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diff --git a/marginal.tex b/marginal.tex index ccfe579..5c3e033 100644 --- a/marginal.tex +++ b/marginal.tex @@ -17,14 +17,13 @@ \affiliation{Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, Rome, Italy 00184} \begin{abstract} - Marginal optima are minima or maxima of a function with many asymptotically + Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors are a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points on their marginality, and - apply it in three isotropic settings with different typical forms for the - Hessian at optima: in the spherical spin-glasses, where the Hessian is GOE; + apply it in three isotropic settings with qualitatively different structure: in the spherical spin-glasses, where the Hessian is GOE; in a multispherical spin glasses, which are Gaussian but non-GOE; and in a model of random nonlinear sum of squares, which is non-Gaussian. In these problems we are able to fully characterize the distribution of marginal @@ -137,6 +136,7 @@ $A$ (or indeed more complicated averages) in order to condition that the minimum eigenvalue is zero. \subsection{Simple example: shifted GOE} +\label{sec:shifted.GOE} We demonstrate the efficacy of the technique by rederiving a well-known result: the large-deviation function for pulling an eigenvalue from the bulk of the @@ -330,7 +330,7 @@ configuration. We can therefore choose $\mu=\mu_\textrm m$ such that \begin{equation} 0=\frac\partial{\partial\lambda^*}G_{\lambda^*}(\mu_\mathrm m)\bigg|_{\lambda^*=0} \end{equation} -In the previous problem, this corresponds precisely to $\mu_\mathrm m=2\sigma$, +In the example problem of section \ref{sec:shifted.GOE}, this corresponds precisely to $\mu_\mathrm m=2\sigma$, the correct marginal shift. Note that when we treat the Dirac $\delta$ function using its Fourier representation with auxiliary parameter $\hat\lambda$, as in the previous subsection, this condition corresponds with choosing $\mu$ such @@ -461,11 +461,14 @@ more nontrivial settings. The procedure to treat the complexity of the spherical models has been made in detail elsewhere \cite{Kent-Dobias_2023_How}. Here we will merely sketch the steps that are standard. We start by translating elements of the Kac--Rice measure into terms more familiar to physicists. This means writing \begin{align} + \label{eq:delta.grad} \delta\big(\nabla H(\mathbf x_a,\pmb\omega_a)\big) &=\int\frac{d\hat{\mathbf x}_a}{(2\pi)^N}e^{i\hat{\mathbf x}_a^T\nabla H(\mathbf x_a,\pmb\omega_a)} \\ + \label{eq:delta.energy} \delta\big(NE-H(\mathbf x_a)\big) &=\int\frac{d\hat\beta_a}{2\pi}e^{\hat\beta_a(NE-H(\mathbf x_a))} \\ \delta\big(N\lambda^*-\mathbf s^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega)\mathbf s\big) + \label{eq:delta.eigen} &=\int\frac{d\hat\lambda_a}{2\pi}e^{\hat\lambda_a(N\lambda^*-\mathbf s^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega)\mathbf s)} \end{align} for the Dirac $\delta$ functions. At this point we will also discuss an @@ -657,6 +660,7 @@ $\Omega=S^{N-1}\times S^{N-1}$ \end{equation} \subsection{Random nonlinear least squares} +\label{sec:least.squares} In this subsection we consider perhaps the simplest example of a non-Gaussian landscape: the problem of random nonlinear least squares optimization. Though, @@ -700,45 +704,17 @@ Applying the Lagrange multiplier method detailed above to enforce the spherical \\ \operatorname{Hess}H(\mathbf x,\omega)=\partial V_k(\mathbf x)\partial V_k(\mathbf x)+V_k(\mathbf x)\partial\partial V_k(\mathbf x)+\omega I \end{align} -\begin{widetext} -\begin{equation} - \begin{aligned} - &\mathcal S - =-\frac1n\frac\alpha2\left\{\log\det\left[ - \hat\beta f(C)+\Big( - f'(C)\odot D+(G\odot G-R\odot R)\odot f''(C) - \Big)f(C) - +(I+R\odot f'(C))^2 - \right]-\log\det(I+G\odot f'(C))^2\right\} \\ - &+\frac1n\frac12\Big(\log\det(CD+R^2)-\log\det G^2\Big) - +\hat\beta E+(g_d-r_d)\mu - \end{aligned} -\end{equation} -where $\odot$ gives the Hadamard or componentwise product between the matrices, while other products and powers are matrix products and powers. - -\begin{equation} - \begin{aligned} - &\hat\beta E+\mu(g_d-r_d)+\frac12\log\frac{d_d+r_d^2}{g_d^2} \\ - &-\frac\alpha2\log\left[ - 1+\hat\beta\big(f(1)-f(0)\big) - \Big(d_d\big(f(1)-f(0)\big)+r_d\big(2+r_df'(1)\big)\Big)f'(1) - +(g_d^2-r_d^2)\big(f(1)-f(0)\big)f''(1) - \right] \\ - &-\alpha f(0)\left( - \big(f(1)-f(0)\big)+\frac{1+r_d\big(2+r_df'(1)\big)f'(1)}{\hat\beta+d_df'(1)+(g_d^2-r_d^2)f''(1)} - \right)^{-1} - \end{aligned} -\end{equation} - -In the case where $\mu$ is not specified, in which the model is supersymmetric, $D=\hat\beta R$ and the effective action becomes particularly simple: -\begin{equation} - \hat\beta e - -\frac12\frac{\alpha f(0)}{1+\hat\beta\big(f(1)-f(0)\big)+r_df'(1)} - -\frac\alpha2\log\left(1+\frac{\hat\beta\big(f(1)-f(0)\big)}{1+r_df'(1)}\right) - +\frac12\log\frac{\hat\beta+r_d}{r_d} -\end{equation} - -\cite{DeWitt_1992_Supermanifolds} +As in the spherical and multispherical models, fixing the trace of the Hessian +at largest order in $N$ is equivalent to constraining the value of the Lagrange +multiplier $\omega=\mu$, since the trace of the random parts of the Hessian +matrix contribute typical values at a lower order in $N$. + +The derivation of the marginal complexity for this model is complicated, but +can be made schematically like that of the derivation of the equilibrium free +energy by use of superspace coordinates \cite{DeWitt_1992_Supermanifolds}. +The use of superspace coordinates in the geometry and dynamics of disordered +systems is well-established. Here, we introduce a novel extension of the +traditional approach to incorporate the marginality condition. Consider supervectors in the $\mathbb R^{N|4}$ superspace of the form \begin{equation} \pmb\phi_{a\alpha}(1,2) @@ -747,32 +723,53 @@ Consider supervectors in the $\mathbb R^{N|4}$ superspace of the form +i\hat{\mathbf x}_a\bar\theta_1\theta_1 +\mathbf s_{a\alpha}(\bar\theta_1\theta_2+\bar\theta_2\theta_1) \end{equation} -The Kac--Rice measure with the eigenvalue-fixing term included is +The traditional complexity problem, outlined in the appendix +\ref{sec:dominant.complexity}, involves a supervector without the last term. +\begin{widetext} + The replicated number of stationary points conditioned on energy $E$, trace $\mu$, and minimum eigenvalue $\lambda^*$ is then given by \begin{equation} \begin{aligned} \mathcal N(E,\mu,\lambda^*)^n - &=\int\prod_{a=1}^n\prod_{\alpha=1}^{m_a}d\pmb\phi_{a\alpha} + &=\int\prod_{a=1}^n\lim_{m_a\to0}\prod_{\alpha=1}^{m_a}d\pmb\phi_{a\alpha} \exp\left\{ \delta_{\alpha1}N(\hat\beta_aE+\hat\lambda_a\lambda^*) +\int d1\,d2\,B_{a\alpha}(1,2)\left[H(\pmb\phi_{a\alpha})+\frac12\mu(\|\pmb\phi_{a\alpha}\|^2-N)\right] \right\} \end{aligned} \end{equation} +where we use the compact notation $d1=d\theta_1\,d\bar\theta_1$ for the +measures associated with the Grassmann directions. Here we have also defined \begin{equation} B_{a\alpha}(1,2)=\delta_{\alpha1}\bar\theta_2\theta_2 (1-\hat\beta_a\bar\theta_1\theta_1) -\delta_{\alpha1}\hat\lambda_a-\beta \end{equation} +which encodes various aspects of the complexity problem, and the measure \begin{align} d\pmb\phi_{a\alpha} =d\mathbf x_a\,\delta(\|\mathbf x_a\|^2-N)\,\frac{d\hat{\mathbf x}_a}{(2\pi)^N}\,d\pmb\eta_a\,d\bar{\pmb\eta}_a\, d\mathbf s_{a\alpha}\,\delta(\|\mathbf s_{a\alpha}\|^2-N)\, \delta(\mathbf x_a^T\mathbf s_{a\alpha}) \end{align} - -\begin{equation} - i\int d1\,d2\,\hat v_{a\alpha}^k(1,2)(V^k(\pmb\phi_{a\alpha}(1,2))-v_{a\alpha}^k(1,2)) -\end{equation} +encoding the measures of all the superfield's constituent variables. Expanding +functions of the superfield in the coordinates $\theta$ and performing the +integrals, this expression is equivalent to that of the replicated Kac--Rice +integrand \eqref{eq:min.complexity.expanded} with the substitutions of the +Dirac $\delta$ functions of \eqref{eq:delta.grad}, \eqref{eq:delta.energy}, and +\eqref{eq:delta.eigen}. + +The first step to evaluate this expression is to linearize the dependence on the random functions $V$. This is accomplished by inserting into the integral a Dirac $\delta$ function fixing the value of the energy for each replica, or +\begin{equation} + \delta\big( + V^k(\pmb\phi_{a\alpha}(1,2))-v_{a\alpha}^k(1,2) + \big) + = + \int\prod_{a\alpha k}d\hat v_{a\alpha}^k\exp\left[ + i\int d1\,d2\,\hat v_{a\alpha}^k(1,2) + \big(V^k(\pmb\phi_{a\alpha}(1,2))-v_{a\alpha}^k(1,2)\big) + \right] +\end{equation} +where we have introduced auxiliary fields $\hat v$. \begin{equation} -\sum_{ab}\sum_{\alpha\gamma}\sum_k\frac12\int d1\,d2\,d3\,d4\, \hat v_{a\alpha}^kf\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)\hat v_{b\gamma}^k @@ -841,6 +838,58 @@ We further take a planted replica symmetric structure for the matrix $Q$, identical to that in \eqref{eq:Q.structure}. \end{widetext} +\appendix + +\section{Complexity of dominant optima in the least-squares problem} +\label{sec:dominant.complexity} + +Here we share an outline of the derivation of formulas for the complexity of +dominant optima in the random nonlinear least squares problem of section +\ref{sec:least.squares}. While in this paper we only treat problems with a +replica symmetric structure, formulas for the effective action are generic to +any structure and provide a starting point for analyzing the challenging +full-RSB setting. + +\begin{widetext} +\begin{equation} + \begin{aligned} + &\mathcal S + =-\frac1n\frac\alpha2\left\{\log\det\left[ + \hat\beta f(C)+\Big( + f'(C)\odot D+(G\odot G-R\odot R)\odot f''(C) + \Big)f(C) + +(I+R\odot f'(C))^2 + \right]-\log\det(I+G\odot f'(C))^2\right\} \\ + &+\frac1n\frac12\Big(\log\det(CD+R^2)-\log\det G^2\Big) + +\hat\beta E+(g_d-r_d)\mu + \end{aligned} +\end{equation} +where $\odot$ gives the Hadamard or componentwise product between the matrices, while other products and powers are matrix products and powers. + +\begin{equation} + \begin{aligned} + &\hat\beta E+\mu(g_d-r_d)+\frac12\log\frac{d_d+r_d^2}{g_d^2} \\ + &-\frac\alpha2\log\left[ + 1+\hat\beta\big(f(1)-f(0)\big) + \Big(d_d\big(f(1)-f(0)\big)+r_d\big(2+r_df'(1)\big)\Big)f'(1) + +(g_d^2-r_d^2)\big(f(1)-f(0)\big)f''(1) + \right] \\ + &-\alpha f(0)\left( + \big(f(1)-f(0)\big)+\frac{1+r_d\big(2+r_df'(1)\big)f'(1)}{\hat\beta+d_df'(1)+(g_d^2-r_d^2)f''(1)} + \right)^{-1} + \end{aligned} +\end{equation} + +In the case where $\mu$ is not specified, in which the model is supersymmetric, $D=\hat\beta R$ and the effective action becomes particularly simple: +\begin{equation} + \hat\beta e + -\frac12\frac{\alpha f(0)}{1+\hat\beta\big(f(1)-f(0)\big)+r_df'(1)} + -\frac\alpha2\log\left(1+\frac{\hat\beta\big(f(1)-f(0)\big)}{1+r_df'(1)}\right) + +\frac12\log\frac{\hat\beta+r_d}{r_d} +\end{equation} + +\end{widetext} + \bibliography{marginal} \end{document} |