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We present a framework for compactly summarizing many recent results in efficient and/or biologically plausible online training of recurrent neural networks (RNN). The framework organizes algorithms according to several criteria: (a) past vs. future facing, (b) tensor structure, (c) stochastic vs. deterministic, and (d) closed form vs. numerical. These axes reveal latent conceptual connections among several recent advances in online learning. Furthermore, we provide novel mathematical intuitions for their degree of success. Testing various algorithms on two synthetic tasks shows that performances cluster according to our criteria. Although a similar clustering is also observed for gradient alignment, alignment with exactmore »methods does not alone explain ultimate performance, especially for stochastic algorithms. This suggests the need for better comparison metrics.« less

A bstract Charged lepton flavor violation is forbidden in the Standard Model but possible in several new physics scenarios. In many of these models, the radiative decays τ ± → ℓ ± γ ( ℓ = e, μ ) are predicted to have a sizeable probability, making them particularly interesting channels to search at various experiments. An updated search via τ ± → ℓ ± γ using full data of the Belle experiment, corresponding to an integrated luminosity of 988 fb − 1 , is reported for charged lepton flavor violation. No significant excess over background predictions from the Standardmore »Model is observed, and the upper limits on the branching fractions, $$ \mathcal{B} $$ B ( τ ± → μ ± γ ) ≤ 4 . 2 × 10 − 8 and $$ \mathcal{B} $$ B ( τ ± → e ± γ ) ≤ 5 . 6 × 10 − 8 , are set at 90% confidence level.« less

A bstract We measure the branching fractions and CP asymmetries for the singly Cabibbo-suppressed decays D 0 → π + π − η , D 0 → K + K − η , and D 0 → ϕη , using 980 fb − 1 of data from the Belle experiment at the KEKB e + e − collider. We obtain $$ {\displaystyle \begin{array}{c}\mathcal{B}\left({D}^0\to {\pi}^{+}{\pi}^{-}\eta \right)=\left[1.22\pm 0.02\left(\mathrm{stat}\right)\pm 0.02\left(\mathrm{syst}\right)\pm 0.03\left({\mathcal{B}}_{\mathrm{ref}}\right)\right]\times {10}^{-3},\\ {}\mathcal{B}\left({D}^0\to {K}^{+}{K}^{-}\eta \right)=\left[{1.80}_{-0.06}^{+0.07}\left(\mathrm{stat}\right)\pm 0.04\left(\mathrm{syst}\right)\pm 0.05\left({\mathcal{B}}_{\mathrm{ref}}\right)\right]\times {10}^{-4},\\ {}\mathcal{B}\left({D}^0\to \phi \eta \right)=\left[1.84\pm 0.09\left(\mathrm{stat}\right)\pm 0.06\left(\mathrm{syst}\right)\pm 0.05\left({\mathcal{B}}_{\mathrm{ref}}\right)\right]\times {10}^{-4},\end{array}} $$ B D 0 → π + π − η = 1.22 ± 0.02 stat ± 0.02more »syst ± 0.03 B ref × 10 − 3 , B D 0 → K + K − η = 1.80 − 0.06 + 0.07 stat ± 0.04 syst ± 0.05 B ref × 10 − 4 , B D 0 → ϕη = 1.84 ± 0.09 stat ± 0.06 syst ± 0.05 B ref × 10 − 4 , where the third uncertainty ( $$ \mathcal{B} $$ B ref ) is from the uncertainty in the branching fraction of the reference mode D 0 → K − π + η . The color-suppressed decay D 0 → ϕη is observed for the first time, with very high significance. The results for the CP asymmetries are $$ {\displaystyle \begin{array}{c}{A}_{CP}\left({D}^0\ {\pi}^{+}{\pi}^{-}\eta \right)=\left[0.9\pm 1.2\left(\mathrm{stat}\right)\pm 0.5\left(\mathrm{syst}\right)\right]\%,\\ {}{A}_{CP}\left({D}^0\to {K}^{+}{K}^{-}\eta \right)=\left[-1.4\pm 3.3\left(\mathrm{stat}\right)\pm 1.1\left(\mathrm{syst}\right)\right]\%,\\ {} ACP\ \left({D}^0\to \phi \eta \right)=\left[-1.9\pm 4.4\left(\mathrm{stat}\right)\pm 0.6\left(\mathrm{syst}\right)\right]\%.\end{array}} $$ A CP D 0 π + π − η = 0.9 ± 1.2 stat ± 0.5 syst % , A CP D 0 → K + K − η = − 1.4 ± 3.3 stat ± 1.1 syst % , ACP D 0 → ϕη = − 1.9 ± 4.4 stat ± 0.6 syst % . The results for D 0 → π + π − η are a significant improvement over previous results. The branching fraction and A CP results for D 0 → K + K − η , and the ACP result for D 0 → ϕη , are the first such measurements. No evidence for CP violation is found in any of these decays.« less