Throughout this section, we will assume the same variables as in Eq.1-5 and Section 3 by default, We omit the soft constraints in original objective. "draw the optimal point back" from too far, and thus we can ignore the points outside the When we consider the effect of violating constraints, we only consider non-degraded activated cone, i.e.,, for an activated cone in a Thus it is straightforward to apply our proofs to the general form of Eq.5. A.2 Lemma 7 In order to prove Theorem 2, we give a crucial lemma. Since K is non-degraded, we have r + q n . With equations in Eq. 17, we get See Theorem 3 for the meaning of union.

MPC has the closest discrepancy to the given non-optimal demonstrations (e.g., when the task loss is

We propose Differentiable Temporal Logic (DTL), a model-agnostic framework that introduces temporal constraints to deep networks. DTL treats the outputs of a network as a truth assignment of a temporal logic formula, and computes a temporal logic loss reflecting the consistency between the output and the constraints. We propose a comprehensive set of constraints, which are implicit in data annotations, and incorporate them with deep networks via DTL. We evaluate the effectiveness of DTL on the temporal action segmentation task and observe improved performance and reduced logical errors in the output of different task models. Furthermore, we provide an extensive analysis to visualize the desirable effects of DTL. Figure 1: A video of activity "coffee preparation". The colored bars, from the top to the bottom, show the ground truth (GT), the predictions from a baseline [ 15 ], and the predictions from the baseline trained with DTL, respectively.

Work done while at Harvard University.

We thank all reviewers for their insightful comments. "repulsive/diverse functions" also makes it more likely to violate a constraint; for example, Figure S1 below shows that (instead of 50). We acknowledge points about similarity to [18] made by R4. Equation (1) in Definition 4.2 should read: