Akio Fujiwara

Takara Nomura, Koichi Yamagata, Akio Fujiwara

We study the CHSH inequality from an informational, timing-sensitive viewpoint using game-theoretic probability, which avoids assuming an underlying probability space. The locality loophole and the measurement-dependence (``freedom-of-choice'') loophole are reformulated as structural constraints in a sequential hidden-variable game between Scientists and Nature. We construct a loopholes-closed game with capital processes that test (i) convergence of empirical conditional frequencies to the CHSH correlations and (ii) the absence of systematic correlations between measurement settings and Nature's hidden-variable assignments, and prove that Nature cannot satisfy both simultaneously: at least one capital process must diverge. This yields an operational winning strategy for Scientists and a game-theoretic probabilistic interpretation of experimentally observed CHSH violations.

Yuzuru Sato, Daiji Tsutsui, Akio Fujiwara

In order to elucidate the plateau phenomena caused by vanishing gradient, we herein analyse stability of stochastic gradient descent near degenerated subspaces in a multi-layer perceptron. In stochastic gradient descent for Fukumizu-Amari model, which is the minimal multi-layer perceptron showing non-trivial plateau phenomena, we show that (1) attracting regions exist in multiply degenerated subspaces, (2) a strong plateau phenomenon emerges as a noise-induced synchronisation, which is not observed in deterministic gradient descent, (3) an optimal fluctuation exists to minimise the escape time from the degenerated subspace. The noise-induced degeneration observed herein is expected to be found in a broad class of machine learning via neural networks.