Mapping machine-learned physics into a human-readable space

We present a technique for translating a black-box machine-learned classifier operating on a high-dimensional input space into a small set of human-interpretable observables that can be combined to make the same classification decisions. We iteratively select these observables from a large space of high-level discriminants by finding those with the highest decision similarity relative to the black box, quantified via a metric we introduce that evaluates the relative ordering of pairs of inputs. Successive iterations focus only on the subset of input pairs that are misordered by the current set of observables. This method enables simplification of the machine-learning strategy, interpretation of the results in terms of well-understood physical concepts, validation of the physical model, and the potential for new insights into the nature of the problem itself. As a demonstration, we apply our approach to the benchmark task of jet classification in collider physics, where a convolutional neural network acting on calorimeter jet images outperforms a set of six well-known jet substructure observables. Our method maps the convolutional neural network into a set of observables called energy flow polynomials, and it closes the performance gap by identifying a class of observables with an interesting physical interpretation that has been previously overlooked in the jet substructure literature.

Figure 1

Schematic of the black-box guided search in Sec. 2b. In each iteration of this strategy, the relative decision ordering of signal/background pairs between the fixed black-box network (BBN, black triangle) and a trainable network of HL observables (white triangle) is used to identify the subset (red box) in which pairs are differently ordered. From a large space of HL observables (circles), the one with the largest ADO in the misordered space (blue circle) is selected for the next iteration. The schematic above corresponds to the $n=4$ iteration. Note that the BBN is not retrained in each iteration, but the network of HL observables is.

Figure 2

Similarity of the classification decisions between the six HL observables, as quantified by the ADO. A value of $\mathrm{ADO}=1$ indicates identical decision ordering for all signal/background pairs, while $\mathrm{ADO}=\frac{1}{2}$ corresponds to no similarity. In this way, the ADO has a similar interpretation to the AUC, but with respect to classification decisions instead of ground truth.

Figure 3

Classification performance of the six HL observables in Table 1 on the diagonal entries, along with AUC values for pairwise classification between coupled HL inputs on the off-diagonal entries.

Figure 4

The top three EFPs from the black-box guided search for a seventh HL observable; see Table 2. Shown are the EFP distributions for signal and background events, both in the full set of events $X_{\mathrm{all}}$ (left column) and in $X_6$ (right column), i.e., the differently ordered space between the 6HL and the CNN. The top two observables, while not identical, have very similar functional forms, up to an overall rescaling. The third observable is the jet constituent multiplicity.

Figure 5

Performance of the black-box guided search strategy to map the CNN solution into human-interpretable observables. Here, we start from just the basic jet features $p_{\mathrm{T}}$ and $M_{\text{jet}}$ and iteratively add one EFP at a time. The performance is shown in terms of the AUC (top) and the ADO (bottom) as a function of the scan number. The performance of a brute force scan of the EFP space (Sec. 5b) and a truth-label guided search (Sec. 5c) are also shown. For reference, the performance of the CNN and of the existing 6HL features are indicated by horizontal lines.

Figure 6

The same as Fig. 5, but now plotted in terms of the cumulative computing time.

Figure 7

The first four EFP graphs selected by the black-box guided strategy beginning from the minimal set of HL observables, $p_{\mathrm{T}}$ and $M_{\text{jet}}$; see Table 3. Shown are the EFP distributions for signal and background events, both in the full set of events $X_{\mathrm{all}}$ (left column) and in $X_n$ (right column), i.e., the differently ordered space between the ${\mathrm{HLN}}_n$ and the CNN after $n$ iterations.