On the Uncomputability of Partition Functions in Energy-Based Sequence Models

1 Introduction . Modeling discrete sequences is central to natural language processing and bioinformatics . Many common parametric sequence models ? ̃ are ( or can be cast as ) energy-based ( LeCun et al. , 2006 ) : they yield a weight ? ̃ ( x ) for any given string x . Although energy-based models ( EBMs ) of sequences need not represent probability distributions , they have been proposed as such to counter the inexpressivity of autoregressive sequence models ( Bakhtin et al. , 2021 ; Lin et al. , 2021 , §4.1 ) . EBMs with finite partition functions / ? ̃ , ∑ x ? ̃ ( x ) ∈ R > 0 define distributions ? ( x ) , ? ̃ ( x ) // ? ̃ over strings , which are often queried under the principle of probabilistic inference ( Ghahramani , 2015 ) .1 Energy-based sequence models are often parametrized as neural models . Each parameter vector ) in some parametric neural network family ⊆ R3 identifies a parametric model ? ̃ ) , which then defines a parametric distribution over strings ? ) , assuming / ? ̃ ) exists ( Chen et al. , 2018 ) . Contemporary neural networks have been shown to be very powerful . In particular , some popular parametric neural sequence model families , such as RNNs ( Siegelmann & Sontag , 1995 ) and Transformers ( Bhattamishra et al. , 2020 ; Pérez et al. , 2021 ) , have been formally shown to recognize recursively enumerable languages : given any Turing machine `` , there is a parameter vector ) ∈ such that the parametrized sequence model # ) recognizes the same language as `` does . In other words , these sequence model families are Turing-complete . It is therefore intuitive that energy-based sequence models , backed by such powerful neural networks , form expressive families of string distributions ( §3.1 ) : for any decision algorithm `` , there is a parameter vector ) '' in a Turing-complete parametric family , such that ? ) '' exists , and ? ) '' ( x ) is high if and only if `` accepts x . It would seem assuring to work with such expressive family of sequence distributions , : assuming the true string probabilities indeed can be computed in polytime , is well-specified . Moreover , one may assume that given we can sample from ? ) '' , we would be able to use consistent estimators to find ) ′ ∈ , where ) ′ ≈ ) '' . Unfortunately , we find that with such an expressive distribution family , whether the identifiability assumption holds— required for most consistent estimators— itself is undecidable ( Turing , 1937 ) . 1Many popular energy-based sequence models compute normalized probabilities directly ( i.e. , ? ̃ ( x ) = ? ( x ) ) ( Jelinek , 1980 ; Katz , 1987 ; Bengio et al. , 2003 ; Brown et al. , 2020 , inter alia ) . This makes both training and querying with string prefixes much easier , at the cost of expressiveness ( Lin et al. , 2021 ) . Moreover , model selection on any held-out data is also undecidable . Even worse , as long as a parametric family ′ contains one specific vector ) ′ ∈ , model selection will not be possible for ′ either— even when ′ itself is not necessarily expressive ( e.g. , ′ can be fixed-size Transformer EBMs , which can not parametrize all EBMs that require more parameters ) . We construct one such ‘ pathological ’ distribution ? ) ′ as a Transformer EBM . These negative results stem from the uncomputability and inapproximability of / . A main technical contribution of this paper is that there is no algorithm ( either deterministic or randomized ) that can approximate / well . Since statistical procedures are also algorithms , common randomized estimates are not useful in this scenario , either—however , we will see that for less expressive model families , such uncomputability issues do not arise . Our negative results are summarized in Table 1 . To ensure that model selection is still possible ( such that we can compare different parameter vectors with confidence ) ,2 we have no choice but to resort to less expressive string distributions . The paper is structured as follows . In §2 we review definitions and known results of weighted languages , sequence model families , and formalize weighted Turing machines , EC-complete parametric families , and computable estimates . In §§3–5 we describe our main technical results : there exist pathological EBM sequence models that have uncomputable partition functions , which can not be approximated well under stochastic estimates , and do not have asymptotic estimates that have any good guarantees . In §6 we that argue our negative results make model selection impossible for expressive model families , and discuss why common estimation methods fail . Finally , we discuss three ‘ palliative ’ parametrization choices in §7 , which all guarantee computable partition functions , at the cost of expressiveness . 2 Background . 2.1 Energy-based sequence models . Energy-based models ( EBMs ) of sequences ( LeCun et al. , 2006 ; Rosenfeld et al. , 2001 ; Sandbank , 2008 ; Huang et al. , 2018 ; Bakhtin et al. , 2021 ) are a family of discrete sequence models . Given a string x ∈ +∗ over a finite vocabulary + , an EBM ? ̃ computes string weight ? ̃ ( x ) , but not the ( normalized ) string probability ? ( x ) = ? ̃ ( x ) /∑x ? ̃ ( x ) . In this work , we focus on EBMs whose weight is efficiently computable , i.e. , polytime in the length of x . Following Bakhtin et al . ( 2021 ) and Lin et al . ( 2021 ) , we wish to use our EBMs as probability distributions ; this requires normalization . While EBMs are often intractable to normalize ( e.g. , the Ising model ) , the infinite domain of finite strings opens the door to uncomputable probabilities , which invite the impossibility of model selection and comparison . 2.2 Weighted languages . EBMs give weights to strings . Here we formally characterize these weighted strings as a weighted language . An unweighted language ! ⊆ +∗ is a set of finite strings x over a finite vocabulary + . A weighted language ? ̃ is a function ? ̃ : +∗ → R≥0.3 In this work , we discuss Boolean languages , such that + = B , { 0 , 1 } . We also focus on weighted languages where distributions over strings exist . We say a weighted language ? ̃ is normalizable when / ? ̃ , ∑ x∈B∗ ? ̃ ( x ) ∈ R > 0 : the sum over all possible 2Of course , model selection ( and computing / ? ̃ ) are not always needed— for example , simply deciding whether ? ̃ ( x1 ) > ? ̃ ( x2 ) for two given strings x { 1,2 } . In such case the uncomputability issues we discuss are not a concern . 3With a slight abuse of notation , we define support ( ? ̃ ) , { x : ? ̃ ( x ) > 0 } . strings is finite and positive . / ? ̃ is also called the partition function of ? ̃.4 We can then normalize ? ̃ into a distribution ? over B∗ such that ? ( x ) = ? ̃ ( x ) // ? ̃ , and thereby ∑ x∈B∗ ? ( x ) = 1 . The efficiently computable ( weighted ) languages ( EC ; Lin et al. , 2021 ) are those weighted languages ? ̃ , where a string ’ s weight ( which must be non-negative ) is a polytime function of the string . Weighted languages defined by most EBM sequence models fall into this class , since they score every string in finite time ( and usually in polytime ) ; and the scores ultimately are computed by some algorithm . By definition , string weights of an EC language can be obtained as a side product from a Turing machine that recognizes x ( in polytime ) . However , Lin et al . ( 2021 ) did not precisely describe how a Turing machine maps input string x to its ( rational ) weight ? ̃ ( x ) in finite time . In this work , such a construction ( out of many possible ones ) is necessary , as we need to show that this string weighting can be done by parametric sequence model families ( see Appendix A ) . 2.3 Locally normalized distributions . A popular subclass of weighted languages is locally normalized weighted languages ( LN ) , where conditional local distributions given prefix x̂ : ? ( · | x̂ ) can be computed in finite time . Since they automatically define a distribution over strings , we use the term locally normalized distributions interchangeably . If ? ( · | x̂ ) can be computed in polynomial time , we call such distributions efficiently locally normalized distributions ( ELN ) — this is the weighted language class of most autoregressive models , just as its superset EC is the weighted language class of most energy-based sequence models . Locally normalized distributions over sequences have ? ̃ ( x ) = ? ( x ) , and / = ∑x∈B∗ ? ̃ ( x ) = 1 . They are consistent : the probability that a string is infinitely long under such distributions is zero . Equivalently , given any n > 0 , we can approximate / with a finite sum of string weights : Proposition 1 ( Consistency of LN distributions ( Booth & Thompson , 1973 ; Chen et al. , 2018 ) ) . Let ? ∈ LN be a locally normalized distribution over strings . All strings of a given length or longer have their probabilities bounded . That is , for all positive real numbers n , there is a length = at which all strings x of length at least = have ? ( x ) < n . The consistency property of locally normalized distributions implies that they have an exact sampling procedure that almost surely terminates in finite time . Therefore , they ( and in particular ELN distributions ) are an attractive choice as proposal distributions in parameter estimation procedures that alternate between expectation and maximization steps ( Dempster et al. , 1977 ) . 2.4 EC-complete parametric families . We have introduced energy-based sequence models ( §2.1 ) and their characterization as weighted languages ( §2.2 ) . However , we usually do not work with Turing machines directly in machine learning . Here the most common models of computation are ( neural ) sequence model families , such as RNNs and Transformers . While those computational models can appear quite dissimilar to state-machine-based models of computation ( e.g. , Turing machines ) , they have been shown to possess the same computation power ( Siegelmann & Sontag , 1995 ; Pérez et al. , 2021 ) . That is , they are Turing-complete . Just as we extend the definition of Turing machines to weighted Turing machines , we likewise formalize weighted sequence model families . We thus introduce EC-complete parametric families as a sequence model counterpart of the weighted language class EC . At a high level , a parametric family is EC-complete if given any ? ̃ ∈ EC ( as a description of a weighted Turing machine ) , we can construct a parameter vector ) ∈ ⊆ Q∗ which defines ? ̃ ’ s corresponding sequence model . The model produces an output embedding which we then decide in polytime to be either a ‘ halting embedding ’ or not . If it is , we can then extract the weight ? ̃ ( x ) . ( A rigorous exposition is in Appendix H. ) We show that with a slight modification to positional embeddings , the family of arbitrary precision , hard attention Transformers defined in Pérez et al . ( 2021 ) is EC-complete:5 Theorem 1 . The class of one-encoder-layer four-decoder-layer Transformer networks with positional encodings ( = , 1/= , 1/=2 , 2= ) is EC-complete . 4We use the convention that / ? ̃ is the partition function of ? ̃ , and / @ ̃ of @ ̃ , etc . The subscript is omitted in unambiguous cases . 5All proofs omitted from the main text are included in the appendices . Proof sketches for major theorems are in the main text . Proof sketch . We extend the construction from Pérez et al . ( 2021 ) , adding an additional layer to accumulate the string weight over time steps .

Summary: This paper studies "efficiently computable (EC) energy-based sequence models (EBMs)" which are simply sets of strings equipped with nonnegative weights that can be computed by a poly-time Turing machine (upon normalization, the weights induce a probability distribution over the strings). In practice, it is common for these weights to be computed by neural networks. In fact, certain neural sequence model families like RNNs and Transformers are expressive enough to be Turing-complete. In this paper, the authors find that the expressivity of such sequence model families, so-called "EC-complete parametric families", comes at a significant cost in terms of *computability/decidability* of various primitives for inference. For instance, they show that if one could actually take in any vector of parameters specifying a model in such a family and output the corresponding partition function (sum of the weights of all strings) even approximately in expectation, then would be able to decide the halting problem (Theorem 2, 4). They also exhibit a single efficiently computable EBM for which proving (within ZFC) that the partition function is one of two possible values would either disprove Godel's second incompleteness theorem or disprove consistency of ZFC (Theorem 3); this can be extended to show that *asymptotic* estimates like rejection sampling and importance sampling will also fail (Theorems 5, 6). Along similar lines, they can reduce from the halting problem to other tasks like deciding whether two given parameter vectors give rise to the same EBM ("parameter identifiability", Theorem 7), or deciding which of two given parameter vectors gives rise to an EBM which is distributionally closer to some other given EBM ("model selection", Theorem 8).  In terms of techniques, the reductions from the halting problem are all based on a simple weight function that is tiny unless if the string corresponds to a valid accepting trace of an input-free Turing machine M. The point is that the weight for any string x can be computed by M simply by simulating M on x, but the partition function is large iff M halts. The ZFC results follow by taking M which enumerates all provable propositions under ZF and halts iff it proves 1 = 0.