RiteeshGrandhi/sciclaimeval-shared-task huggingface dataset:
Theoretical Analysis
In this section, we analyze MPGD via three different angles. We start by presenting results on superdiffusive limit (homogenization [ 29 ] ) for a class of fast-slow deterministic systems that include MPGD as a special case in Subsection 4.1 . We then study generalization properties of MPGD by deriving a generalization bound for the limiting dynamics in Subsection 4.2 . Lastly, we study the regularizing effects of MPGD via the lens of implicit regularization in Subsection 4.3 .
Empirical Results
In this section, we illustrate the advantages of MPGD compared to other schemes on the tasks of (1) minimizing the widening valley loss, (2) regression on the Airfoil Self-Noise Dataset, and (3) classification on CIFAR-10. We also provide additional results and details in Appendix F .
Minimizing the Widening Valley Loss
We consider the problem of minimizing the widening valley loss, given by \ell(u,v)=v^{2}\|u\|^{2}/2 . The gradient of l is given by \nabla\ell(u,v)=(v^{2}u,\|u\|^{2}v) , and the trace of the Hessian is dv^{2}+\|u\|^{2} , where d is the dimension of u . The trace of the Hessian measures the flatness of the minima, which is monotonously changing in this case. The loss has a valley of minima with zero loss for all (u,v) with v=0 . The smaller the norm of u , the flatter the minimum. Vanilla GD gets stuck when it first enters this valley, and the intuition is that injecting suitable perturbations should help convergence to a flat (with small \|u\| ) part of the valley. All (u,v) with v=0 are minima, but we also need \|u\| to be minimized in order to minimize the trace of the Hessian. In high dimension (large d ), the GD path is biased towards making v small and not optimizing u since the direction along v is the most curved.

For the experiments, we start optimizing from the point (u_{0},0) , where u_{0}\sim 5\cdot\mathcal{U}(d) with d=10 , and use the learning rate \eta=0.01 . We study and compare the behavior of the following schemes: (i) baseline (vanilla GD), (ii) GD with uncorrelated Gaussian noise injection instead, and (iii) MPGD. Figure 2 demonstrates that MPGD can lead to successful optimization of the widening loss whereas the baseline GD and GD with Gaussian perturbations lead to poor solutions. This is in agreement with our analysis of implicit regularization for MPGD, showing that the injected perturbations effectively favor small trace of the loss Hessian, thereby biasing the solution to flatter region of the loss landscape.
Airfoil Self-Noise Prediction
We consider the Airfoil Self-Noise Dataset [ 10 ] from the UCI repository. It comprises of differently sized airfoils at various wind tunnel speeds and angles of attack. This is a regression problem aiming to predict scaled sound pressure level, in decibels, of the airfoil based on the the features: frequency, angle of attack, chord length, free-stream velocity, and suction side displacement thickness. We use 1202 samples for training and 301 samples for testing.

For the training, we use a fully connected shallow neural network of width 16 with ReLU activation and train for 3000 epochs with the learning rate \eta=0.1 , using mean square error (MSE) as the loss and choosing \beta=0.5 . Table 1 reports the average root MSE (RMSE) and the RMSE gap (defined as test RMSE - train RMSE) evaluated for models that are trained with 5 different seed values for this task. We can see that MPGD leads to both lower test RMSE and RMSE gap when compared with vanilla GD (baseline) and GD with uncorrelated Gaussian perturbations (see the results not in parenthesis in Table 1 ; here \mu=0.01 , \sigma=0.02 ). Using the form of the perturbations in ( 11 ) instead can also give lower RMSE gap (see the results in parenthesis in Table 1 ; here \sigma=\mu=0.01 ). Overall these results support our generalization theory for MPGD. Lastly, we note that using larger values of \gamma naively does not guarantee better test performance: one has to fine tune the parameters \beta , \mu , \sigma , \eta appropriately to achieve favorable trade-off between training stability and test performance.
Conclusion
In this paper, we introduce and study a class of slow-fast deterministic dynamical systems which homogenize to a limiting SDE driven by a heavy-tailed Lévy-stable process in an appropriate scaling limit as a rigorous framework for perturbed GDs. Within this framework, we introduce MPGD, a novel version of perturbed GD, which we show to have good generalization and regularization properties. We further demonstrate the advantages of MPGD empirically in various optimization tasks. Our framework can provide useful tools for identifying implicit randomness in deterministic optimization algorithms and inspire other promising algorithms. It would be interesting to investigate the interactions of the chaotic perturbations in MPGD with the noise arising from mini-batch sampling in SGD [ 37 ] and other optimization tricks. Since this is a theoretical paper studying a framework for perturbed GDs, there are no potential negative societal impacts of our work.
B.1.1 Skorokhod-type topologies
The Skorokhod \mathcal{J}_{1} topology on D([0,1],\mathbb{R}^{d}) is induced by the following metric.

The Skorokhod distance on D([0,1],\mathbb{R}^{d}) , the space of càdlàg functions, is defined by

\sigma_{\infty}(X_{1},X_{2})=\inf\|\lambda-\mathrm{id}\|_{\infty}\vee\|X_{1}\circ\lambda,X_{2}\|_{\infty},

where the inf is taken on all increasing bijections \lambda from [0,1] to itself.

Another important topology on D([0,1],\mathbb{R}^{d}) is the \mathcal{SM}_{1} topology defined as follows.

The Skorokhod \mathcal{SM}_{1} topology on D([0,1],\mathbb{R}^{d}) , the space of càdlàg functions, is defined by the metric

d_{\mathcal{SM}_{1}}(X_{1},X_{2})=\inf\|(\lambda_{1},\gamma_{1})-(\lambda_{2},\gamma_{2})\|_{\infty},

where the inf is taken on all pairs (\lambda_{i},\gamma_{i})\in C([0,1],[0,1]\times\mathbb{R}^{d}) such that (\lambda_{i},\gamma_{i})(0)=(0,X_{i}(0)) , (\lambda_{i},\gamma_{i})(1)=(1,X_{i}(1)) and \gamma_{i}(t)\in[X_{i}(\lambda_{i}(t)_{-}),X_{i}(\lambda_{i}(t))] , i=1,2 . It is equivalent to the metric on the graph of functions in D , with discontinuities connected by the straight segments.

Recall that for 1\leq p<\infty , the p -variation of u:[0,1]\to\mathbb{R}^{d} is given by

\|u\|_{p-var}=\sup_{0=t_{0}<t_{1}<\ldots<t_{k}=1}\left(\sum_{j=1}^{k}|u(t_{j})-u(t_{j-1})|^{p}\right)^{1/p},

and the subspace of D([0,1],\mathbb{R}^{d}) with finite p -variation is denoted by D^{p\text{-var}}([0,1],\mathbb{R}^{d}) . Let us define the following p -variation generalisations of the Skorokhod topology.

\sigma_{p-var}(X_{1},X_{2})=\inf\max\{\|\lambda-\mathrm{id}\|_{\infty},\|X_{1}\circ\lambda-X_{2}\|_{p-var}\},

where the inf is taken on all increasing bijections \lambda from [0,1] to itself.
B.1.2 Generalised Skorokhod topologies with interpolations
For discontinuous càdlàg functions, one can interpolate jumps with path functions.

A path function on \mathbb{R}^{d} is a map \phi:J\to C([0,1],\mathbb{R}^{d}) , where J\subset\mathbb{R}^{d}\times\mathbb{R}^{d} , for which \phi(x,y)(0)=x and \phi(x,y)(1)=y for all (x,y)\in J .

A pair (X,\phi) is called admissible if all the jumps of X are in the domain of definiton J of \phi , i.e. (X(t-),X(t))\in J for all jump times t of X . Denote by \mathscr{D}([0,1],\mathbb{R}^{d}) the space of admissible pairs, modulo the equivalence (X_{1},\phi_{1})\sim(X_{2},\phi_{2}) if X_{1}=X_{2} and \phi_{1}(X_{1}(t-),X_{1}(t)) is a reparametrization of \phi_{2}(X_{1}(t-),X_{1}(t)) for all jump times t of X_{1} .

For admissible pair (X,\phi)\in\mathscr{D}([0,1],\mathbb{R}^{d}) , we now construct a continuous path X^{\phi,\delta} as follows.

Given a sequence r_{1},r_{2},\ldots>0 with r:=\sum_{j}r_{j}<\infty ,

Let \tau:[0,1]\to[0,1+r] given by \tau(t)=t+\sum_{k}\delta r_{k}1_{\{t_{k}\leq t\}}.

Define an intermediate process \hat{X}\in C([0,1+\delta r],\mathbb{R}^{d}) ,

\hat{X}(t)=\left\{\begin{aligned} &X(s)& if t=\tau(s) for some s\in[0,1]\ &\phi(X(t_{k}-),X(t_{k}))\left(\frac{s-\tau(t_{k}-)}{\delta r_{k}}\right)& if t\in[\tau(t_{k}-),\tau(t_{k})) for some k.\end{aligned}\right.

Finally, let X^{\phi,\delta}(t)=\hat{X}(t(1+\delta r)) . We will drop the superscript \delta if \delta=1 .

The (pseudo)metric \alpha_{p\text{-var}} on \mathscr{D}([0,1],\mathbb{R}^{d}) is defined by

\alpha_{p-var}((X,\phi),(\overline{X},\overline{\phi})):=\lim_{\delta\to 0}\sigma_{p-var}(X^{\phi,\delta},\overline{X}^{\overline{\phi},\delta})

independent of the choice of the series \sum_{k=1}^{\infty}r_{k} . Denote by \mathscr{D}^{p\text{-var}}([0,1],\mathbb{R}^{d}) the subspace of \mathscr{D}([0,1],\mathbb{R}^{d}) with finite \alpha_{p\text{-var}} distance to 0 . Conventionally, write

\alpha_{\infty}((X,\phi),(\overline{X},\overline{\phi})):=\lim_{\delta\to 0}\sigma_{\infty}(X^{\phi,\delta},\overline{X}^{\overline{\phi},\delta})

If \phi is the linear path function, then \alpha_{\infty} induces the \mathcal{SM}_{1} topology on the space D([0,1],\mathbb{R}^{d}) .
B.2 Background on rough differential equations (RDEs)
We say that X is a solution to the forward RDE \mathrm{d}X_{t}=b(X)^{-}_{t}\mathrm{d}W_{t} if

X_{t}=X_{0}+\int_{0}^{t}b(X_{s-})\mathrm{d}W_{s},

where the integral above denotes a limit of Riemann-Stieltjes sums with b(X(s-)) evaluated at the left limit points of the partition intervals:

\int_{0}^{t}b(X_{s-})\mathrm{d}W_{s}=\lim_{|\mathcal{P}|\to 0}\sum_{[s,s^{\prime}]\in\mathcal{P}}b(X_{s-})(W_{s^{\prime}}-W_{s}),

where the \mathcal{P} are partitions of [0,t] into intervals, and |\mathcal{P}| is the size of the longest interval.

We make the following two observations.

If W has finitely many jumps at times 0<t_{1}<\ldots<t_{n}\leq 1 , then the forward solution of \mathrm{d}X_{t}=b(X)_{t}^{-}\mathrm{d}W_{t},X_{0}=x can be obtained by solving the canonical RDE on each of the intervals on which W is continuous, i.e., [0,t_{1}),[t_{1},t_{2}),\ldots,[t_{n},1) , and requiring that at jump times t_{k} , k=1,\ldots,,n :

X_{t_{k}}=X_{t_{k}-}+b(X_{t_{k-}})(W_{t_{k}}-W_{t_{k-}}).

If in ( 7 ), we write X^{(m)}_{t}:=x^{(m)}_{\lfloor mt\rfloor} and V^{(m)}_{t}=\lfloor tm\rfloor/m , then the first equation of ( 7 ) is nothing but the forward RDE

\mathrm{d}X^{(m)}_{t}=a_{m}(X^{(m)}_{t})^{-}\mathrm{d}V^{(m)}_{t}+b_{m}(X^{(m)}_{t})^{-}\mathrm{d}W^{(m)}_{t},\qquad X^{(m)}_{0}=x^{(m)}_{0}\in\mathbb{R}^{d}.
B.2.1 Young’s integral and
Marcus SDE
For the purpose of defining Young’s integral for driving functions with finite q -variation, q\geq 1 , we fix x\in C^{p\text{-var}}([0,1],\mathbb{R}^{d}) and y\in C^{q\text{-var}}([0,1],\mathbb{R}^{e\times d}) with \theta=\frac{1}{p}+\frac{1}{q}>1 .

There exists a sequence x^{n}\in C^{1\text{-var}}([0,T],\mathbb{R}^{d}) such that x^{n}\to x in C^{p\text{-var}}([0,T],\mathbb{R}^{d}) and a sequence y_{n}\in C^{1\text{-var}}([0,T],\mathbb{R}^{e\times d}) such that y_{n}\to y in C^{q\text{-var}}([0,T],\mathbb{R}^{e\times d}) . For every s<t , the limit of \int_{s}^{t}y^{n}(u)dx^{n}(u) exists, which we denote by \int_{s}^{t}y(u)dx(u) the Young’s integral of y against x on the interval [s,t] . The Young’s integral does not depend on the choices of sequences x^{n} and y^{n} , and we have

\left|\int_{s}^{t}y(u)dx(u)-y(s)(x(t)-x(s))\right|\leq\frac{1}{1-2^{1-\theta}}\|x\|_{p-var;[s,t]}\|y\|_{q-var;[s,t]}.

We record the following Young-Lóeve estimate, which will be useful for our purposes.

Let x\in C^{p\text{-var}}([0,T],\mathbb{R}^{d}) and y\in C^{q\text{-var}}([0,T],\mathbb{R}^{e\times d}) with \frac{1}{p}+\frac{1}{q}>1 . The integral path t\to\int_{0}^{t}y(u)dx(u) is continuous with a finite p -variation and

\left\|\int_{0}^{.}y(u)dx(u)\right\|_{p-var,[s,t]}\leq C\|x\|_{p-var;[s,t]}(\|y\|_{q-var;[s,t]}+\|y\|_{\infty;[s,t]})

Solutions to Marcus rough differential equation (RDE) are defined via Young’s integrals and the linear path function defined in Definition B.6 .

The solution of a Marcus RDE

\mathrm{d}X=b(X)\diamond\mathrm{d}W,\qquad X(0)=x_{0}

is obtained by

first solving the continuous RDE \mathrm{d}\tilde{X}=b(\tilde{X})\mathrm{d}W^{\phi,1} , with \phi=l_{d} being the linear path function on \mathbb{R}^{d} ;

then the càdlàg solution path X[0,1]\to\mathbb{R}^{d} is given by X(t)=\tilde{X}(\tau(t)) . Recall that \tau:[0,1]\to[0,1+r]\text{ given by }\tau(t)=t+\sum_{k}\delta r_{k}1_{\{t_{k}\leq t\}} .
B.3 Example of a Marcus RDE
Here is an example of the solution of a rough differential equation in the Marcus sense [ 6 ] .
Let \theta>0 and W^{(m)}:[0,1]\to\mathbb{R} be the deterministic process which are 0 on [0,1/2] , \theta on [1/2+1/m,1] and linear on [1/2,1/2+1/m] . Denote by X^{(m)}_{t}=(x^{(m)}_{t},y^{(m)}_{t}) the solution to the ordinary differential equation

\begin{pmatrix}\mathrm{d}x^{(m)}_{t}\ \mathrm{d}y^{(m)}_{t}\end{pmatrix}=\begin{pmatrix}-y^{(m)}_{t}\ x^{(m)}_{t}\end{pmatrix}\mathrm{d}W^{(m)}_{t},\quad\begin{pmatrix}x^{(m)}_{0}\ y^{(m)}_{0}\end{pmatrix}=\begin{pmatrix}1\ 0\end{pmatrix}

It is not hard to see that that X^{(m)}_{t}=(\cos{W^{(m)}_{t}},\sin{W^{(m)}_{t}}) . Therefore, we have that pointwise

W^{(m)}_{t}\to W_{t}=\theta 1_{t\in(\frac{1}{2},1]}\, and \,X^{(m)}_{t}\to X_{t}=\begin{pmatrix}x_{t}\ y_{t}\end{pmatrix}=\begin{pmatrix}1\ 0\end{pmatrix}1_{t\in[0,\frac{1}{2}]}+\begin{pmatrix}\cos{\theta}\ \sin{\theta}\end{pmatrix})1_{t\in(\frac{1}{2},1]}.

If \theta\neq 0 , (X_{t})_{t\in[0,1]} fails to be the right-hand side of the limiting forward RDE since

\int_{0}^{1}\begin{pmatrix}-y_{s-}\ x_{s-}\end{pmatrix}\mathrm{d}W_{s}=\lim_{|\mathcal{P}|\to 0}\sum_{[s,s^{\prime}]\in\mathcal{P}}\begin{pmatrix}-y_{s-}\ x_{s-}\end{pmatrix}(W_{s^{\prime}}-W_{s})=\begin{pmatrix}-0\ 1\end{pmatrix}\times\theta=\begin{pmatrix}0\ \theta\end{pmatrix} eq X_{1}-X_{0}.

Effectively, (X_{t})_{t\in[0,1]} is the solution to the Marcus RDE in the limit.
C.2 Proof of Theorem C.1
If we write V_{n}(t)=\lfloor tn\rfloor/n , the assumptions of Theorem C.1 and the same treatment in [ 6 ] implies that up to subtracting a subsequence, (V_{n},W_{n})\to(\mathrm{id},\mathrm{L}) a.s. in \alpha_{p\text{-var}} for any p>\alpha . Then it follows from Theorem C.2 that for p^{\prime}>p and along each subsequencial limit of (V_{n},W_{n}) as n\to\infty ,

((W_{n},X_{n}),l_{r+d})\to((\mathrm{L},X),\phi_{a,b})\, in \,(\mathcal{D}([0,1],\mathbb{R}^{r+d}),\alpha_{p^{\prime}-var}).

Therefore ((W_{n},X_{n}),l_{r+d})\overset{\mathrm{(d)}}{\to}((\mathrm{L},X),\phi_{a,b})\,\text{ in }\,(\mathcal{D}([0,1],\mathbb{R}^{r+d}),\alpha_{p\text{-var}}) for any p>\alpha . This completes the proof.

∎
F.1 Electrocardiogram (ECG) Classification
We consider the Electrocardiogram (ECG) binary classification task that aims to discriminate between normal and abnormal heart beats of a patient that has severe congestive heart failure [ 12 ] . We use 500 sequences of
length 140 for training, and 4000 sequences for testing. We use a fully connected shallow neural network of width 32 with sigmoid activation and train for 1000 epochs, with the binary cross-entropy as the loss. We choose \eta=0.05 . For the MPGD and the Gaussian scheme, we inject the perturbations in the first 500 epochs instead, and stop injecting after that to allow the algorithm to converge. We use the perturbation level \mu=\sigma=0.2 . The experimental setup is implemented in PyTorch, and all experiments are run on Google Colab.

Table 3 shows the average test accuracy (evaluated for 5 models that are trained with different seed values) for this task.
We see that overall MPGD improves the test accuracy when compared to the vanilla GD and Gaussian perturbations. In fact, the vanilla GD gets stuck in the loss landscape. While Gaussian perturbations can help to mobilize the GD iterates in the landscape, MPGD is more effective in steering the iterates to achieve higher test accuracy (see Figure 3 ), which is the main goal of the learning task. This illustrates that adding the perturbations of MPGD can help improve the outcome of the optimization process in situations where the vanilla GD and Gaussian perturbations fail to make meaningful progress.

Note that the higher values of the perturbation levels used in the MPGD schemes lead to instabilities and fluctuations of large magnitudes in the earlier epochs. These are necessary to boost the test accuracy; see the jump in the improvement of the test accuracy after epoch 500 in Figure 3 . The instabilities and fluctuations could be reduced if lower perturbation levels were used, but this would lower the test accuracy (which would still be higher than the baseline and the Gaussian results) obtained. Again, we emphasize that we are not going after competitive test performance here, but rather demonstrating the effectiveness of MPGD in improving the test performance in situations when other training schemes fail to make substantial progress in optimization.
F.2 Details and Additional Results for the Airfoil Self-Noise Prediction Task
The experimental setup is implemented in PyTorch, and all experiments are run on 4x NVIDIA Tesla T4 GPUs with 16 GB VRAM belonging to an internal SLURM cluster.
Table 4 reports the sample standard deviation for the results obtained in Table 1 .
Table 5 shows the corresponding results when different values of \beta are used instead, illustrating the trade-offs induced by the selection of the stability parameter \gamma and the skewness parameter \beta for this particular setting. Here, we see that using the MPGD variant ( 11 ) with \gamma=0.6 and \beta=0.5 (see Table 1 ) leads to the lowest RMSE gap.
F.3 Details for the CIFAR-10 Classification Task
We use the implementation of ResNet18 and modify the implementation of the full-batch GD training provided at https://github.com/JonasGeiping/fullbatchtraining to set up MPGD and GD with Gaussian perturbations. Please refer to Section 5 and [ 13 ] for the relevant details. The experimental setup is in PyTorch, and all experiments are run on an NVIDIA A100-SXM4 GPU with 40 GB VRAM belonging to an internal SLURM cluster. Table 2 reports the mean validation accuracies over runs of 5 models trained with different seed values, whereas Table 6 reports the sample standard deviation for the results in Table 2 .
Figure 1: A realization of the Thaler iterates (left) and the observable v (middle) for \gamma=0.6 , \beta=0.1 . Empirical distribution of the corresponding Birkhoff sum \frac{1}{k^{\gamma}}\sum_{j=0}^{k-1}v^{(j)} with k=1000 (right).; A realization of the Thaler iterates (left) and the observable v (middle) for \gamma=0.6 , \beta=0.1 . Empirical distribution of the corresponding Birkhoff sum \frac{1}{k^{\gamma}}\sum_{j=0}^{k-1}v^{(j)} with k=1000 (right).
Figure 2: Evolution of GD with Gaussian perturbations (left plot) vs. that of MPGD with \gamma=0.7 , \beta=0.5 (middle plot), using \mu=0.02 , \sigma=0.05 , and \eta=0.01 ( m=100 ). Here we see that MPGD leads to successful optimization of the widening valley loss whereas that with Gaussian perturbations fails to converge. Moreover, MPGD effectively reduces the trace of loss Hessian (see right plot), steering the GD iterates to flatter region of the loss landscape.; Evolution of GD with Gaussian perturbations (left plot) vs. that of MPGD with \gamma=0.7 , \beta=0.5 (middle plot), using \mu=0.02 , \sigma=0.05 , and \eta=0.01 ( m=100 ). Here we see that MPGD leads to successful optimization of the widening valley loss whereas that with Gaussian perturbations fails to converge. Moreover, MPGD effectively reduces the trace of loss Hessian (see right plot), steering the GD iterates to flatter region of the loss landscape.
Figure 3: Mean test accuracy obtained under various GD schemes over 1000 epochs for the ECG classification task. Here we choose the learning rate \eta=0.05 . MPGD refers to the scheme ( 8 ), whereas MPGD variant refers to ( 11 ), both using \mu=\sigma=0.2 . The MPGDs considered in the left plot use \gamma=0.55,\beta=0 , whereas those in the right plot use \gamma=0.55,\beta=0.5 . We see that applying the MPGD schemes in the first 500 epochs helps to boost the test accuracy significantly when compared to the other two schemes.; Mean test accuracy obtained under various GD schemes over 1000 epochs for the ECG classification task. Here we choose the learning rate \eta=0.05 . MPGD refers to the scheme ( 8 ), whereas MPGD variant refers to ( 11 ), both using \mu=\sigma=0.2 . The MPGDs considered in the left plot use \gamma=0.55,\beta=0 , whereas those in the right plot use \gamma=0.55,\beta=0.5 . We see that applying the MPGD schemes in the first 500 epochs helps to boost the test accuracy significantly when compared to the other two schemes.
Shallow neural nets trained on the airfoil data set. The results in parenthesis are achieved with the variant ( 11 ).
All the results are averaged over 5 models trained with different seed values.
ResNet-18 trained on CIFAR-10 for 1000 epochs. Here, accuracy gap = training accuracy - validation accuracy. The results in parenthesis are achieved with the variant of MPGD ( 11 ). All the results are averaged over 5 models trained with different seed values.
Shallow neural networks trained on the ECG5000 Data Set for 1000 epochs. The results in parenthesis are achieved with the variant of MPGD ( 11 ). All the results are averaged over 5 models trained with different seed values. Here \rm{std} denotes sample standard deviation. We use \eta=0.05 , and \mu=\sigma=0.2 for the perturbation schemes.
Statistics for the results obtained in Table 1 . We report the sample standard deviation, denoted \rm{std} , of the test RMSE and RMSE gap. The values in parenthesis refer to the sample standard deviation results for the MPGD ( 11 ).
Mean and standard deviation of RMSE gap under the setting used for obtaining the results in Table 1 but with \beta=0 (left) and \beta=0.25 (right) for the MPGDs instead. The values in parenthesis refer to the corresponding results for the MPGD ( 11 ). Here \rm{std} denotes sample standard deviation, and \eta,\mu and \sigma are the same as the ones used for obtaining the results in Table 1 .
Statistics for the results obtained in Table 2 . We report the sample standard deviation (in %), denoted \rm{std} , for the validation accuracy and the accuracy gap. The values in parenthesis refer to the sample standard deviation results for the MPGD variant ( 11 ).