Joint learning of constraint weights and gradient inputs in Gradient Symbolic Computation with constrained optimization (2020)

Abstract: This paper proposes a method for the joint optimization of constraint weights and symbol activations within the Gradient Symbolic Computation (GSC) framework. The set of grammars representable in GSC is proven to be a subset of those representable with lexically-scaled faithfulness constraints. This fact is then used to recast the problem of learning constraint weights and symbol activations in GSC as a quadratically-constrained version of learning lexically-scaled faithfulness grammars. This results in an optimization problem that can be solved using Sequential Quadratic Programming.

## introduction and background this paper proposes a method for the joint optimization of constraint weights and symbol activations within the gradient symbolic computation (gsc) framework. the set of grammars representable in gsc is proven to be a subset of those representable with lexically-scaled faithfulness constraints. this fact is then used to recast the problem of learning constraint weights and symbol activations in gsc as a quadratically-constrained version of learning lexically-scaled faithfulness grammars. this results in an optimization problem that can be solved using sequential quadratic programming. the remainder of this paper proceeds as follows. the rest of this section provides the relevant background on gsc, previous approaches to the same problem, and maximum entropy grammars which are used in the proposed model. §2 describes and proves the relationship between gsc grammars and lexically-scaled faithfulness constraints and then uses this proof to develop the proposed learning algorithm. §3 illustrates with a minimal test case of an example used through the gsc literature, french liaison. §4 provides a brief discussion and concludes. ## an example to illustrate the promise of the proposed approach to learning gs grammars, this section applies it to a minimal example of the french liaison problem that @xcite use to motivate the use of gradient representations in the phonological grammar. liaison is a phenomenon in which, in certain syntactic contexts, a consonant surfaces between vowel-final and vowel-initial words when hiatus would otherwise occur. the identity of this consonant, the liaison consonant, is not phonologically predictable. there is a long literature on the phonological analysis on liaison and its interacting processes, including competing analyses that propose that the liaison consonant is specified by the first word @xcite and by the second word in the sequence @xcite . there is a class of words which are phonologically vowel initial but exceptionally do not trigger the surfacing of a liaison consonant in environments where it is otherwise predicted to surface. these words are always the second word in the pair and are called the h-aspiré words, referencing the fact that they are orthographically h-initial. consider the following set of french surface forms. when petit comes together with ami, a vowel-initial word, the liaison consonant [t] surfaces. the adjective [pøti] petit is associated with a liaison t. when it occurs in isolation the liaison consonant does not surface, however when it occurs before the vowel-initial [ami] ami the liaison consonant surfaces, preventing two adjacent vowels from surfacing. despite being vowel-initial, the h-aspiré word [eko] héros does not trigger the surfacing of the liaison consonant when it surfaces after peti. smolensky and goldrick (2016) offer an analysis of this phenomenon couched in gradient symbolic computation, which suggests that the liaison consonant is specified by both the first and second word in the pair. in their analysis both words contain partially active edge consonants. when the words surface together the combined activation is enough to get cause the liaison consonant to surface. in this analysis h-aspiré words differ from their liaisonparticipating counterparts in that they have no or minimal activation on liaison consonants at their left edge, preventing them from contributing to the combined activation. in terms of the minimal dataset above, they propose that there is a partially-activated /t/ in the input at both the right edge of peti and at the left edge of ami. when either word occurs in isolation there is not sufficient activation of the /t/ for it to surface. when the two words surface adjacent to one another the combined activation of /t/ in both words overcomes a threshold and liaison @xcite surfaces. in the h-aspiré héros there is little to no activation on an input /t/ at the left edge. despite the consequence of realizing a marked vowel-vowel sequence, the liaison [t] does not surface between @xcite because the combined activation of the input /t/s is not enough to justify its realization. they argue that this analysis overcomes empirical shortcomings of analyses which place the onus of specifying the liaison consonant on exclusively the first or second consonant, see @xcite and @xcite for detailed discussion. as proof of concept a gs grammar was fit to these data using the procedure described above. model parameters include the weight of three constraints, hiatus, max(t) and dep(t), as well as the activation levels of liaison /t/ at the left edge of petit and at the right edge of ami and héros. max(t) is a prop constraint and dep(t) is a comp constraint. in every tableau there are two competing candidates, one in which [t] surfaces and one in which it does not. activations were constrained to being positive by adding the constraint in eq. ( 9 ) to the optimization procedure. in practice the jacobian and hessian of the objective function are estimated analytically, so the algorithm described above is non-deterministic. the quadratically-constrained optimization problem is also generally non-convex, so variation is expected across runs. consequently 10 models were fit with weights randomly initialized in @xcite ). an l2 prior is included with λ = 0.01. the activations suggest that the model may be converging on a solution that resembles the analysis proposed by smolensky and goldrick. petit and ami both have a partially-activated /t/ in the at the relevant edge, while the activation of liaison /t/ in héros is approximately 0. the individual tableaux confirm that the learned analysis resembles smolensky and goldrick's. for simplicity, and consistency with previous work, all tableaux will be presented without probabilities, as hg tableaux. while petit and ami both have partially-activated underlying /t/s, the activation is low enough that when either of these words occur in isolation the /t/ is not realized. this is demonstrated in tableaux (3) and ( 4 ). the presented learning algorithm for gs grammars reliably converges on the analysis of french liaison offered by @xcite as a motivating pattern for the inclusion of gradient inputs in the phonological grammar. this serves to illustrate the fact that the proposed learning algorithm is capable of learning interpretable gs grammars and has promising application in future work, both in finding gsc analyses of linguistic phenomena and in evaluating the learnability of phenomena in the gsc framework. ## discussion and conclusions this paper has presented a method for the joint optimization of blended inputs and constraint weights in gradient symbolic grammars. the proposed method leverages the fact that the set of functions representable by gs grammars is a subset of those representable by lexically-scaled faithfulness grammars to cast the gs grammar learning problem as a constrained version of the lsf grammar learning problem. the primary aim of this work is to introduce and justify the method, rather than discuss its implications for linguistic theory, however points of interest to linguistic theory will be briefly addressed here. the subset-superset relationship that was shown to hold between gs and lsf grammars does not make predictions regarding the expressivity of the two theories in terms of the linguistic phenomena they are capable of representing. it does, however, highlight differences between the two theories which may provide a starting point for comparing their linguistic expressivity. for example, representing gs grammars in the lsf framework requires a set of faithfulness constraints which make reference to every position in every input. this differs from standard approaches to positional faithfulness, where faithfulness constraints make reference to prosodic positions @xcite , and may yield pathological predictions. consequently, despite the fact that lsf grammars represent a greater range of functions, it is likely that there are phenomena that can be captured with gs grammars but not with lsf grammars given a limited constraint set. this is left to future work. this work has also shown that the optimization problem for gs grammars is likely more difficult than the analogous problem in other frameworks designed to capture the same types of phonological phenomena. for example, grammars with lexicallyscaled constraints like those mentioned throughout this paper have also been shown to capture lexical exceptionality and subregularity but, as described, they correspond to a convex optimization problem. similarly, grammars with underlying representation constraints have also been shown to be a viable approach to capturing these phonological phenomena @xcite and, in learning problems like that described in this paper present a convex optimization problem. the critical difference between these approaches and gs grammars is that harmony function for gs grammars is quadratic, consequently the optimization problem is not guaranteed to be convex. it is not necessarily the case that the complexity of the related optimization problems is a valid metric along which to compare linguistic theories. previous work however, has made strong claims regarding the relationship between the numerical optimization of max-ent/hg grammars and the learning trajectories of human language learners @xcite @xcite , in which case there may be merit in comparing the optimization procedure for competing theories. the broader gsc framework offers a novel theory of phonological grammars, the expressivity and restrictiveness of which has not been thoroughly explored. this work hopes to facilitate further research by introducing a method for simultaneously learning constraint weights and input activations of gs grammars which both relates gs grammars to an existing phonological framework and serves as a tool in finding gs analyses of phonological phenomena.