Automating the construction of jet observables with machine learning

Machine-learning assisted jet substructure tagging techniques have the potential to significantly improve searches for new particles and Standard Model measurements in hadronic final states. Techniques with simple analytic forms are particularly useful for establishing robustness and gaining physical insight. We introduce procedures to automate the construction of a large class of observables that are chosen to completely specify $M$-body phase space. The procedures are validated on the task of distinguishing $H\to b\overline{b}$ from $g\to b\overline{b}$, where $M=3$ and previous brute-force approaches to construct an optimal product observable for the $M$-body phase space have established the baseline performance. We then use the new methods to design tailored observables for the boosted $Z^{\prime }$ search, where $M=4$ and brute-force methods are intractable. The new classifiers outperform standard two-prong tagging observables, illustrating the power of the new optimization method for improving searches and measurement at the LHC and beyond.

Figure 1

A schematic diagram of the network architecture used to produce the probability distribution of ${\beta }_M$ for a given set of input parameters $\{a,b,c,\dots \}$. In this case, $N=25,000$.

Figure 2

(a) Comparison of the probability density function of the new ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}}$ observables for ungroomed $H\to b\overline{b}$ discrimination, using $\sim 500,000$ signal and background samples, and the distributions of the regression DNN prediction. The distributions are rescaled by a constant for the sake of visual comparison. (b) Probability densities of ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}}$ obtained via linear regression. (c) Comparison of discrimination power of ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}}$ and ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}}$ to standard observables. (d) Comparison of ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}}$ and ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}}$ to ${\beta }_3$ proposed in Ref. [8]; we note that three observables provide essentially the same discrimination power.

Figure 3

$M$-body discrimination results for ungroomed $Z^{\prime }$ vs QCD jets. Discrimination power is effectively saturated at 4-body phase space for each case.

Figure 4

Top panels [(a)–(c)]: Comparison of the probability density function of the new ${\beta }_4^{\mathrm{ML}}$ observables for ungroomed $Z^{\prime }$ discrimination, calculated for $\sim 500,000$ signal and background samples, and the distributions of the regression DNN predictions of 25,000 observable values. The distributions are rescaled for the sake of visual comparison. Bottom panels [(d)–(f)]: Distributions of the ${\hat{\beta }}_4^{\mathrm{ML}}$ observables for ungroomed $Z^{\prime }$ discrimination that were obtained via linear regression.

Figure 5

Top panels [(a)–(c)]: Comparison of discrimination power of ${\beta }_4^{\mathrm{ML}}$ observables to standard observables; the latter are calculated with an angular exponent of 1, for which they were observed to perform best. Bottom panels [(d)–(f)]: Comparison of ${\beta }_4^{\mathrm{ML}}$ to discrimination power of neural networks trained on the $M$-body observable bases; the observables seem to capture increasing amounts of the discrimination power of the 3- and 4-body neural networks with increasing $m_{Z^{\prime }}$.

Figure 6

Here we plot results for the calculation of the AUC from the distributions generated by the signal and background regression DNNs for the ungroomed $m_{Z^{\prime }}=90\mathrm{GeV}$ case (blue dashed curve) along with the true AUC (red solid) computed on the same statistics. These are seen to be in good agreement with each other. The results demonstrate the usefulness of the networks to accurately reproduce the change in the signal and background PDFs, represented via the accurate reproduction of the AUCs calculated from them, as a function of the variation of each individual input parameter $\{a,\dots ,h\}$.

Figure 7

(a) Comparison of PDFs of ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$ for mMDT groomed $H\to b\overline{b}$ discrimination, using $\sim 250,000$ signal and background samples, and the distributions of the regression DNN prediction. The distributions are rescaled by a constant for the sake of visual comparison. (b) Probability density distributions of ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$ obtained via linear regression. (c) Comparison of discrimination power of ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$, ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$ and ${\widehat{\beta }}_{4,H\to b\overline{b}}^{\mathrm{ML}(g)}$ to standard observables, where the 4-body product observable is seen to perform best for groomed $H\to b\overline{b}$ discrimination. (d) Comparison of ${\widehat{\beta }}_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$ and ${\beta }_{3,H\to b\overline{b}}^{\mathrm{ML}(g)}$ to ${\beta }_3^{(g)}$ proposed in [8]; we note that the latter two 3-body product observables provide essentially the same discrimination power while the 3- and 4-body ones obtained with linear regression outperform them.

Figure 8

$M$-body discrimination results of mMDT groomed $Z^{\prime }$ vs QCD jets. Here, discrimination power is again seen to effectively saturate at 4-body phase space for all considered values of $m_{Z^{\prime }}$.

Figure 9

Top panels [(a)–(c)]: Comparison of the probability density function of the new ${\beta }_4^{\mathrm{ML}(\mathrm{g})}$ observables for mMDT groomed $Z^{\prime }$ discrimination, calculated for $\sim 300,000$ signal and background samples, and the distribution of the regression DNN predictions of 25,000 observable values. The distributions are rescaled for the sake of visual comparison. Bottom panels [(d)–(f)]: Distributions of the ${\widehat{\beta }}_4^{\mathrm{ML}(\mathrm{g})}$ observables for ungroomed $Z^{\prime }$ discrimination that were obtained via linear regression.

Figure 10

Top panels [(a)–(c)]: Comparison of discrimination power of ${\beta }_4^{\mathrm{ML}(\mathrm{g})}$ observables to standard observables; the latter are computed with an angular exponent of 2, for which they were observed to perform best for mMDT groomed samples. Bottom panels [(d)–(f)]: Comparison of ${\beta }_4^{\mathrm{ML}(\mathrm{g})}$ to discrimination power of neural networks trained on the $M$-body observable bases; the observables capture almost all of the discrimination power of the 4-body neural networks.

Figure 11

Here we plot results for the new observables on $Z^{\prime }$ samples with a different mass point to that which they were optimized on, within the mass windows appropriate for the corresponding signal. We note that for all cases, all the new observables demonstrate very similar discrimination power, and outperform standard observables.

Figure 12

Here we compare the discrimination power of ${\widehat{\beta }}_{M,Z_{90}^{\prime }}^{\mathrm{ML}}$ for ungroomed $Z^{\prime }$ discrimination for values of $M=3,\dots ,8$. We note that while discrimination power of the product form does increase with higher $M$ (until the inclusion of 7- or 8-body phase space variables), it can only capture a limited amount of useful discriminating information from inclusion of variables from beyond the basis of the point of saturation of a DNN classifier (dark gray).