiation with the turbulent cascade
can produce the emission with the time delay less than 1 s. It is also possible to explain the emission
with the time delay larger than 1 s by the jitter radiation
with the turbulent cascade, if we consider the emission is at a large fireball radius and GRB has a
relatively small bulk Lorentz factor. The details were presented in {{cite:998da0d23c36c9a5bffb15ff523c34c76a52243d}}. In the magnetic-dominated
case, particles are not effectively accelerated by relativistic shocks. Furthermore, as presented in
Section 1, it seems that the electron cooling by both synchrotron and IC/SSC mechanisms are not sufficient
to dissipate magnetic field energy in a short time-scale. {{cite:386bcbd032b62b73a97c8cf06e42ac89ffb3380f}}
suggested a continuous reacceleration for electrons to halt the fast cooling rate.
We note that a part of the magnetic energy released by the magnetic reconnection can be transferred to the electron kinetic energy.
It is important to note that the turbulence cascade in the small lengthscales can be induced by the magnetic reconnection {{cite:4e16adb736d7667721f511408051ab783d1923dd}}. The particle acceleration in the kinetic length-scale was further investigated in the magnetically dominated plasmas when the turbulence is well developed {{cite:1a1b680bb2961f75861be55b9293d180a6a91251}}. These works encourage us to further consider the radiation effects. In our opinion, magnetic field energy can
be dissipated by both magnetic turbulence and magnetic reconnection heating at small
length-scales. Particles can be accelerated by turbulence and magnetic reconnection. In the meanwhile, electrons have effective cooling by jitter radiation. The systematical analysis could be performed in detail in the future.
| d | 4f2b4f5262b205f86ef5f43e7c9c8454 |
{{cite:8f5dd21276a0aae124024f63c2539ac490b4bc54}} focused on the idea that these data show that the
hmsfrs are orbiting the Galaxy with a velocity that lags the
circular speed by {{formula:c78b3084-6855-464d-b893-dd573fd9090e}} . Our analysis of similar models
shows that the data are fit better by a slightly smaller offset of
{{formula:95c9b230-5d98-4309-8179-19a22830bd6d}} to the circular speed. We have also shown that an
equivalent fit to the data can be obtained by instead increasing to
{{formula:c10b998a-82e8-4fae-b180-875907067088}} the amount {{formula:fbb0c745-7cd7-47a4-9987-a247dff513bf}} by which the Sun is assumed to
circulate faster than the circular speed. In fact, the sources that
provide the strongest statistical support for {{formula:a6e287e2-ccb4-42a8-a482-bf074676aca7}} are
found near the Sun (Fig. REF ), so a change in {{formula:330bb824-2c62-45ac-b421-b54fb0f3bf3d}} has
a very similar effect to a change in {{formula:41be25a7-43c1-47fa-9b02-ad99680f50e9}} . More maser data at
different Galactic azimuths could break this degeneracy, as well as
reducing the correlation between the values of {{formula:2151da01-b2e6-427e-bf5c-2e8073004e98}} and {{formula:8e6c0f29-e169-4f44-8a4d-d388d1b1cd01}} seen in
Fig. REF .
| d | 7bb4523c2d418c966e98c0836c4d41be |
One of the first zero-shot methods is BM3D {{cite:d329045951f33e1c7873c21b90883f56072e6f22}}. BM3D collects overlapping patches, clusters them and then finds a lower dimensional representation of those clustered patches. BM3D assumes Gaussian noise and requires a user supplied estimate of the standard deviation, therefore it is not blind. A more recent approach to non-blind denoising is {{cite:6c23bb77e135fbd3a1f15248b80a43752b50e6ed}}, which uses a novel modelling strategy to better remove noise between similar patches.
{{figure:b030af1d-dd17-46a8-a3f2-6db65eb0453c}} | m | c0f5d8b6e18ac0b432658822411f9833 |
Following the strange nature of dark energy and considering the similarity of this component with inflaton field, an alternative to cosmological constant for explanation of the late time accelerated expansion is a dynamical scalar field with infinitesimal energy density, called quintessence, which its slow-rolling evolution at the late time can accelerate the universe {{cite:7280847ac61363485274e0f170bf9240777279cd}}. This scalar field avoids the extreme fine-tuning of the cosmological constant {{cite:cf3b082f23a462a8efc924015e042108f8d09ade}}. Although, {{formula:37231c02-052c-4e03-9486-965969f347c5}} CDM model with dark energy equation of state {{formula:aa79428a-d804-4abc-b97f-396faa0d0e02}} satisfies the cosmological data very well and we may not need any other explanation for dark energy, but if future cosmological experiments show a deviation from {{formula:31960116-b073-462d-9c70-ce917d49a044}} for dark energy equation of state, then quintessence may help us to explain this deviation {{cite:b6f5459f4d0e25dbf19f1ef289df1e4a98260268}}.
| i | 1fc7a4d7cf3b8c8f18f718f4ba845b14 |
Fig. REF shows examples of caption generated by SGN and TA {{cite:5cec516fbecd24c9dedc8a533e36d4943aaea257}} - the same pre-trained CNN features are used.
SGN is able to better identify the subject responsible for the action performed in the lengthy video scene. For example, SGN chose to predict “a band is performing” rather than “a man is singing” (Figure REF ), and it provides a more detailed description of the video (Figure REF ).
Overall, SGN seems to understand the context better than TA as shown in Figure REF and REF .
| r | c44501f85aa85ffa0bec0027127b1de1 |
Taking a relatively large perturbation of {{formula:d948f7e9-d7ee-47d2-ad8f-8ec736f8e677}} , we have {{formula:55222a87-6b60-4bd1-b7fa-145d974aed6f}} independent samples for an eigenvalue {{formula:a563c74c-05c9-4fe2-a53d-3c7a57e81df2}} , the order of magnitude of the largest mode {{cite:f293e8b53367e4ce150541f94e133185de043bee}}. Linear growth in the number of required samples, however, restricts us from measuring insensitive modes in a reasonable amount of time (the experiments analyzed here last 8 minutes and collect roughly 80 to 120 independent samples {{cite:a97e960a1374c71d1e3c26c66292bd86debc06fb}}). Though mapping the full local information geometry of {{formula:de789a2f-6228-4a01-83ae-f9130f2dc55c}} neurons would be difficult because the number of pairwise perturbations exceeds {{formula:16604dcf-4b4a-4e53-88ce-c5c29a17e147}} (accounting two distinct pairs of matcher and target neurons), a similar experiment with about {{formula:2cf161aa-3f51-42e0-b1f7-739039f2142f}} neurons seems reasonable when coupled with techniques for incomplete matrix estimation {{cite:ea0e4c9097268f582846d6d19055b6cdd83f71a6}}. The feasibility of such an experiment stems from our argument that perturbative experiments harness analyticity to vastly simplify the range of possible perturbations. We exploit this property to extract a basis for the local information geometry including multi-component perturbations. One goal of such experimental intervention is then not to be more precise but sufficiently varied to span the local basis, an idea that can be generalized to other experimental systems besides the C. elegans model we consider.
| d | 989a2720f61ae7649c8dfcacb5c055a2 |
Having such a Gaussian random field at hand, one usually parametrizes
it by means of the Karhunen-Loève (KL) expansion or a polynomial chaos (PC) expansion
{{cite:4c2ab74879e0272da7d2e155fcd98d9b05a8a975}}, which fits to the {{formula:52ee17c9-c66e-425c-b439-caa973dd9fe2}} setting, i.e.,
{{formula:df785e28-8732-4ad8-bb3c-e867ff46dfb6}}
| i | a86e3cac283b85ba42b79bbf03348a59 |
In order to improve the interpretability of the discovered physical laws, learning explicit dynamics (e.g., closed-form governing equations or their parameters) has recently become more popular in physical scene understanding. Several hybrid methods take a data-driven approach to estimate real mechanical process from video sequences {{cite:a42edfb015ee157f60bb6e72699ccbb7b29f9b5d}}, {{cite:b9267fc46072f6d68213830e6a9ebbde3f0a750d}}, or model Newtonian physics via latent variable to predict motion trajectories in images {{cite:f80723ab9187bd697db700e13f244f5d29dc829c}}. Since the discovery of explicit physical laws requires extracting the motion of moving object, two-step staggered discovery has become the most common strategy, where the physical law is distilled after the moving trajectory being extracted {{cite:3125130a4284503712dd3263f8f737f5da414f78}}, {{cite:f5cc9a7f3e286586b8b7e143ae7b17e0c856458a}}, {{cite:0f4e3baa602a15135fc7e8e0b78fa4bcfe1d0afe}}. Advances in unsupervised object localization techniques like based on spatial transformers (ST) {{cite:fe6d0d95cf810005619f842e10df2e18e8344a2b}}, {{cite:e60759bfbcc32d60df141b1c9797eeac75c06657}}, {{cite:0f4e3baa602a15135fc7e8e0b78fa4bcfe1d0afe}} and Position-Velocity Encoders (PVEs) {{cite:41d88f95f7b7e41103a7e11c15f03028a8012bfa}} have enabled the explicit physical law discovery in an unsupervised scheme {{cite:564a0168497dfa0a19897a8034f7ff6c8aa41872}}, {{cite:7c3bbb40c5a4ea62fa60efc322ff7529a3fe5b26}}. However, these approaches require strong prior knowledge on the structure of the physical law or governing equation (e.g., the equation form is given while the coefficients need to be discovered). Furthermore, for those methods, physics is modeled in pixel coordinates which restricts the discovery for complex dynamical systems (e.g., ODEs) where the physical states need to be described in another physical coordinate system.
| i | 338a273ea81f58995ea6093b80c3d6a1 |
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