Hessian-based toolbox for reliable and interpretable machine learning in physics

The lack of interpretability is now a widely recognized challenge in the computer science community [27–31] especially when ML is applied to real-world problems like medical diagnosis, insurance cost estimation, etc. In science, the lack of interpretability can be disturbing because the black-box behavior of the models prevents us from learning anything about novel physics. Physicists are already addressing the need for interpretation of ML models, but the majority of proposed methods is either restricted to linear and kernel models [5, 6, 25, 32–36] or to the particular model architecture [12, 32, 37] or requires pre-engineering of the data, which limits the results nonuniversally to a specific ML and physical model [21, 38]. An interesting, largely model-agnostic method in the context of lattice quantum field theory was proposed in [16].

Another desired feature of ML models, which is intertwined with interpretability, is their reliability. A reliable ML model informs a user if its decisions are uncertain or result from pure extrapolation [39]. In computer science, the reliability is especially important in the context of safety-critical problems or adversarial attacks, i.e. careful perturbations of input samples that aims to mislead the ML model on the test set [40]. While data sets of physical interest are not endangered by such intentional attacks, adversarial ML tells us that tiny noise in the input may completely derail the model prediction. Moreover, while the reliability of ML in everyday problems can often be controlled by humans checking the predictions, it is improbable for, e.g. unknown phase diagrams. Nevertheless, the reliability of ML is not yet properly addressed in physics. While Bayesian ML, i.e. the most popular approach providing the uncertainty of predictions, proved to be a promising direction in molecular dynamics [41], it is generally difficult and needs a specific model architecture. Therefore, there is a great need for more model-agnostic tools to estimate the uncertainty of ML predictions.

To address the need for interpretability and reliability in the detection of phase transitions with ML methods, in this work, we apply four ML interpretability and reliability tools to the CNN trained in a phase classification problem. To better understand what a ML model learns, we extract the concept of similarity between input data from a machine. As a result, we can find out what is the relation between data according to the ML model and deduce what features are important for the classification. To this end, we employ and compare influence functions [42, 43] and relative influence functions (RelatIFs) [44]. Moreover, we address the need for model-agnostic assessment of the uncertainty of model predictions. We present resampling uncertainty estimation (RUE) [45], which allows for generating analogues of error bars for ML model predictions. Finally, we apply a tool called local ensembles (LEs) [46], which warns a user if a ML model makes predictions with a high level of extrapolation. We present these methods on the neural network trained to detect the quantum phase transition in the one-dimensional (1D) spinless Fermi–Hubbard model. The four methods require a single calculation of the Hessian of the training loss. Together, they form a Hessian-based toolbox that can be applied to any ML model and any learning scheme that relies on the calculation of the test and training loss functions and therefore finds application outside of physics and the detection of phase transitions.

This paper is structured as follows. Section 2 describes the used interpretability and reliability ML methods along with the Hessian and the concept of similarity. Section 3 presents and discusses the numerical results. Sections 3.3 and 3.4 concern tools that increase interpretability of the ML model, i.e. influence functions and relative influence functions (RelatIFs). Sections 3.5 and 3.6 focus on increasing the reliability of the model by assessing the extrapolation score (LEs) and uncertainty of the model predictions (RUE). Section 4 summarizes our paper and presents future possible applications and extensions.

Throughout this work, we focus on supervised ML classification problems with labeled training data $\mathcal{D} = \{z_i\}_{i = 0}^n$, with $z_i = (\mathbf{x}_i, y_i)$ where n is the size of the training set. The input data comes from an input space $\mathbf{x}_i \in \mathcal{X}$, and the labels are $y_i \in \lbrace 0, \ldots, c-1~\rbrace$, where $c \in \mathbb{N}$ is the number of classes in a given problem. In our setup, the inputs x_i are the state vectors of a given physical system, and y_i are the corresponding phase labels.

A model, f (in our case a CNN), is determined by the set of M parameters $\boldsymbol{\theta} = \lbrace \theta_0, \ldots, \theta_{M-1} \rbrace$. The number of parameters, M, in our case is of an order of thousand. For a given input x, the model outputs a real-valued c-dimensional vector, $\mathbf{f}_\mathbf{x} = f(\mathbf{x} ; \boldsymbol{\theta})$. The output encodes the prediction of the model, $y^{^{\prime}} = \mathrm{argmax}(\mathbf{f}_\mathbf{x})$. E.g., for a two-class problem, $\mathbf{f}_\mathbf{x}$ could be $[0.1, 0.9]$, which would correspond to predicting a label $y^{^{\prime}} = 1$. Elements of the output vector $\mathbf{f}_\mathbf{x}$ tend to be connected to probabilities of the input belonging to the corresponding classes. However, this interpretation can be misleading in the presence of data set shift [47, 48] or non-uniformity of errors [49]. Finally, the training process consists in searching the parameter space for the optimal parameters $\boldsymbol{\tilde{\theta}}_{\mathcal{D}} \equiv \boldsymbol{\tilde{\theta}}$ of the ML model, which minimize the empirical risk $\mathfrak{L}(\mathcal{D}, \boldsymbol{\theta}) = \frac{1}{n} \sum_{z \in \mathcal{D}} \mathcal{L}(z, \boldsymbol{\theta})$, also called the training loss, where $\mathcal{L}$ is the loss function.

The four interpretability and reliability methods discussed in this work are all based on local perturbations of the loss function. They study how a particular action, e.g. removal of a training point, changes the model parameters. This change, in turn, impacts the prediction of the model at a test point, $y^{^{\prime}}_{\mathrm{test}}$. The change of the model parameters caused by some action can be approximated using the Hessian matrix of the empirical risk (training loss) calculated at the minimum of the loss landscape at $\boldsymbol{\tilde{\theta}}$, namely $(H_{\boldsymbol{\tilde{\theta}}})_{ij} = \partial^2_{\theta_i \theta_j} \mathfrak{L}(\mathcal{D}, \boldsymbol{\theta}) |_{\boldsymbol{\theta} = \boldsymbol{\tilde{\theta}}}$. $H_{\boldsymbol{\tilde{\theta}}}$ describes the local curvature around the minimum point reached within the training. The eigenvectors of $H_{\boldsymbol{\tilde{\theta}}}$ corresponding to the largest positive eigenvalues indicate directions with the steepest ascent around the minimum. The high curvature implies that the training data strongly determines the model parameters along that direction ⁵ . In contrary to the common intuition, the training of a ML model leads to a local minimum or a saddle point [50, 53, 54]: the vast majority of the eigenvalues is close to zero, indicating various flat directions and some small negative eigenvalues are also present, indicating directions with negative curvature. Such a non-positive curvature around the minimum, in general, does not affect the quality of the model predictions but may cause problems when working with the Hessian.

All the methods we study in this work approximate how the change of parameters impacts the model predictions, but the reason for the change of parameters is different for each method. Influence functions and RelatIF study the removal of a single training point from a training data set, RUE analyzes training on various samples of the training data set, while LEs modify model parameters in the flat directions of the Hessian. These methods aim to answer different questions regarding the reliability and interpretability of the model, and we will discuss them in detail in the following sections.

2.3.1. Leave-one-out training

2.3.2. Influence functions

Retraining the model is, however, expensive, and an approximation of the LOO training was proposed and named influence functions [55–57]. It was then ported to ML applications by Koh and Liang [42, 43]. The influence function reads:

$$\begin{align} \mathcal{I}(z_\textrm{r}, z_{\textrm{test}}) = \frac{1}{n} \nabla_{\boldsymbol{\theta}} \mathcal{L}(z_{\textrm{test}}, \boldsymbol{\tilde{\theta}})^T H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla_{\boldsymbol{\theta}} \mathcal{L}(z_\textrm{r}, \boldsymbol{\tilde{\theta}}) \equiv \frac{1}{n} \nabla \mathcal{L}_{\mathrm{test}}^T H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}} , \end{align} \tag{ 1 }$$

and it estimates the change of the test loss for a chosen test point z_test after the removal of a chosen training point z_r. $\nabla_{\boldsymbol{\theta}} \mathcal{L}(z_\textrm{test}, \boldsymbol{\tilde{\theta}})$ is the gradient of the loss function of the single test point, and $\nabla_{\boldsymbol{\theta}} \mathcal{L}(z_\textrm{r}, \boldsymbol{\tilde{\theta}})$ is the gradient of the loss function of the single training point whose removal's impact is being approximated. Both are calculated at the minimum $\boldsymbol{\tilde{\theta}}$ of the training loss landscape.

2.3.3. Geometrical interpretation

The influence function (1) can be written as the inner product of $- \nabla \mathcal{L}_{\mathrm{test}}$ and $-H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}}$ [44], where the term $- H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}}$ describes an approximated change in parameters $\boldsymbol{\tilde{\theta}} \rightarrow \boldsymbol{\theta}^{^{\prime}}$ due to the removal of the training point z_r (for a derivation, see appendix A of [42]). This formulation emphasizes the geometric interpretation of influence functions, which is a projection of the approximated change in parameters due to the removal of a training point onto the test sample's negative loss gradient (see figure 1(b)). The term $- H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}}$ can also be understood as a Newton step [58] towards a new minimum resulting from a removal of z_r. Note that the same term involves scaling by the inverse of eigenvalues of $H^{-1}_{\boldsymbol{\tilde{\theta}}}$. In other words, we see that the influence function is a scalar product of the gradients $\nabla \mathcal{L}_{\mathrm{test}}$ and $\nabla \mathcal{L}_{\mathrm{r}}$ accounting for a local curvature of the loss landscape described by $H_{\boldsymbol{\tilde{\theta}}}$. The resulting value of influence functions depends on two factors: how similar are the test and the removed training point and how representative they are in the data set.

Zoom In Zoom Out Reset image size

2.3.4. Similarity measure

Firstly, the more similar the test point and the removed training point are, the larger is the value of the influence function between them. More specifically, the largest influence is for the change in parameters which is along the direction of $\nabla \mathcal{L}_{\mathrm{test}}$. It happens when the gradients $\nabla \mathcal{L}_{\mathrm{test}}$ and $\nabla \mathcal{L}_{\mathrm{r}}$ are aligned in the parameter space, corrected for the local curvature of the loss landscape, so when the test point, z_test, is similar to the removed training point, z_r. Note that by similarity here we understand the distance in the model's internal representation, so in the model's parameter space, corrected by the local curvature described by the Hessian. This similarity is different than, e.g. similarity as a distance of input vectors z_r and z_test in the input space $\mathcal{X}$ or the similarity in the Euclidean parameter space. In particular, the predictive model and especially neural networks can be highly nonlinear and may use an internal representation in which similar (close) points are far away in both the input and the Euclidean parameter space. We can then define a model's similarity measure between data points z_i and z_j equal to [45]:

$$\begin{align} S (z_i, z_j) = [\nabla_{\boldsymbol{\theta}} \mathcal{L}(z_i, \boldsymbol{\tilde{\theta}})^T H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla_{\boldsymbol{\theta}} \mathcal{L}(z_j, \boldsymbol{\tilde{\theta}})]^2~\equiv [\nabla \mathcal{L}_i^T H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_j]^2~\propto \mathcal{I}(z_i, z_j)^2. \end{align} \tag{ 2 }$$

2.3.5. Representative data and outliers

2.3.6. Sensitivity to outliers

2.3.7. RelatIF

Barshan et al [44] proposed then a variant of influence functions that takes advantage of the similarity measure but eradicates the influence of unrepresentative data points and outliers. The method is called RelatIF and restricts the pool of influential points to the most similar ones (see the gradients circled in red in figure 1(b)). Mathematically, it amounts to introducing a normalization to the influence function's formula:

$$\begin{align} \mathcal{I}_{\mathrm{R}}(z_\textrm{r}, z_{\textrm{test}}) = \frac{\mathcal{I}(z_\textrm{r}, z_{\textrm{test}})}{|| H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla_{\boldsymbol{\theta}} \mathcal{L}(z_\textrm{r}, \boldsymbol{\tilde{\theta}}) ||} = \frac{\frac{1}{n} \nabla \mathcal{L}_{\mathrm{test}}^T H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}}}{|| H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_\textrm{r} ||} . \end{align} \tag{ 3 }$$

Both tools increase the interpretability of the ML model by indicating what the model regards as similar. The influence functions' focus on the unrepresentative data points also allows one to judge the model's reliability by finding outliers in the training data set. The model's reliability is the central issue addressed by two other methods discussed in this work, namely RUE and LEs.

The RUE [45] aims at assessing the uncertainty of predictions of the ML model. It can be applied to the already trained model and requires no specific architecture or learning scheme in contrast to, e.g. Bayesian methods, which are the most common approach in ML for assessing uncertainty [60, 61]. The RUE method makes use of two important criteria to judge whether a prediction is reliable: the density criterion and local fit criterion [45]. The density criterion states that a prediction at the input z_test is reliable if there are samples in the training data that are similar to z_test. The local fit criterion states that a prediction at the z_test is reliable if the model has a small error on samples in the training data similar to z_test. Both criteria hinge upon a measure of similarity defined in equation (2) and can be addressed with the bootstrap sampling.

To quantify uncertainty, the RUE algorithm makes b 'bootstrap' samples created by sampling with replacement from the uniform distribution over the original training data set. Let us start from the original data set $\mathcal{D}$ containing each training data point once, indicated with the short-hand notation by $\mathcal{D} [1,1, \ldots, 1]$. We can then create a bootstrap sample by drawing the same point more than once and omit others, e.g. $\mathcal{D}_b [2,0,3, \ldots, 1,0]$, which stands for taking twice the first training example, omitting the second training example, etc. If a ML model trained on $\mathcal{D} [1,1, \ldots, 1]$ converges to parameters $\boldsymbol{\tilde{\theta}}$, b ML models trained on b different bootstrap samples converge to similar model parameters $\boldsymbol{\theta}_b^{^{\prime}}$. Now we can make b predictions at the same test point $z_\mathrm{test}$ with b ML models and calculate the variance of the test loss across these b models. Small variance means that a prediction of the original model can be trusted, while large variance means that it is not reliable. Intuitively, this variance estimates how much the model prediction would change if we fitted the model on different data sets drawn from the same distribution as the original training data. This intuition is connected to the idea of the classical bootstrap [62].

As for the LOO training, such a retraining procedure is prohibitively expensive. Therefore, one can make a similar approximation of the change of the model parameters due to the removal of some training examples and adding copies of others to the training set within a bootstrap sample. We can approximate the new parameters via:

$$\begin{align} \boldsymbol{\theta}_b^{^{\prime}} \approx \boldsymbol{\tilde{\theta}} - H^{-1}_{\boldsymbol{\tilde{\theta}}} \cdot L \cdot w_{\Delta_b}, \end{align} \tag{ 4 }$$

where $w_{\Delta_b}$ is a vector of differences between the composition of the original $\mathcal{D} [1,1, \ldots, 1]$ and $\mathcal{D}_b$. E.g., for $\mathcal{D}_b [2,0,3, \ldots, 1,0]$, $w_{\Delta_b}$ is $[1,-1,2,\ldots,0, 1]$. L is a matrix of all the n single training loss gradients w.r.t. every model parameter, so it has the size M × n, and it takes the form $L = [\nabla_{\boldsymbol{\theta}} \mathcal{L}(z_0, \boldsymbol{\tilde{\theta}}),\ldots,\nabla_{\boldsymbol{\theta}} \mathcal{L}(z_{n-1}, \boldsymbol{\tilde{\theta}})]^T$. In the next step, one generates predictions at a test point with b models with approximated parameters $\boldsymbol{\tilde{\theta}}_b$, obtaining b test losses based on $\mathbf{f}_b = f(\boldsymbol{\tilde{\theta}}_b, x_\mathrm{test})$. Finally, we calculate the variance of b test losses, i.e. the average of the squared deviations from the original test loss.

We say the prediction at a test point is underdetermined if many different predictions are equally consistent with the constraints posed by the training data and the learning problem specification (i.e. the model architecture and the loss function). An example of such behavior is when a model trained on the same training data arrives at different predictions depending on the choice of a random seed and, therefore, relies on arbitrary choices outside the learning problem specification. Intuitively, underdetermination can be understood as the model converging during the optimization process to various distinct points in a flat basin forming a minimum. As discussed in section 2.2, the training data puts limited constraints on the flat directions around the minimum. However, changing the model parameters in these directions still can impact predictions at test points drawn from a different distribution than the training set, i.e. so-called out-of-distribution (OOD) test points [63]. A reliable model should warn the user when making a prediction at such a test point.

LEs are a method to detect the underdetermination at test time in a pre-trained model [46]. LE consists of local perturbations of the parameters of the trained model that fit the training data equally well, i.e. have the same value of training loss. In other words, we perturb the parameters of the model only in the directions of the Hessian eigenvectors corresponding to close to zero eigenvalues, meaning we explore only flat basins around the minimum. Analogously to RUE, the next step is to make predictions at the same test point $z_\mathrm{test}$ with LE models and calculate the variance of the test loss within the LE. Madras et al [46] found an even simpler approximation of this variance for a test point, $z_\mathrm{test}$, and named it a LEES:

$$\begin{align} \mathcal{E}_m (z_\mathrm{test}) = || U_m^\top \nabla_{\boldsymbol{\theta}} \mathcal{L} (z_\mathrm{test}, \boldsymbol{\tilde{\theta}}) ||_2 = || U_{m}^\top \nabla \mathcal{L}_\mathrm{test} ||_2. \end{align} \tag{ 5 }$$

U_m is a matrix of $(M - m)$ Hessian eigenvectors spanning a subspace of low curvature, i.e. after removing m eigenvectors corresponding to largest eigenvalues and, therefore, to directions with the highest curvature, which are well-constrained by training data. The authors of the method admit that choosing m is not a trivial task, with the danger of omitting under-constraint directions if m is set too high or including well-constraint directions if m is set too low. In this work, we choose the smallest possible m for which $\mathcal{E}_m$ starts to converge for all test points (m = 12 in figure 5).

A careful reader may have noticed two numerical challenges resulting from equations (1)–(5). Firstly, the calculation of influence functions, RelatIF, and RUE requires inverting the Hessian of a training loss which in deep learning is known to be highly non-convex [64]. As we pointed out in section 2.2, the optimization usually leads to a critical point with a majority of flat or almost flat directions (corresponding to eigenvectors with zero or close to zero eigenvalues) and a small number of directions of negative curvature (corresponding to eigenvectors with negative eigenvalues). The inverse of a matrix exists only if it is positive-definite (has only positive eigenvalues). Therefore, Koh and Liang proposed to add a damping term to the Hessian [42], $\lambda \, I$, with I being the identity matrix and λ being larger than the absolute value of the largest negative Hessian eigenvalue, $|E_0|$. It is equivalent to L₂ regularization [44] and amounts to shifting all eigenvalues by λ, guaranteeing the existence of the Hessian inverse. Regardless of the exact value of the damping, the Hessian-based toolbox keeps giving meaningful results [42]. We usually choose $\lambda \approx |E_0| + 0.01$, except for RUE where the authors explicitly state that the smallest eigenvalue of the damped Hessian needs to be around one, rendering $\lambda_{\mathrm{RUE}} = \lambda + 1$.

Secondly, the calculation of the Hessian matrix for a model with a large number of parameters can be very demanding. Fortunately, we do not need to calculate the full Hessian, only Hessian-vector products (e.g. $H^{-1}_{\boldsymbol{\tilde{\theta}}} \nabla \mathcal{L}_{\mathrm{r}}$ in equations (1) and (3)) or the top part of the Hessian spectrum, which significantly reduces the computational complexity of the problem [65]. The inverse of the Hessian for influence functions, RUE, and RelatIF can be approximated with so-called stochastic approximation with LiSSA [65]. Additionally, the authors of RelatIF approximated the normalization factor in equation (3) with a method related to K-FAC [66]. On the other hand, LE needs no inverse but an ensemble subspace of eigenvectors with zero or close to zero eigenvalues. The authors of LE proposed to use the Lanczos iteration [67] to calculate m eigenvectors with the largest eigenvalues, build an m-dimensional subspace (of highest curvatures), and create its orthogonal complement, namely the ensemble subspace (of flat directions). There is also a Python library called PYHESSIAN designed to tackle Hessian-based problems [68]. Within this paper, however, we calculate the Hessian explicitly, due to the limited size of our neural network.

We show the functionalities of the four interpretability and reliability methods on the example of a CNN trained to recognize phases of the spinless half-filled one-dimensional (1D) Fermi–Hubbard model [69]. This model describes fermions hopping between neighboring sites with amplitudes J (set to 1 throughout the paper) and interacting with nearest neighbors with strength V₁,

$$\begin{align} \hat{H} = - J \sum_{\langle i,j \rangle} c_i^{\dagger} c_j + V_1~\sum_{\langle i,j \rangle} n_i n_j, \end{align} \tag{ 6 }$$

where $c_i^{\dagger}$ and c_i are fermionic creation and annihilation operators at site i, respectively, and $n_i = c_i^{\dagger} c_i$ is the number operator. We calculate eigenvectors with exact diagonalization using QuSpin and SciPy packages [70, 71]. The detailed description of the data set can be found in our previous work [72]. We focus on a single transition line resulting from a competition between hopping and nearest-neighbor interaction, presented in figure 2. It starts in the gapless Luttinger liquid (LL) phase, and the increase of V₁ leads to a phase transition to a gapped charge-density wave (CDW-I) with density patterns 1010 [73, 74]. The order parameter describing the transition, $O_{\mathrm{CDW-I}}$, is the average difference between the nearest-neighbour densities. It is zero in the LL phase and grows to 1 in the CDW-I phase.

Zoom In Zoom Out Reset image size

We feed the CNN with ground states expressed in the Fock basis, labeled with their appropriate phases, calculated for a 12-site or 14-site system. The architectures of the used CNNs are presented in appendix B. Intuitively, we could expect that the most similar quantum states, according to the ML model, are those generated for the most similar V₁. Instead, as we showed in [72], the similarity learned by the ML model is based on the order parameter (or something related to it), as it is a much better discriminator between the phases. Therefore, a well-trained model sees all LL data points as very similar (as they all have a zero order parameter), while the similarity of CDW-I data points depends on V₁ (on this side, the order parameter continuously goes up from 0 to 1).

While in [72] we studied in detail the potential of influence functions for interpretability, we here focus on the differences between influence functions and relative influence functions (RelatIFs). To this end, we train a CNN on the eigenvectors of the 12-site 1D Fermi–Hubbard model with the labels indicating phases they belong to, i.e. LL or CDW-I. We then analyze the trained model with influence functions (equation (1)) and RelatIFs (equation (3)) and present results in figure 3. Each column of figure 3 presents both methods calculated for the same test points, indicated by the orange vertical lines. In the case of influence functions (first row of figure 3), the most helpful training points are the most similar to the test point and the most unrepresentative in the data set. RelatIFs, presented in the second row of figure 3, are expected instead to ignore how representative data is and to indicate the most similar training points to the test point, paying less attention to outliers.

Zoom In Zoom Out Reset image size

3.3.1. Behavior in the CDW-I phase

3.3.2. Model-agnostic RelatIFs

3.3.3. Behavior in the LL phase

We continue working with the data set of the eigenvectors of the 12-site 1D Fermi–Hubbard model with labels. The global sign of such an eigenvector is not a physical observable, and therefore a well-generalizing model may ignore this property. To challenge this intuition, we prepare two data sets differing only in the distribution of the global sign. The first data set is composed of a large majority of positive-sign eigenvectors and several negative-sign eigenvectors. In the second, half of the eigenvectors have a positive global sign and the other half—a negative global sign. We call them the 'sign-imbalanced' and 'sign-balanced' data sets, respectively.

3.4.1. Detection of outliers

The CNN trained on the sign-imbalanced data set has high accuracy on positive-sign test points. Regarding negative-sign test points, the CNN correctly classifies them on the CDW-I side but has lower accuracy on the LL side. This suggests that the ML model grasped the global-sign invariance only to a limited level. Figure 4(a) presents the influence functions between the training data and the single positive-sign test point near the transition (marked with the orange vertical line). The pattern is dominated by two smooth lines on both sides of the phase transition, formed by positive-sign training points, as in figure 3(b) in the previous section. However, there are several single training points that are outside the continuous patterns. These 'outsiders' are negative-sign training points that the model regards as distinct from positive training points, in the sense of the similarity as described in section 2.3. Influence functions then immediately pinpoint anomalies in the training data and improve the reliability of the model. The non-zero influence of the outliers and the decent test accuracy on negative-sign points indicate that the model gains some information from a minority of negative-sign training points. However, a model can develop different similarity measures depending on the training process and its architecture. For example, all anomalies (or outliers) can have zero influence regardless of the test point. It shows that the model ignores them during the training (which may be encouraged by large regularization), and we could further use such knowledge to improve the model.

Zoom In Zoom Out Reset image size

3.4.2. Global sign (in)variance of the ML model

3.4.3. Strategies to make a ML model invariant

We now challenge the concept of a test loss as a reliability measure. As we will show, the extrapolation score with LEs highlights better OOD test points. To do so, we introduce in our test set a percentage of test elements whose components are randomly permuted. A well-trained ML model, combined with a reliability method, should be able to inform us of the OOD test points.

3.5.1. Minimal test loss

Let us start by looking at the loss function for every test point, plotted in figure 5 with a purple line. We here use the minimal version of the loss function (i.e. assuming all predicted labels are correct, more details in appendix A) to mimic a real-life scenario: we ask a trained ML model to make predictions at test points whose labels we do not know. However, here we have access to the ground-truth labels, and we know the model misclassifies the test points generated for $V_1/J \in [1,2.1]$. The model wrongly predicts that the points in this interval belong to the LL phase. We see that the minimum test loss in figure 5 is primarily smooth and reaches a maximum around $V_1/J = 2.1$. This is the predicted transition point of the model, i.e. for this test point, the model outputs values corresponding to the LL and CDW-I class, which are very close to each other.

Zoom In Zoom Out Reset image size

3.5.2. LE-based extrapolation score

3.5.3. Fate of the OOD test points

We finally use the RUE method to analyze the uncertainty of predictions of the ML model and therefore identify phase transitions regions. In particular, it is known that finite size effects play a great role in the analysis of phases and should be taken into account when predicting phase transitions. We, therefore, train two copies of the same CNN architecture on two data sets, so on the eigenvectors of the 12-site and 14-site 1D Fermi–Hubbard model with labels corresponding to the LL or CDW-I phase. We choose a CNN architecture that is invariant to the input size (see appendix B).

3.6.1. Visualization of RUE

To detect the quantum phase transition region, we calculate RUE for the whole test set uniformly spread across the transition line. Figures 6(a) and (b) show the results for the CNNs trained on the 12- and 14-site systems, respectively. We represent RUE as error bars as they are the variance of the test loss' change under the different sampling of the original training data ⁶ . It is important to note that we calculate RUE using the minimal version of the test loss (see the discussion in appendix A), assuming that predicted labels are the correct ones.

Zoom In Zoom Out Reset image size

3.6.2. Transition regime and error bars

3.6.3. Error bars for 12- and 14-site systems