Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference

Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work, we carry out an analytic study of the performance of the algorithm most commonly considered in physics, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked mixed matrix-tensor model. The typical behavior of this algorithm is described by a system of integrodifferential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is suboptimal with respect to the one given by AMP. This phenomenon is due to the residual glassiness present in that region of parameters. We also present a simple heuristic expression of the transition line, which appears to be in agreement with the numerical results.

Popular Summary

Many tasks of current interest in science and technology are solved using machine-learning methods. While machine-learning methods work impressively well in many cases of practical interest, so far, theoretical understanding of the properties of the algorithms is poor. Our work is driven by the hope that better theoretical understanding will lead to an improvement of existing methods. We introduce a model for which the performance of one of the commonly considered algorithms, the Langevin algorithm, can be fully analyzed, and we locate the region of parameters for which the algorithm reaches optimal performance.

Using methods designed to study the dynamics of glassy materials, we write and analyze a closed set of equations describing the quality of the result found by the Langevin algorithm. We deduce a very simple formula for a threshold of signal-to-noise ratio below which the Langevin algorithm fails. Comparing it to the performance of an alternative algorithm known as approximate message passing, we show that, remarkably, the algorithmic threshold of the Langevin algorithm is largely suboptimal.

We hope that the insight gained in our work can be extended to a better understanding of similar “gradient-based” algorithms in models that are closer to currently used neural networks.

Figure 1

Phase diagram of the spiked $2+3$-spin model (the matrix and order-3 tensor are observed). In the easy (green) region, the AMP achieves an optimal error smaller than the random pick from the prior. In the impossible region (red), the optimal error is as bad as the random pick from the prior, and the AMP achieves it as well. In the hard region (orange), the optimal error is low, but the AMP does not find an estimator better than the random pick from the prior. In the case of the Langevin algorithm, the performance is strictly worse than that for the AMP in the sense that the hard region increases up to the line $1/{\mathrm{\Delta }}_2^{*}=\max (1,\sqrt{{\mathrm{\Delta }}_3/2})$, depicted by green dots. The green circles are obtained by numerical extrapolation of the Langevin state evolution equations.

Figure 2

Evolution of the correlation with the signal $\overline{C}(t)$ starting from $\overline{C}(t=0)={10}^{-4}$ in the Langevin algorithm at fixed noise on the matrix (${\mathrm{\Delta }}_2=0.7$) and different noises on the tensor (${\mathrm{\Delta }}_p$). As we decrease ${\mathrm{\Delta }}_p$, the time required to jump to the solution appears to diverge. Inset: The behavior of $\overline{C}(t)$ as a function of the iteration time for the AMP algorithm for the same values of ${\mathrm{\Delta }}_p$ and the same initialization.

Figure 3

Extrapolation of the Langevin relaxation time. The inset presents the relaxation time for fixed ${\mathrm{\Delta }}_p=1$. The main panel presents a fit using a power law consistent with a divergence at ${\mathrm{\Delta }}_2^{*}\approx 0.72$. The circles are obtained with a numerical solution of LSE that uses the dynamical grid, while crosses are obtained using a fixed grid; the initial condition is $\overline{C}(t=0)={10}^{-40}$ (details are given in Appendix pp4-s5).

Figure 4

Correlation with the signal of the AMP and Langevin at the $k$th iteration (at time $t$) for fixed ${\mathrm{\Delta }}_2=0.7$, where both the evolutions start with initial overlap ${10}^{-4}$.

Figure 5

Dynamical evolution of the energy starting from a configuration with overlap ${10}^{-40}$ at ${\mathrm{\Delta }}_p=1.0$ and for various ${\mathrm{\Delta }}_2$. The system first tends to the threshold energy [Eq. (12), horizontal lines] and then, for sufficiently small ${\mathrm{\Delta }}_2$, finds the direction toward the signal.

Figure 6

The factor graph representation of the posterior measure of the matrix-tensor factorization model. The variable nodes represented with white circles are the components of the signal, while black squares are factor nodes that denote interactions between the variable nodes that appear in the interaction terms of the Boltzmann distribution in Eqs. (a6) and (a7). There are three types of factor nodes: $P_X$ is the prior that depends on a single variable, $P_Y$ is the probability of observing a matrix element $Y_{ij}$ given the values of the variables $x_i$ and $x_j$, and, finally, $P_T$ is the probability of observing a tensor element $T_{i_1,\dots ,i_p}$. The posterior, apart from the normalization factor, is simply given by the product of all the factor nodes.

Figure 7

Left panel: Phase diagram of the spiked matrix-tensor model for $p=3$. The phase diagram identifies four regions: easy (green), impossible (red), and hard (orange). The lines correspond to different phase transitions, namely, the stability threshold (dashed black), the information-theoretic threshold (solid red), the algorithmic threshold (solid cyan), and the dynamical threshold (dotted orange). The vertical cuts represent the section along which the magnetization is plotted in Fig. 9. Right panel: Phase diagram of the spiked matrix-tensor model for $p=4$. The main difference with respect to the case $p=3$ is that the algorithmic spinodal (solid cyan) is strictly above the stability threshold (dashed black). The hybrid-hard phase appears between these two lines (combined green and orange color). The vertical cuts represent the section along which the magnetization is plotted in Fig. 10.

Figure 8

Left panel: Phase diagram of the spiked matrix-tensor model for $p=5$. Right panel: Phase diagram of the spiked matrix-tensor model for $p=10$. In both cases, we observe qualitatively the same scenario found in the right panel of Fig. 7.

Figure 9

Fixed points of Eq. (b37) as a function of ${\mathrm{\Delta }}_2$ for $p=3$ and several fixed values of ${\mathrm{\Delta }}_p$. The values of ${\mathrm{\Delta }}_p$ correspond to the vertical cuts in the left panel of Fig. 7. Solid lines are stable fixed points; dashed lines are unstable fixed points. The blue line represents informative fixed points with positive overlap with the signal, while the orange line represents uninformative fixed points with no overlap with the signal. Starting from high ${\mathrm{\Delta }}_2$, an informative fixed point appears at the dynamical threshold (vertical dashed line) but is energetically disfavored until it reaches the information-theoretic threshold (vertical dotted line); finally, it becomes the only stable solution crossing the algorithmic threshold (vertical dotted-dashed line). When the transition is continuous, the three vertical threshold lines merge, and we have a single second-order phase transition, which occurs here at ${\mathrm{\Delta }}_p\ge 1$.

Figure 10

Fixed points, Eq. (b37), as a function of ${\mathrm{\Delta }}_2$ for $p=4$ and several fixed values of ${\mathrm{\Delta }}_p$. The values of ${\mathrm{\Delta }}_p$ correspond to the vertical cuts of the right panel of Fig. 7. The situation is qualitatively similar to Fig. 9, the difference being only the presence of the hybrid-hard phase. We can observe that when the transition is discontinuous, from (a) to (e), for $1/{\mathrm{\Delta }}_2>1.0$, the uninformative solution becomes unstable and continuously goes to a stable-informative solution, which is not the optimal one.

Figure 11

Representation of the initialization and the first two iterations for the evaluation of a two-times observable using the dynamic-grid algorithm. The empty circles represent slots allocated in memory but not associated with any specific value, while the full circles are memory slots already associated with a specific value. For any two-times function, it first allocates the memory (a), and then it fills half of the grid by linear propagation (b). Still using linear propagation, it fills the slots with $t-t^{\prime }\ll 1$ (c), and it sets the other values by imposing self-consistency (d). Finally, it halves the grid (e), doubles the time step, and it allocates the memory (f). Then, the algorithm loops follow the same scheme as in panels (b)–(e).

Figure 12

Evolution of the correlation with the signal starting from the solution, ${\overline{C}}_0=1.0$ at fixed ${\mathrm{\Delta }}_2$ for different ${\mathrm{\Delta }}_p$. The dotted red line overlapping with other lines is the same quantity evaluated using the fixed grid algorithm up to time 100. We have started the LSE from an informative initial condition.

Figure 13

Correlation with the signal of AMP (dotted lines) and Langevin (solid lines) at the $k$th iteration and $t$ time, respectively, starting in both cases with an initial overlap of ${10}^{-4}$. The black dashed line is the asymptotic value predicted with AMP. In the easy region, provided there is enough running time, Langevin dynamics finds the same alignment as AMP. The figures show qualitatively the same behavior for different values of ${\mathrm{\Delta }}_2$.

Figure 14

Left panel: Phase diagram of the spiked matrix-tensor model for $p=3$ as presented in the left panel of Fig. 7, with the additional boundary of the Langevin hard phase (green circles and dotted green line). The data points (circles) have been obtained numerically by fitting the relaxation time at fixed ${\mathrm{\Delta }}_p$ and increasing ${\mathrm{\Delta }}_2$. The green dotted line shows fixed points of expression (15). The blue dashed-dotted line marks a region above which we no longer observe a stable positive 1RSB complexity. Finally, we plot with orange and yellow dots on the dynamical transition line as extracted from the relaxation time of the Langevin algorithm coming from the hard and impossible phases, respectively. Right panel: Phase diagram of the spiked matrix-tensor model for $p=4$ as presented in the right panel of Fig. 7 with the additional Langevin hard phase boundary. The data points are obtained by fixing ${\mathrm{\Delta }}_2$ and decreasing ${\mathrm{\Delta }}_p$. Also in this case, we observe that the Langevin hard phase extends to the AMP easy phase. Interestingly, the Langevin hard phase here folds and presents a reentrant behavior; investigating the precise character of this reentrance is hampered by the vicinity of the critical point and is left for future work. The blue dashed line marks a region above which we no longer observe a stable positive 1RSB complexity.

Figure 15

Estimated divergence point ${\mathrm{\Delta }}_2^{*}$ at fixed ${\mathrm{\Delta }}_p=0.9$, with $p=3$ a function of the initial condition $\overline{C}$, ranging from ${10}^{-40}$ to ${10}^{-4}$. The vertical axis is in the linear scale, while the horizontal axis is in the log scale. We observe that the dependence of estimated divergence points on the initial condition is consistent with the asymptotic value $1/{\mathrm{\Delta }}_2^{*}=\sqrt{2/{\mathrm{\Delta }}_p}\approx 1.491$ following from Eq. (15) and depicted by the dashed line. The dotted line instead represents the AMP threshold, for comparison.

Figure 16

Phase diagram of the $p=3$ model, where we compare the threshold of the Langevin hard region using different values of the initial conditions, respectively: green circles ${\overline{C}}_0={10}^{-40}$, yellow circles ${\overline{C}}_0={10}^{-30}$, red circles ${\overline{C}}_0={10}^{-20}$.

Figure 17

Relaxation time obtained from the LSE starting from an informative initial condition ${\overline{C}}_0=1$. The four cases refer to the $2+3$ model and are fitted with a power law, and the relaxation time appears to diverge very close to the point predicted by AMP (the AMP prediction is given in the captions).

Figure 18

The figures show the correlation with the signal in time obtained using different annealing protocols, whose details are reported in the legend of the first figure. All of the protocols have $C=100$, which means that all the dynamics start with close effective ${\mathrm{\Delta }}_p\sim 100$, $T_p(0){\mathrm{\Delta }}_p\simeq 100$ and an initial overlap ${\overline{C}}_0={10}^{-4}$. What changes among the different lines is the relaxation speed, from the fastest (drawn in blue) to the slowest (drawn in brown). They are compared with the asymptotic value of AMP (dotted line) and the Langevin dynamics without tensor annealing (dashed line).

Figure 19

Relaxation times of Langevin dynamics at ${\mathrm{\Delta }}_p=100$ without using protocols starting from ${\overline{C}}_0={10}^{-4}$.

Figure 20

Complexity as a function of the Parisi parameter $x$ for $p=3$ on the line ${\mathrm{\Delta }}_p=0.5$. The solid line characterizes the stable part of the complexity, while the dashed line is the unstable one.

Figure 21

The stable part of the 1RSB complexity as a function of the free energy for $p=3$ and ${\mathrm{\Delta }}_p=0.5$.

Figure 22

Left panel: Parametric plot of the integrated response function with respect to the correlation function for $p=3$, ${\mathrm{\Delta }}_2=0.8$, and ${\mathrm{\Delta }}_p=0.2$. The different lines represent different waiting times, $t^{\prime }$. The black dashed line corresponds to the FDT prediction $x=1$. The vertical dotted line is the point where we observe a kink, which we denote by $\mathcal{C}={\widehat{q}}_{EA}$ and should be equal to the saddle-point value of $q_M$ as extracted from the 1RSB threshold states in the replica computation [27]: ${\widehat{q}}_{EA}=0.633$ and $q_M=0.638$. For $\mathcal{C}$ smaller than $q_{EA}$, the FDT is violated and is replaced by a generalized version as in Eq. (e10). We can obtain the value of the FDT ratio from a fit of the slope of the asymptotic curves for $\mathcal{C}<{\widehat{q}}_{EA}$. We obtain $\widehat{x}=0.397$, which should be compared with the Parisi parameter that corresponds to 1RSB marginally stable states obtained from the replica computation, that is, $x=0.408$. Right panel: Parametric plot of the integrated response as a function of the correlation for $p=3$ and ${\mathrm{\Delta }}_2=1.4$ and ${\mathrm{\Delta }}_p=0.2$. In this case, the value of the FDT ratio extracted from fitting the data is $\widehat{x}=0.397$, to be compared with the value of the Parisi parameter for the 1RSB threshold states, that is, $x=0.408$. At the same time, data give ${\widehat{q}}_{EA}=0.633$, while the replica computation gives $q_M=0.638$.