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Fictitious play (FP) is one of the earliest and most fundamental learning procedures in games, where each agent independently chooses a best response to the empirical distribution of opponents' past actions by maximizing the expected payoff with respect to it. However, learning agents may not always be expected-utility maximizers: they may use nonlinear evaluation functionals to account for the uncertainty induced by the opponents, e.g., via risk measures. In this work, we study nonlinear fictitious play (NL-FP), in which each pure action is evaluated by a general, possibly nonlinear functional of the payoff distribution induced by the opponent's empirical belief. To analyze NL-FP, we first identify a single-crossing property (SCP) of evaluation functionals that preserves the classical geometry of FP in $2\times2$ games. Furthermore, we conduct a novel geometric analysis of the convergence behavior of NL-FP in $2\times2$ games, even when the SCP fails. Beyond $2\times2$ games, we show that the convergence of NL-FP in $2\times n$ ($n\ge3$) games may fail even for continuous-time NL-FP, when the two-action player uses conditional value-at-risk (CVaR) in a $2\times3$ game. In addition, we identify a finite-crossing condition that ensures convergence of the selected continuous-time dynamics and the discrete-time iterates in general $2\times n$ games, which also provides new insights into the global landscape of continuous-time classical FP in such games.

Stochastic optimization theory often assumes that gradient noise has uniformly bounded variance. However, this assumption can be inadequate even in simple settings; e.g., it does not hold for linear least squares regression. The Blum-Gladyshev ($\mathsf{BG}$-0) noise model instead allows the stochastic gradient variance to grow quadratically with the distance from the initialization and has recently been identified as the weakest known viable variance assumption. We study nonconvex stochastic optimization under $\mathsf{BG}$-0 noise and prove that normalized momentum converges with one stochastic gradient per iteration, reaching an $\varepsilon$-stationary point with the optimal $O(\varepsilon^{-6})$ complexity. We then show that, under mean-square smoothness, a variance-reduced normalized STORM method achieves the minimax optimal complexity of $O(\varepsilon^{-4})$. We further extend our analysis to the symmetric generalized smoothness framework, which captures objectives whose local curvature can scale with the gradient norm. To our knowledge, these are the first convergence guarantees for normalized methods in nonconvex stochastic optimization under $\mathsf{BG}$-0 noise without bounded domains.