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+# Abductive Reasoning in Logical Credal Networks
+
+Radu Marinescu
+
+IBM Research, Ireland
+
+radu.marinescu@ie.ibm.com
+
+Junkyu Lee
+
+IBM Research, USA
+
+junkyu.lee@ibm.com
+
+Debarun Bhattacharjya
+
+IBM Research, USA
+
+debarunb@us.ibm.com
+
+Fabio Cozman
+
+Universidade de Sao Paulo, Brazil
+
+fgcozman@usp.br
+
+Alexander Gray
+
+Centaur AI Institute, USA
+
+alexander(gray@centaurinstitute.org
+
+# Abstract
+
+Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge. Unlike many existing formalisms, LCNs have the ability to represent cycles and allow specifying marginal and conditional probability bounds on logic formulae which may be important in many realistic scenarios. Previous work on LCNs has focused exclusively on marginal inference, namely computing posterior lower and upper probability bounds on a query formula. In this paper, we explore abductive reasoning tasks such as solving MAP and Marginal MAP queries in LCNs given some evidence. We first formally define the MAP and Marginal MAP tasks for LCNs and subsequently show how to solve these tasks exactly using search-based approaches. We then propose several approximate schemes that allow us to scale MAP and Marginal MAP inference to larger problem instances. An extensive empirical evaluation demonstrates the effectiveness of our algorithms on both random LCN instances as well as LCNs derived from more realistic use-cases.
+
+# 1 Introduction
+
+Probabilistic logic which combines probability and logic in a principled manner has emerged over the past decades as a unified representational and reasoning framework capable of dealing effectively with complex real-world applications that require efficient handling of uncertainty and compact representations of domain expert knowledge [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Logical Credal Networks or LCNs [11] were introduced recently as a probabilistic logic designed for representing and reasoning with imprecise knowledge. Unlike many existing probabilistic logics, LCNs have the ability to represent cycles (e.g., feedback loops) as well as allow specifying marginal and conditional probability bounds on logic formulae which may be important in many realistic usecases.
+
+Up until now, the work on LCNs has focused exclusively on marginal inference, i.e. efficiently computing posterior lower and upper probability bounds on a query formula. However, abductive reasoning tasks such as explaining the evidence observed in an LCN are equally important in many real-world applications. In probabilistic graphical models, these tasks are commonly known as MAP and Marginal MAP (MMAP) inference and have received extensive attention over the past decades [12, 13]. They are typically tackled efficiently with dynamic programming (e.g., variable elimination) or heuristic search (e.g., depth-first branch and bound) based algorithms [13, 14, 15, 16].
+
+Contribution. In this paper, we consider solving MAP and Marginal MAP inference queries in LCNs. Unlike in graphical models, an LCN encodes a set of probability distributions over its interpretations. Therefore, a complete or a partial explanation of the evidence which represents a complete or a partial truth assignment to the LCN's propositions may correspond to more than one distribution. Our work builds on very recent work on Marginal MAP inference for credal networks, a class of probabilistic graphical models that allow reasoning with imprecise probabilities [17]. We formally introduce the MAP and Marginal MAP tasks for LCNs as finding a complete or a partial truth assignment to the LCN's propositions with maximum lower (respectively, upper) probability, given some evidence in the LCN. We show how to evaluate such MAP assignments using exact marginal inference for LCNs and, subsequently, propose several search schemes based on depth-first search, limited discrepancy search and simulated annealing to solve these tasks in practice. We then extend a recent message-passing scheme for approximate marginal inference in LCNs [18] to handle effectively the MAP and MMAP inference tasks in LCNs as well as adapt the limited discrepancy search and simulated annealing methods to use an approximate evaluation of the MAP assignments during search. We experiment and evaluate our proposed exact and approximate algorithms on several classes of LCNs including random as well as more realistic LCN instances. Our results show that the search methods based on exact evaluation of the MAP assignments are limited to solving small size problems in practice, while the approximate message-passing scheme and, to some extent, the approximate search-based methods can scale to much larger problem instances. This is important because it allows us to tackle practical problems involving hundreds and possibly many thousands of propositions. The supplementary material includes additional details and experiments.
+
+# 2 Background
+
+We provide next a brief overview of basic concepts about LCNs and marginal inference in these models. Throughout the paper we will use the following notations. Logical propositions are denoted by uppercase letters (e.g.,  $A, B, C, \ldots$ ) while for sets of propositions we use boldfaced uppercase letters (e.g., A, B, C, ...). Truth assignments to propositions (i.e., literals) are denoted by either lowercase or uppercase letters, namely we use  $a$  or  $A$  to indicate that proposition  $A$  holds true, and  $\neg a$  or  $\neg A$  if  $A$  is false. Sets of literals are denoted by boldfaced lowercase letters (e.g., a, b, c, ...).
+
+# 2.1 Logical Credal Networks
+
+A Logical Credal Network (LCN) [11] is defined by a tuple  $\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$ , where  $\mathbf{A} = \{A_1,\dots ,A_n\}$  is a set of propositions (or atoms), and  $\mathcal{C}$  is a set of probability labeled sentences (or constraints) having the following two forms:
+
+$$
+\alpha \leq P (\phi) \leq \beta \tag {1}
+$$
+
+$$
+\alpha \leq P (\phi | \varphi) \leq \beta \tag {2}
+$$
+
+Here,  $\phi$  and  $\varphi$  are arbitrary propositional logic formulae<sup>1</sup> involving propositions in  $\mathcal{A}$  and logical connectives such as negation, disjunction and conjunction, and  $0 \leq \alpha \leq \beta \leq 1$  are lower and upper probability bounds, respectively.
+
+An LCN is associated with primal graph which is a directed graph  $G$  containing formula nodes and proposition nodes, as well as directed edges from each proposition node in a formula  $\phi$  to the formula node representing  $\phi$  (for type 1 sentences), and directed edges from each of the proposition nodes in  $\varphi$  to  $\varphi$ , a directed edge from  $\varphi$  to  $\phi$ , and bi-directed edges from  $\phi$  to the proposition nodes in  $\phi$ , respectively (for type 2 sentences) [11]. A parent of a proposition  $A$  in  $G$  is a proposition  $B$  such that there is a directed path in  $G$  from  $B$  to  $A$  in which all intermediate nodes are formulae. A descendant of a proposition  $A$  in  $G$  is a proposition  $B$  such that there is a directed path in  $G$  from  $A$  to  $B$  in which no intermediate node is a parent of  $A$  [11].
+
+An LCN is endowed with a Local Markov Condition (LMC) where a proposition node  $A$  is independent, given its parents, of all proposition nodes that are not  $A$  itself nor descendants of  $A$  nor parents of  $A$  [11]. Therefore, an LCN represents a set of probability distributions over all interpretations of its formulae that satisfy the constraints represented by the type (1) and (2) sentences as well as the constraints induced by the independence relations given by the local Markov condition [11].
+
+$$
+0. 0 5 \leq P (B) \leq 0. 1
+$$
+
+$$
+0. 3 \leq P (S) \leq 0. 4
+$$
+
+$$
+0. 1 \leq P (B \vee C | S) \leq 0. 2
+$$
+
+$$
+0. 6 \leq P (D | B \wedge C) \leq 0. 7
+$$
+
+$$
+0. 7 \leq P (\neg (X \oplus D) | C) \leq 0. 8
+$$
+
+![](images/1c84774e900dcb17de70aeba0463be7b70c1918764106acfa1b72cd06fa72c0b.jpg)  
+(a) LCN sentences  
+(b) Primal graph  
+Figure 1: A simple LCN and its primal graph.
+
+Example 1. Figure 1 describes a simple LCN whose sentences shown in Figure 1a state that: Bronchitis  $(B)$  is more likely than Smoking  $(S)$ ; Smoking may cause Cancer  $(C)$  or Bronchitis; Dyspnea  $(D)$  or shortness of breadth is a common symptom for Cancer and Bronchitis; in case of Cancer we have either a positive X-Ray result  $(X)$  and Dyspnea, or a negative X-Ray and no Dyspnea. Figure 1b shows the primal graph where the formula and proposition nodes are displayed as rectangles and shaded circles, respectively.
+
+# 2.2 Marginal Inference in Logical Credal Networks
+
+$$
+\sum_ {i = 1} ^ {m} p _ {i} = 1 \tag {3}
+$$
+
+$$
+p _ {i} \geq 0, \forall i = 1, \dots , m \tag {4}
+$$
+
+$$
+\alpha \leq \vec {I} _ {\phi} \odot \vec {p} \leq \beta \tag {5}
+$$
+
+$$
+\alpha \cdot \vec {I} _ {\varphi} \odot \vec {p} \leq \vec {I} _ {\phi \wedge \varphi} \odot \vec {p} \leq \beta \cdot \vec {I} _ {\varphi} \odot \vec {p} \tag {6}
+$$
+
+$$
+\left(\vec {I} _ {a} \odot \vec {p}\right) \cdot \left(\vec {I} _ {b} \odot \vec {p}\right) - \left(\vec {I} _ {c} \odot \vec {p}\right) \cdot \left(\vec {I} _ {d} \cdot \vec {p}\right) = 0 \tag {7}
+$$
+
+minimize/maximize  $\vec{l}_{\psi} \odot \vec{p}$
+
+Given an LCN  $\mathcal{L}$  with  $n$  propositions, the marginal inference task is to compute lower and upper bounds on the posterior probability  $P(\psi)$  of a query formula  $\psi$ , which we denote by  $\underline{P}(\psi)$  and  $\overline{P}(\psi)$ , respectively. This is achieved by solving a non-linear program given by Equations (3)-(8) and defined by a set of non-negative real-valued variables representing the probabilities of  $\mathcal{L}$ 's interpretations, a set of linear constraints derived from  $\mathcal{L}$ 's sentences, a set of non-linear constraints corresponding to the independence
+
+assumptions given by the local Markov condition, and a linear objective function encoding the query  $P(\psi)$  which is minimized and maximized to yield the desired bounds. More specifically, let  $\vec{p} = (p_1, \dots, p_m)$  be the vector of real-valued variables representing the probabilities of  $\mathcal{L}$ 's interpretations, where  $m = 2^n$ , and let  $\vec{I}_{\phi} = (a_1^{\phi}, \dots, a_m^{\phi})$  be a binary vector, called an indicator vector, such that  $a_i^\phi$  is 1 if formula  $\phi$  is true in the  $i$ -th interpretation and 0 otherwise. Since the probability of a formula  $\phi$  is the sum of the probabilities of the interpretations in which  $\phi$  is true, we can write  $P(\phi)$  as  $\vec{I}_{\phi} \odot \vec{p}$  where  $\odot$  is the dot-product of two vectors. Therefore, Equations (3) and (4) ensure that  $\vec{p}$  is a valid probability distribution, Equations (5) and (6) encode the type (1) and (2) sentences in  $\mathcal{L}$  while Equation 7 encodes the conditional independencies of the form  $P(X_j|\mathbf{S}_j, \mathbf{T}_j) = P(X_j|\mathbf{S}_j)$ , where  $X_j$  is a proposition,  $\mathbf{S}_j = \{S_{j1}, \dots, S_{jk}\}$  and  $\mathbf{T}_j = \{T_{j1}, \dots, T_{jl}\}$  are  $X_j$ 's parents and non-descendants in the primal graph of  $\mathcal{L}$ ,  $\vec{I}_{\phi}$  and  $\vec{I}_{\phi \wedge \varphi}$  are the indicator vectors for formulae  $\phi$  and  $\phi \wedge \varphi$  involved in  $\mathcal{L}$ 's sentences, and  $\vec{I}_a, \vec{I}_b, \vec{I}_c$  and  $\vec{I}_d$  are the indicator vectors corresponding to the formulae  $a = (x_j \wedge s_{j1} \wedge \dots \wedge s_{jk} \wedge t_{j1} \wedge \dots \wedge t_{jl}), b = (s_{j1} \wedge \dots \wedge s_{jk}), c = (x_j \wedge s_{j1} \wedge \dots \wedge s_{jk}), d = (s_{j1} \wedge \dots \wedge s_{jk} \wedge t_{j1} \wedge \dots \wedge t_{jl})$ , respectively (see also [11] for more details).
+
+# 3 MAP and Marginal MAP Inference in LCNs
+
+Maximum A Posteriori (MAP) and Marginal MAP (MMAP) inference are well known abductive reasoning tasks in probabilistic graphical models such as Bayesian networks and Markov networks [12, 13, 14, 15, 16]. Specifically, the MAP task calls for finding a complete assignment to all variables having maximum probability, given the evidence. Marginal MAP generalizes MAP and looks for a partial variable assignment that has maximum marginal probability, given the evidence. MAP
+
+and MMAP inference tasks appear in many real-world applications such as diagnosis, abduction and explanation and are typically tackled with dynamic programming (e.g., variable elimination) or heuristic search (e.g., depth-first branch and bound) based algorithms [13, 14, 15, 16].
+
+In this section, we present our novel approach for solving the MAP and Marginal MAP inference tasks in Logical Credal Networks. Unlike in graphical models, a (partial) variable assignment (or interpretation) in an LCN may correspond to more than one distribution. Therefore, we begin by formally defining two MAP and MMAP inference tasks for LCNs, called maximin MAP (resp. maximin MMAP) and maximax MAP (resp. maximax MMAP). Subsequently, we develop several exact and approximation schemes for solving these tasks efficiently in practice.
+
+# 3.1 The MAP and Marginal MAP Tasks in LCNs
+
+Let  $\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  be an LCN with  $n$  propositions and let  $\mathbf{E} = \{E_1,\dots ,E_k\} \subseteq \mathbf{A}$  be a subset of  $k$  propositions, called evidence, for which the truth values  $\mathbf{e} = \{e_1,\ldots ,e_k\}$  are known. Let  $\mathbf{Y} = \{Y_{1},\ldots ,Y_{m}\} \subseteq \mathbf{A}\setminus \mathbf{E}$  be a subset of  $m$  propositions called MAP propositions. A truth assignment to  $\mathbf{Y}$  is called a MAP assignment and is denoted by  $\mathbf{y} = \{y_1,\dots ,y_m\}$ , respectively. Clearly, if  $\mathbf{Y} = \mathbf{A}\setminus \mathbf{E}$  (i.e.,  $m = n - k$ ) then we have a MAP task, otherwise we have a MMAP task (i.e.,  $m < n - k$ ). The maximin and maximax MAP/MMAP tasks are defined as follows:
+
+Definition 1 (maximin). Given an  $LCN\mathcal{L}$  with  $n$  propositions, evidence  $\mathbf{e}$ , and MAP propositions  $\mathbf{Y}$ , the maximin MAP (or maximin MMAP if  $m < n - k$ ) task is finding a truth assignment  $\mathbf{y}^*$  to  $\mathbf{Y}$  having maximum lower probability, given evidence  $\mathbf{e}$ , namely:
+
+$$
+\mathbf {y} ^ {*} = \underset {\mathbf {y} \in \Omega (\mathbf {Y})} {\operatorname {a r g m a x}} \underline {{P}} \left(\psi_ {\mathbf {y} \wedge \mathbf {e}}\right) \tag {9}
+$$
+
+where  $\Omega (\mathbf{Y})$  is the set of all truth assignments to the MAP propositions, and  $\psi_{\mathbf{y}\wedge \mathbf{e}} = y_1\wedge \dots \wedge y_m\wedge e_1\wedge \dots \wedge e_k$  is the conjunction of the literals in  $\mathbf{y}$  and  $\mathbf{e}$ , respectively.
+
+Definition 2 (maximax). Given an  $LCN\mathcal{L}$  with  $n$  propositions, evidence  $\mathbf{e}$ , and MAP propositions  $\mathbf{Y}$ , the maximax MAP (or maximax MMAP if  $m < n - k$ ) task is finding a truth assignment  $\mathbf{y}^*$  to  $\mathbf{Y}$  having maximum upper probability, given evidence  $\mathbf{e}$ , namely:
+
+$$
+\mathbf {y} ^ {*} = \underset {\mathbf {y} \in \Omega (\mathbf {Y})} {\operatorname {a r g m a x}} \bar {P} \left(\psi_ {\mathbf {y} \wedge \mathbf {e}}\right) \tag {10}
+$$
+
+where  $\Omega (\mathbf{Y})$  is the set of all truth assignments to the MAP propositions, and  $\psi_{\mathbf{y}\wedge \mathbf{e}} = y_1\wedge \dots \wedge y_m\wedge e_1\wedge \dots \wedge e_k$  is the conjunction of the literals in  $\mathbf{y}$  and  $\mathbf{e}$ , respectively.
+
+# 3.2 Search Algorithms Using Exact MAP Assignment Evaluations
+
+We present next three search-based schemes for solving the MAP and MMAP tasks in LCNs. These methods employ different search strategies for exploring the search space defined by the MAP propositions while evaluating exactly each complete or partial MAP assignment.
+
+Exact Evaluation of a MAP Assignment. Clearly, computing the lower and upper probabilities  $\underline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  and  $\overline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  of a MAP assignment  $\mathbf{y}$  given evidence  $\mathbf{e}$  can be done easily by minimizing and, respectively maximizing the non-linear program defined by Equations (3)-(8), where the query formula is the conjunction of positive or negative literals in  $\mathbf{y}$  and  $\mathbf{e}$ , namely  $\psi_{\mathbf{y}\wedge \mathbf{e}} = y_1\wedge \dots \wedge y_m\wedge e_1\wedge \dots \wedge e_k$ . Therefore, evaluating a MAP assignment in case of both MAP and Marginal MAP inference in LCNs is quite difficult as it involves solving a marginal inference problem for LCNs which is known to be NP-hard [11]. This is in contrast with graphical models where, at least for MAP inference, the evaluation of a MAP assignment is linear in the number of variables [13].
+
+Example 2. For illustration, consider the LCN example from Figure 1 and assume that we have evidence  $\mathbf{e} = \{x,\neg s\}$ , namely a patient has a positive  $X$ -Ray result  $(X = x)$  and is not smoking  $(S = \neg s)$ . The MAP propositions in this case are  $\mathbf{Y} = \{B,C,D\}$  and the MAP assignment  $\mathbf{y} = (b,\neg c,\neg d)$  corresponds to the query formula  $\psi_{\mathbf{y}\wedge \mathbf{e}} = b\wedge \neg c\wedge \neg d\wedge x\wedge \neg s$ . The lower and upper probabilities  $\underline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  and  $\overline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  of the MAP assignment are 9.9e-09 and 0.1, respectively.
+
+Algorithm 1 Depth-First Search for MAP and Marginal MAP Inference in LCNs  
+1: procedure DFS  $(\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle ,\mathbf{E} = \mathbf{e},\mathbf{Y})$  10: score(y)  $\leftarrow \overline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$    
+2: initialize  $\mathbf{y}^*\gets \emptyset$  best  $\leftarrow -\infty$  11: if score(y)  $>$  best then   
+3: SEARCH(0, Y) 12:  $\mathbf{y}^{*}\gets \mathbf{y}$  best  $\leftarrow$  score(y)   
+4: return y\* 13: else   
+5: procedure SEARCH(y, Y) 14: select unassigned proposition  $Y_{i}\in \mathbf{Y}$    
+6: if size(y)  $= =$  size(Y) then 15: for all values  $y\in \{y_i,\neg y_i\}$  do   
+7: if maximin then 16:  $\mathbf{y}\leftarrow \mathbf{y}\cup \{Y_i = y\}$    
+8: score(y)  $\leftarrow P(\psi_{\mathbf{y}\wedge \mathbf{e}})$  17: SEARCH(y, Y)   
+9: else
+
+Algorithm 2 Limited Discrepancy Search for MAP and Marginal MAP Inference in LCNs  
+1: procedure LDS  $(\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle ,\mathbf{E} = \mathbf{e},\mathbf{Y},\delta)$  12: score(y)  $\leftarrow \overline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$    
+2: initialize y0 randomly and let  $\mathbf{y}^{*}\gets \mathbf{y}_{0}$  13: if score(y) > best then   
+3: best  $\leftarrow$  score(y*) 14:  $\mathbf{y}^{*}\gets \mathbf{y}$  best  $\leftarrow$  score(y)   
+4: for all  $\theta = 1\dots \delta$  do 15: else   
+5: SEARCH(y*, Y,  $\theta ,1$  16: for all values  $y\in \{y_i,\neg y_i\}$  do   
+6: return y*,best 17: if y[i] == y then   
+7: procedure SEARCH(y, Y,  $\theta ,i$  18: z  $\leftarrow$  SEARCH(y, Y, i + 1,  $\theta$  )   
+8: if  $\theta = = 0$  or  $i > |\mathbf{Y}|$  then 19: else   
+9: if maximin then 20: y'  $\leftarrow$  y; y'[i]  $\leftarrow$  y   
+10: score(y)  $\leftarrow \underline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  21: z  $\leftarrow$  SEARCH(y', Y, i + 1,  $\theta -1$  1   
+11: else 22: return z
+
+Depth-First Search. Our first approach for solving the MAP and MMAP tasks, called DFS, is described by Algorithm 1. It takes as input an LCN  $\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$ , evidence  $\mathbf{E} = \mathbf{e}$  and a set of MAP propositions  $\mathbf{Y}\subseteq \mathbf{A}\setminus \mathbf{E}$  and outputs the optimal MAP assignment  $\mathbf{y}^*$ . The method conducts a depth-first search over the space of partial assignments to the MAP propositions, and, for each complete MAP assignment  $\mathbf{y}$  computes its score as the exact lower probability  $\underline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  for maximin tasks, and respectively, the upper probability  $\overline{P} (\psi_{\mathbf{y}\wedge \mathbf{e}})$  for maximax tasks, given the evidence  $\mathbf{e}$ . This way, the optimal solution  $\mathbf{y}^*$  corresponds to the MAP assignment with the highest score.
+
+Theorem 1 (complexity). Given an  $LCN\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  with  $n$  propositions, evidence  $\mathbf{E} = \mathbf{e}$  and MAP propositions  $\mathbf{Y}\subseteq \mathbf{A}\setminus \mathbf{E}$ , algorithm DFS is sound and complete. The time and space complexity of the algorithm is  $O(2^{m + 2^n})$  and  $O(2^{n})$ , respectively, where  $m$  is the number of MAP propositions.
+
+Example 3. Consider again the LCN from Figure 1 with evidence  $\mathbf{e} = \{x, \neg s\}$ . In this case, the exact maximin MAP assignment found by algorithm DFS is  $\mathbf{y}^* = \{\neg b, c, d\}$  with value 9.99e-09, while the exact maximax MAP assignment is  $\mathbf{y}^* = \{\neg b, \neg c, d\}$  with value 0.7, respectively.
+
+Limited Discrepancy Search. Our second approach for MAP and MMAP inference in LCNs uses Limited Discrepancy Search (LDS) [19, 20] to explore the search space and is described by Algorithm 2. Specifically, LDS is a depth-first search strategy that searches for new solutions by iteratively increasing the number of discrepancy values, where a discrepancy value indicates the maximum number of allowed variable-value assignment changes to an initial solution [19]. Function SEARCH (lines 7-22) performs the actual exploration of the search space limited by discrepancy  $\theta$ . If the selected truth value  $y \in \{y_i, \neg y_i\}$  is different from the one corresponding to proposition  $Y_i \in \mathbf{Y}$  at position  $i$  in the assignment  $\mathbf{y}$ ,  $\theta$  is decremented to reduce the number of changes allowed to the remaining MAP propositions. Otherwise, the truth value for proposition  $Y_i$  remains unchanged and the  $\theta$  value is preserved. As before, complete MAP assignments are evaluated exactly (lines 9-12) and the best solution found so far is maintained (lines 13-14).
+
+Algorithm 3 Simulated Annealing for MAP and Marginal MAP Inference in LCNs  
+1: procedure SA  $(\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle ,\mathbf{E} = \mathbf{e},\mathbf{Y})$  10: if  $\Delta >0$  then  $\mathbf{y}\gets \mathbf{y}'$    
+2: initialize  $\mathbf{y}_0$  randomly and let  $\mathbf{y}^{*}\leftarrow \mathbf{y}_{0}$  11: else  
+3: best  $\leftarrow$  score(y*) 12: sample randomly  $p\in (0,1)$    
+4: for all iterations  $i = 1\dots N$  do 13: if  $p <   e^{\frac{\Delta}{T}}$  then  $\mathbf{y}\gets \mathbf{y}'$    
+5: set  $\mathbf{y}\gets \mathbf{y}^*,T\gets T_{init}$  14: if score(y)  $>$  best then  
+6: for all flips  $j = 1\dots M$  do 15:  $\mathbf{y}^{*}\gets \mathbf{y},$  best  $\leftarrow$  score(y)   
+7: let  $\mathcal{N}$  be y's neighbors 16:  $T\gets T*\sigma$    
+8: select random neighbor  $\mathbf{y}'\in \mathcal{N}$    
+9:  $\Delta \leftarrow \log score(\mathbf{y}') - \log score(\mathbf{y})$  17: return  $\mathbf{y}^*$
+
+Algorithm 4 Approximate MAP and Marginal MAP Inference in LCNs  
+1: procedure  $\mathrm{AMAP}(\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle ,\mathbf{E} = \mathbf{e},\mathbf{Y})$  9: else  $\mathbf{y}^{*}\gets \mathbf{y}^{*}\cup \{\neg y\}$    
+2: Create factor graph  $\mathcal{F}$  of  $\mathcal{L}$  10: else  
+3: Apply the ARIEL scheme from [18] on  $\mathcal{F}$  11:  $\overline{P} (y) = \min_{f\in N(Y)}u_f\to Y$    
+4: for all MAP propositions  $Y\in \mathbf{Y}$  do 12:  $\overline{P} (\neg y) = 1 - \overline{P} (y)$    
+5: if maximin then 13: if  $\overline{P} (y) > \overline{P} (\neg y)$  then  $\mathbf{y}^{*}\leftarrow \mathbf{y}^{*}\cup \{y\}$    
+6:  $\underline{P} (y) = \max_{f\in N(Y)}l_f\rightarrow Y$  14: else  $\mathbf{y}^{*}\gets \mathbf{y}^{*}\cup \{\neg y\}$    
+7:  $\underline{P} (\neg y) = 1 - \underline{P} (y)$    
+8: if  $\underline{P} (y) > \underline{P} (\neg y)$  then  $\mathbf{y}^{*}\leftarrow \mathbf{y}^{*}\cup \{y\}$  15: return  $\mathbf{y}^*$
+
+Theorem 2 (complexity). Given an  $LCN\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  with  $n$  propositions, evidence  $\mathbf{E} = \mathbf{e}$  and MAP propositions  $\mathbf{Y}\subseteq \mathbf{A}\setminus \mathbf{E}$ , algorithm LDS is sound and complete. The time and space complexity of the algorithm is  $O(2^{m + 2^n})$  and  $O(2^{n})$ , respectively, where  $m$  is the number of MAP propositions.
+
+Simulated Annealing. The third approach for solving MAP and MMAP tasks in LCNs is described by Algorithm 3 and employs a form of stochastic local search known as Simulated Annealing (SA) [21] to explore the search space defined by the MAP propositions. The algorithm starts from an initial guess  $\mathbf{y}$  as a truth assignment to the MAP propositions  $\mathbf{Y}$ , and iteratively tries to improve it by moving to a better neighbor  $\mathbf{y}'$  that has a higher score. A neighbor  $\mathbf{y}'$  of  $\mathbf{y}$  is defined as a new assignment  $\mathbf{y}'$  which results from changing the truth value of a single proposition  $Y$  in  $\mathbf{Y}$ . At each step, the transition from the current state  $\mathbf{y}$  to a neighboring state  $\mathbf{y}'$  is decided probabilistically using an acceptance probability function  $P(\mathbf{y}',\mathbf{y},T)$  that depends on the scores of the two states as well as a global time-varying parameter  $T$  called temperature which is decreased using a cooling schedule  $\sigma < 1$  [21]. We chose  $P(\mathbf{y}',\mathbf{y},T) = e^{\frac{\Delta}{T}}$ , where  $\Delta = \log \text{score}(\mathbf{y}') - \log \text{score}(\mathbf{y})$ .
+
+Theorem 3 (complexity). Given an  $LCN\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  with  $n$  propositions, evidence  $\mathbf{E} = \mathbf{e}$  and MAP propositions  $\mathbf{Y}\subseteq \mathbf{A}\setminus \mathbf{E}$ , the time and space complexity of algorithm SA is  $O(N\cdot M\cdot 2^{2^n})$  and  $O(2^{n})$ , respectively, where  $N$  is the number of iterations and  $M$  is the number of flips per iterations.
+
+# 3.3 Approximate MAP and Marginal MAP Inference
+
+The main bottleneck in the proposed search algorithms is the exact evaluation of the MAP assignments which is computationally very expensive [11]. This limits the applicability of these methods to relatively small LCNs. Therefore, in order to be able to tackle larger LCNs, we extend a recent message-passing approximation scheme for marginal inference in LCNs [18] to solve the MAP and MMAP tasks in LCNs. Subsequently, we also adapt the limited discrepancy search and simulated annealing methods to use an approximate evaluation of the MAP assignments during search.
+
+Algorithm 4 describes our message-passing based approximation scheme for MAP and MMAP inference in LCNs which we denote hereafter by AMAP. We build upon a recent scheme for approximate marginal inference in LCNs, called ARIEL [18], which propagates messages along the edges of a factor graph associated with the input LCN until convergence. The factor graph  $\mathcal{F}$  of an
+
+LCN  $\mathcal{L}$  is a bi-partite graph the connects proposition nodes labeled by the propositions in  $\mathcal{L}$  with factor nodes associated with sentences that involve the same set of propositions [18]. The messages propagated between the nodes of  $\mathcal{F}$  are intervals representing lower and upper bounds on the marginal probabilities of  $\mathcal{L}$ 's propositions and are computed as follows: the message sent from a proposition to a factor node tightens these bounds based on the incoming messages from the factor nodes connected to it; the message sent from a factor to a proposition node computes new bounds by solving a local non-linear program defined by the factor's sentences and the constraints encoding the assumption that the factor's propositions are independent of each other and the marginal probabilities of the factor's propositions are within the bounds given by the incoming proposition-to-factor messages (see also [18] for more details). Upon convergence, the maximin MAP assignment  $\mathbf{y}^*$  can be obtained as follows: for each MAP proposition  $Y \in \mathbf{Y}$  we compute the tightest lower probability bound  $\underline{P}(y)$  by maximizing the lower bound of all incoming factor-to-proposition messages to  $Y$ , and, subsequently, select  $y$  as the most likely value assignment to  $Y$  if  $\underline{P}(y) > \underline{P}(\neg y)$  and  $\neg y$  otherwise (for the maximax tasks we use the upper probability bounds  $\overline{P}(y)$  and  $\overline{P}(\neg y)$ , respectively).
+
+Theorem 4 (complexity). Given an  $LCN\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  with  $n$  propositions, evidence  $\mathbf{E} = \mathbf{e}$  and MAP propositions  $\mathbf{Y}\subseteq \mathbf{A}\setminus \mathbf{E}$ , the time and space complexity of algorithm AMAP is  $O(N\cdot M\cdot 2^{2^r})$  and  $O(2^{r})$ , where  $N$  is the number of iterations,  $M$  bounds the number of factor-to-node messages per iteration and  $r$  bounds the number of propositions in the factor nodes, respectively.
+
+# 3.4 Search Algorithms Based on Approximate MAP Evaluations
+
+The main assumption behind algorithm AMAP is that all MAP propositions are independent of each other and therefore the solution  $\mathbf{y}^*$  returned by AMAP is likely to correspond to a local maxima. One way to escape such a local optima and obtain a potentially better solution is to employ a search scheme based on either limited discrepancy search or simulated annealing that continues the exploration of the search space starting from  $\mathbf{y}^*$ . However, in order to scale to larger LCNs, we would like the search schemes to rely on an approximate rather than an exact evaluation of the MAP assignments.
+
+Approximate Evaluation of a MAP Assignment. Estimating the lower and upper probabilities of a MAP assignment  $\mathbf{y}$  can be done by approximate marginal inference on an augmented LCN as follows. Let  $\mathcal{L} = \langle \mathbf{A},\mathcal{C}\rangle$  be the input LCN and let  $\mathbf{y} = (y_{1},\ldots ,y_{m})$  be a MAP assignment to propositions  $\mathbf{Y} = \{Y_1,\dots ,Y_m\}$  (for simplicity, we include the evidence e in  $\mathbf{y}$ ). The augmented LCN  $\mathcal{L}' = \langle \mathbf{A}',\mathcal{C}'\rangle$  is constructed by adding a set of auxiliary propositions  $\mathbf{W} = \{W_1,\dots ,W_m\}$ , one for each MAP proposition, and additional constraints of the following two forms:  $P(W_{1}|Y_{1})$  and  $P(W_{j}|W_{j - 1}\wedge Y_{j})$ , for all  $2\leq j\leq m$ , such that  $P(w_{1}|y_{1}) = 1$ ,  $P(w_{1}|\neg y_{1}) = 0$ ,  $P(w_{j}|w_{j - 1}\wedge y_{j}) = 1$ ,  $P(w_{j}|w_{j - 1}\wedge \neg y_{j}) = 0$ ,  $P(w_{j}|\neg w_{j - 1}\wedge y_{j}) = 0$  and  $P(w_{j}|\neg w_{j - 1}\wedge \neg y_{j}) = 0$ , respectively. Then, we can estimate  $\underline{P} (\psi_{\mathbf{y}})$  and  $\overline{P} (\psi_{\mathbf{y}})$ , where  $\psi_{\mathbf{y}} = y_1\wedge \dots \wedge y_m$ , by computing the posterior marginals  $\underline{P} (w_m)$  and  $\overline{P} (w_m)$  in the augmented LCN  $\mathcal{L}'$  using the method from [18].
+
+Limited Discrepancy Search and Simulated Annealing. Our approximate LDS and SA based algorithms denoted by ALDS and ASA can be obtained from Algorithms 2 and 3 by replacing the score(y) function with the approximate MAP evaluation scheme described above. These algorithms can start the search either from a random MAP assignment or from the solution found by algorithm AMAP. Finally, the time complexity of algorithms ALDS and ASA can be bounded by  $O(2^{m + 2^r})$  and  $O(N \cdot M \cdot 2^{2^r})$ , respectively, where  $m$  is the number of MAP propositions,  $N$  is the number of iterations used by ASA,  $M$  is the maximum number of flips per iteration, and  $r$  bounds the number of propositions in the factor nodes of the factor graph associated with the input LCN [18].
+
+# 4 Experiments
+
+In this section, we empirically evaluate the proposed exact and approximate schemes for MAP and MMAP inference in LCNs. All competing algorithms were implemented<sup>2</sup> in Python 3.10 and used the ipopt 3.14 solver [22] with default settings to handle the non-linear constraint programs. We ran all experiments on a 3.0GHz Intel Core processor with 128GB of RAM.
+
+Table 1: Results for MAP tasks obtained on small/large scale polytree, dag, and random LCNs. Average CPU time in seconds and number of problem instances solved. Time limit is 2 hours.  
+
+<table><tr><td rowspan="2">size n</td><td colspan="3">exact MAP eval</td><td rowspan="2">AMAP</td><td colspan="2">approx MAP eval</td></tr><tr><td>DFS</td><td>LDS(3)</td><td>SA</td><td>ALDS(3)</td><td>ASA</td></tr><tr><td colspan="7">polytree</td></tr><tr><td>5</td><td>15.30 (10)</td><td>26.07 (10)</td><td>20.18 (10)</td><td>2.87 (10)</td><td>174.17 (10)</td><td>188.27 (10)</td></tr><tr><td>8</td><td>3246.28 (4)</td><td>3072.18 (4)</td><td>1199.51 (10)</td><td>8.05 (10)</td><td>1054.53 (10)</td><td>518.18 (10)</td></tr><tr><td>10</td><td>-</td><td>-</td><td>-</td><td>11.81 (10)</td><td>2273.16 (10)</td><td>813.30 (10)</td></tr><tr><td>30</td><td>-</td><td>-</td><td>-</td><td>31.55 (10)</td><td>-</td><td>3091.74 (10)</td></tr><tr><td>50</td><td>-</td><td>-</td><td>-</td><td>52.30 (10)</td><td>-</td><td>5324.71 (10)</td></tr><tr><td>70</td><td>-</td><td>-</td><td>-</td><td>79.28 (10)</td><td>-</td><td>7279.56 (10)</td></tr><tr><td colspan="7">dag</td></tr><tr><td>5</td><td>21.09 (10)</td><td>15.66 (10)</td><td>24.04 (10)</td><td>5.54 (10)</td><td>163.02 (10)</td><td>156.34 (10)</td></tr><tr><td>8</td><td>1633.38 (8)</td><td>1958.16 (9)</td><td>633.77 (10)</td><td>13.05 (10)</td><td>1339.71 (10)</td><td>571.55 (10)</td></tr><tr><td>10</td><td>-</td><td>-</td><td>-</td><td>15.55 (10)</td><td>2903.05 (10)</td><td>944.17 (10)</td></tr><tr><td>30</td><td>-</td><td>-</td><td>-</td><td>49.94 (10)</td><td>-</td><td>3593.71 (10)</td></tr><tr><td>50</td><td>-</td><td>-</td><td>-</td><td>89.13 (10)</td><td>-</td><td>5639.90 (10)</td></tr><tr><td>70</td><td>-</td><td>-</td><td>-</td><td>132.34 (10)</td><td>-</td><td>6093.28 (10)</td></tr><tr><td colspan="7">random</td></tr><tr><td>5</td><td>19.51 (10)</td><td>17.56 (10)</td><td>20.37 (10)</td><td>5.26 (10)</td><td>152.99 (10)</td><td>143.60 (10)</td></tr><tr><td>8</td><td>3152.57 (1)</td><td>3209.54 (5)</td><td>1226.88 (10)</td><td>10.29 (10)</td><td>954.46 (10)</td><td>444.17 (10)</td></tr><tr><td>10</td><td>-</td><td>-</td><td>-</td><td>12.21 (10)</td><td>2150.27 (10)</td><td>717.75 (10)</td></tr><tr><td>30</td><td>-</td><td>-</td><td>-</td><td>40.54 (10)</td><td>-</td><td>3335.14 (10)</td></tr><tr><td>50</td><td>-</td><td>-</td><td>-</td><td>76.83 (10)</td><td>-</td><td>5276.93 (10)</td></tr><tr><td>70</td><td>-</td><td>-</td><td>-</td><td>105.70 (10)</td><td>-</td><td>6059.57 (7)</td></tr></table>
+
+Random LCNs. We generated three classes of random LCNs with  $n$  propositions  $\{X_1, \ldots, X_n\}$  and sentences of the following types: (a)  $l \leq P(x_i) \leq u$ , (b)  $l \leq P(x_i | x_j) \leq u$ ,  $i \neq j$  and (c)  $l \leq P(x_i | X_j \wedge X_k) \leq u$ ,  $i \neq j \neq k$ , such that the corresponding primal graph is a polytree, a dag or a random graph. The type (c) sentences were generated for all truth values of propositions  $X_j$  and  $X_k$ , namely  $P(x_i | x_j)$ ,  $P(x_i \lnot x_j)$ ,  $P(x_i | x_j \wedge x_k)$ ,  $P(x_i | x_j \wedge \neg x_k)$ ,  $P(x_i \lnot x_j \wedge x_k)$  and  $P(x_i \lnot x_j \wedge \neg x_k)$ , respectively. The probability bounds  $l$  and  $u$  were selected uniformly at random between 0 and 1 such that  $u - l \leq 0.6$ , and we ensured that all instances with  $n \leq 10$  were consistent.
+
+Table 1 summarizes the results obtained for maximax MAP queries on the random LCNs.
+
+![](images/281652f59af3a00deb1dc6e3b5a51ca8014e6b8e101d1ddbaed8a269ed447e70.jpg)  
+Figure 2: Wins for LCNs with  $n = 10$ .
+
+For each problem class we consider both smaller  $(5\leq n\leq 10)$  and larger  $(30\leq n\leq$  70) scale instances, respectively. We report the average CPU time in seconds and number of problem instance solved (out of 10) for each problem size. A  $\cdot^{\prime}$  indicates that the respective algorithm exceeded the 2 hour time limit. The maximum discrepancy value use by algorithms LDS and ALDS was set to  $\delta = 3$  while algorithms SA and ASA used up to 30 flips over a single iteration. We can see that the algorithms using exact MAP assignment evaluations (i.e., DFS, LDS and SA) are limited to small scale problem instances with up to 8 propositions and they
+
+run out of time on the larger instances. This is caused by the prohibitively large computational overhead associated with the exact evaluation of the MAP assignments during search. In contrast, the approximate search algorithms ALDS and specially ASA can scale to much larger problem instances due to the less expensive approximate MAP assignment evaluations. AMAP is the best performing algorithm in terms of running time and number of problems solved for all reported problem sizes. However, since the solution found by AMAP is only a local maxima, in Figure 2 we report on the solution quality found by algorithms AMAP, ALDS and ASA on LCN instances of size 10. Specifically, we show the number of wins as the number of times (out of 10) each algorithm found the best solution. In this case, algorithms ALDS and ASA were initialized with the MAP assignment
+
+![](images/3397d81dcd8769887b582add4056a3b090114a1206be6e2573a5367eefb42b33.jpg)  
+Figure 3: Average CPU time in seconds and standard deviation vs discrepancy  $\delta$  for ALDS(δ).
+
+![](images/e5392d3603900f3ecf9a2eac0ec373f7125a0ea49e9742a4eee45dcf4caaf4a7.jpg)
+
+![](images/8741d577ad761b43de232dcb8db50161d0b3ef332dd0b99c7dda03bf1fa22fa2.jpg)
+
+Table 2: Results for MMAP tasks on realistic LCNs. CPU time in seconds. Time limit is 2 hours.  
+
+<table><tr><td rowspan="2">LCN</td><td colspan="3">exact MAP eval</td><td rowspan="2">AMAP</td><td colspan="2">approx MAP eval</td></tr><tr><td>DFS</td><td>LDS(3)</td><td>SA</td><td>ALDS(3)</td><td>ASA</td></tr><tr><td>Toy</td><td>2.20</td><td>3.18</td><td>1.85</td><td>0.85</td><td>134.83</td><td>141.17</td></tr><tr><td>Earth</td><td>9.19</td><td>7.67</td><td>2.75</td><td>1.28</td><td>150.99</td><td>162.35</td></tr><tr><td>Cancer</td><td>16.34</td><td>14.09</td><td>8.52</td><td>2.64</td><td>157.92</td><td>159.66</td></tr><tr><td>Asia</td><td>811.82</td><td>800.18</td><td>312.10</td><td>4.07</td><td>187.44</td><td>201.76</td></tr><tr><td>Credit</td><td>-</td><td>6719.30</td><td>2976.55</td><td>5.09</td><td>204.77</td><td>222.52</td></tr><tr><td>Engine</td><td>4786.12</td><td>4502.34</td><td>2033.77</td><td>6.57</td><td>212.61</td><td>235.70</td></tr><tr><td>Suicide</td><td>-</td><td>-</td><td>-</td><td>5.99</td><td>220.31</td><td>203.68</td></tr><tr><td>Tank</td><td>-</td><td>-</td><td>-</td><td>8.04</td><td>263.65</td><td>281.73</td></tr><tr><td>Alarm</td><td>-</td><td>-</td><td>-</td><td>4.28</td><td>216.19</td><td>186.67</td></tr><tr><td>Hepatitis</td><td>-</td><td>-</td><td>-</td><td>8.22</td><td>260.38</td><td>250.45</td></tr></table>
+
+found by AMAP. We can see that almost always the search-based approaches ALDS and ASA are able to find better solutions than AMAP. This is important in practice, particularly on larger scale problems where we can use AMAP to find a MAP solution quickly, and subsequently refine that solution using a search-based algorithm like ALDS or ASA if the time budget allows it. Finally, in Figure 3 we show the impact of the maximum discrepancy value  $\delta$  on the running time of algorithm  $\mathrm{ALDS}(\delta)$ . It is easy to see that as the discrepancy value  $\delta$  increases, the search space explored by  $\mathrm{ALDS}(\delta)$  becomes larger, and therefore its corresponding running time increases as well.
+
+Realistic LCNs. We experimented with a set of more realistic LCNs which were first introduced in [18]. These LCNs were derived from real-world Bayesian networks [23] and contain up to 10 propositions as well as up to 24 sentences of the form  $l \leq P(x_i) \leq u$  and  $l \leq P(x_i | \pi_i) \leq u$ , respectively, where  $x_i$  is the positive literal of proposition  $X_i$  and  $\pi_i = y_{i1} \wedge \dots \wedge y_{ik}$  is the conjunction of the positive or negative literals corresponding to a particular configuration of the parents  $\{Y_{i1}, \ldots Y_{ik}\}$  of  $X_i$  in the Bayesian network. The specification of these LCNs is included in the supplementary material. Table 2 reports the results obtained on 10 LCN instances for the maximax MMAP task with 4 MAP propositions selected randomly. As before, algorithms DFS, LDS(3) and SA which rely on exact evaluations of the MAP assignments during search can only solve the smallest problem instances within the 2 hour time limit. In contrast, algorithms ALDS(3) and ASA solve all problem instances due to a much reduced overhead associated with the approximate MAP assignment evaluations. In this case, the search spaces explored by ALDS(3) and ASA are approximately the same in size and therefore the corresponding running times are comparable. AMAP is the fastest algorithm in this case as well.
+
+Application to Factuality in Large Language Models. We consider an application of MMAP inference in LCNs to assess the factuality of the output  $A$  generated by a large language model (LLM) in response to a user query  $Q$  with respect to an external source of knowledge  $C$  that may contain contradicting factual information (e.g., Wikipedia) [24]. The goal is to compute a factuality score for response  $A$ , denoted by  $f_{C}(A)$ , in the context of the information from  $C$ . In the following, we assume that  $A$  can be decomposed into a set of  $n$  atomic facts (or just atoms)  $A = \{A_1, \dots, A_n\}$  (e.g., one way to do that is to split  $A$  into sentences) and, for each atom  $A_i$ , up to  $k$  relevant passages  $\{C_{i1}, \dots, C_{ik}\}$  called contexts can be retrieved from  $C$ . A natural language inference (NLI) classifier such as SBERT [25] can be used to infer the entailment, contradiction and neutrality relationships between the texts corresponding to the atoms and contexts together
+
+Table 3: Results for factuality LCNs. Average CPU time in seconds and number of problem instances solved. Time limit is 2 hours.  
+
+<table><tr><td rowspan="2">size
+n, k = 2</td><td colspan="3">exact MAP eval</td><td rowspan="2">AMAP</td><td colspan="2">approx MAP eval</td></tr><tr><td>DFS</td><td>LDS(2)</td><td>SA</td><td>ALDS(2)</td><td>ASA</td></tr><tr><td>2</td><td>56.95 (10)</td><td>57.37 (10)</td><td>60.09 (10)</td><td>0.31 (10)</td><td>5.25 (10)</td><td>4.13 (10)</td></tr><tr><td>4</td><td>-</td><td>-</td><td>-</td><td>0.98 (10)</td><td>80.07 (10)</td><td>54.15(10)</td></tr><tr><td>6</td><td>-</td><td>-</td><td>-</td><td>1.97 (10)</td><td>453.88 (10)</td><td>219.57 (10)</td></tr><tr><td>10</td><td>-</td><td>-</td><td>-</td><td>7.33 (10)</td><td>2713.90 (10)</td><td>928.28 (10)</td></tr><tr><td>20</td><td>-</td><td>-</td><td>-</td><td>28.42 (10)</td><td>-</td><td>3809.23 (10)</td></tr><tr><td>50</td><td>-</td><td>-</td><td>-</td><td>379.18 (10)</td><td>-</td><td>-</td></tr><tr><td>100</td><td>-</td><td>-</td><td>-</td><td>1807.10 (10)</td><td>-</td><td>-</td></tr></table>
+
+with their corresponding probabilities (or scores). Specifically, we consider relationships between an atom and a context  $r(A_i, C_{ij})$ , and between two contexts  $r(C_{ij}, C_{pq})$ , respectively, where  $r \in \{\text{entailment}, \text{contradiction}\}$ . We define an LCN  $\mathcal{L}$  containing  $n + n \times k$  propositions for each of the atoms and contexts, and two types of sentences corresponding to the entailment and contradiction relationships as follows:  $l \leq P(Y|X) \leq u$  if  $X$  entails  $Y$ , and  $l \leq P(\neg Y|X) \leq u$  if  $X$  contradicts  $Y$ , where  $X$  and  $Y$  are the propositions corresponding to a context and an atom, or to two different contexts, respectively. The lower and upper bounds  $l$  and  $u$  can be calculated easily from the probabilities obtained by running multiple NLI classifiers. Finally, the factuality score  $f_C(A)$  is the proportion of true atoms in the MAP assignment obtained by solving a maximax MMAP task over  $\mathcal{L}$  where the MAP propositions are those corresponding to  $A$ 's atoms.
+
+Table 3 displays the results obtained on randomly generated factuality LCNs. More specifically, for each reported problem size  $n \in \{2,4,6,10,20,50,100\}$ , we generated 10 random instances with  $n$  atoms and  $k = 2$  contexts per atom such that  $10\%$  of all possible pairwise relationships between atoms and contexts were selected to be either entailment or contradiction with probability 0.5 while the remaining relationships were labeled as neutral and thus ignored. The lower and upper probability bounds  $l$  and  $u$  in the corresponding LCN sentences were also generated randomly between 0 and 1 such that  $u - l \leq 0.6$ . In this case, the maximum discrepancy value was set to 2 and simulated annealing was allowed a single iteration and 30 flips. We observe again that algorithms DFS, LDS(2) and SA can only solve the smallest instances due to large computational overhead associated with exact evaluation of the MAP assignments. In contrast, algorithms ALDS(2) and ASA which rely on less expensive approximate evaluations of the MAP assignments can scale to larger problems with up to 20 atoms. Algorithm AMAP outperforms its competitors and solves all problem instances.
+
+In summary, our empirical evaluation showed that the exact search-based MAP/MMAP algorithms are limited to solving relatively small problem instances. In contrast, the approximate MAP/MMAP schemes based on either message-passing or search can scale to much larger LCN instances.
+
+# 5 Conclusions
+
+In this paper, we address abductive reasoning tasks such as generating MAP and Marginal MAP (MMAP) explanations in Logical Credal Networks (LCNs), a recently introduced probabilistic logic framework for reasoning with imprecise knowledge. Since an LCN encodes a set of distributions over its interpretations, a complete or partial explanation of the evidence (i.e., a MAP assignment) may correspond to more than one distribution. Therefore, we define the maximin/maximax MAP and MMAP tasks for LCNs as finding complete or partial MAP assignments that have maximum lower/upper probability given the evidence. We propose several search algorithms that combine depth-first search, limited-discrepancy search or simulated annealing with exact evaluations of the MAP assignments using marginal inference for LCNs. We also develop an approximate message-passing scheme as well as extend limited discrepancy search and simulated annealing to use an approximate evaluation of the MAP assignments during search. Our experiments with random LCNs and LCNs derived from realistic use-cases demonstrate conclusively that the search methods based on exact evaluations of the MAP assignments are limited to small size problems, while the approximation schemes can scale to much larger problems. For future work we plan to investigate more advanced depth-first branch-and-bound and best-first search techniques. However, these kinds of methods require developing novel heuristic bounding schemes to guide the search more effectively [16].
+
+# Acknowledgements
+
+Fabio Cozman thanks CNPq (grant 305753/2022-3) and the Center for AI at Universidade de Sao Paulo, funded by FAPESP (grant 2019/07665-4) and IBM.
+
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+[16] Radu Marinescu, Junkyu Lee, Rina Dechter, and Alexander Ihler. AND/OR search for marginal MAP. Journal or Artificial Intelligence Research (JAIR), 63(1):875 - 921, 2018.  
+[17] Radu Marinescu, Debarun Bhattacharjya, Junkyu Lee, Alexander Gray, and Fabio Cozman. Credal marginal map. In 37th Conference on Neural Information Processing Systems (NeurIPS), 2023.  
+[18] Radu Marinescu, Haifeng Qian, Alexander Gray, Debarun Bhattacharjya, Francisco Barahona, Tian Gao, and Ryan Riegel. Approximate inference in logical credal networks. In 32nd International Joint Conference on Artificial Intelligence (IJCAI), 2023.  
+[19] William Harvey and Matthew Ginsberg. Limited discrepancy search. In International Joint Conference on Artificial Intelligence (IJCAI), pages 607-613, 1995.
+
+[20] Richard Korf. Improved limited discrepancy search. In AAAI Conference on Artificial Intelligence (AAAI), pages 286-291, 1996.  
+[21] Scott Kirkpatrick, Daniel Gelatt, and Mario Vecchi. Optimization by simulated annealing. Science, 220:671-680, 1983.  
+[22] Andreas Wächter and Lorenz Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25-57, 2006.  
+[23] Anthony Constantinou, Yang Liu, Kiattikun Chobtham, Zhigao Guo, and Neville Kitson. The bayesys data and bayesian network repository. Technical report, Bayesian Artificial Intelligence research lab, Queen Mary University of London, London, UK, 2020.  
+[24] Sewon Min, Kalpesh Krishna, Xinxi Lyu, Mike Lewis, Wen-tau Yih, Pang Koh, Mohit Iyyer, Luke Zettlemoyer, and Hannaneh Hajishirzi. FActScore: Fine-grained atomic evaluation of factual precision in long form text generation. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 12076-12100, 2023.  
+[25] Nils Reimers and Iryna Gurevych. Sentence-bert: Sentence embeddings using siamese bert-networks. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing, pages 3973-3983, 2019.
+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and precede the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS paper checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Sections 3 and 4
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Section 3, 4 and 5. Essentially the exact inference methods proposed in this paper are limited to small size problems with up to 8 propositions/variables while the proposed approximate inference methods can scale to much larger problems with tens and even hundreds of variables.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
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+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: Section 3
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: Section 4 and supplementary material
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
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+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Section 4 and supplementary material
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.
+
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Section 4 and the supplementary material
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: Section 4 and the supplementary material
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
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+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Section 4
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification:
+
+Guidelines:
+
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+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
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+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification:
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification:
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: Section 4
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: Section 4
+
+# Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification:
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification:
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

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+# Abrupt Learning in Transformers: A Case Study on Matrix Completion
+
+Pulkit Gopalani
+
+University of Michigan gopalani@umich.edu
+
+Ekdeep Singh Lubana
+
+Harvard University ekdeeplubana@fas.harvard.edu
+
+Wei Hu
+
+University of Michigan vvh@umich.edu
+
+# Abstract
+
+Recent analysis on the training dynamics of Transformers has unveiled an interesting characteristic: the training loss plateaus for a significant number of training steps, and then suddenly (and sharply) drops to near-optimal values. To understand this phenomenon in depth, we formulate the low-rank matrix completion problem as a masked language modeling (MLM) task, and show that it is possible to train a BERT model to solve this task to low error. Furthermore, the loss curve shows a plateau early in training followed by a sudden drop to near-optimal values, despite no changes in the training procedure or hyper-parameters. To gain interpretability insights into this sudden drop, we examine the model's predictions, attention heads, and hidden states before and after this transition. Concretely, we observe that (a) the model transitions from simply copying the masked input to accurately predicting the masked entries; (b) the attention heads transition to interpretable patterns relevant to the task; and (c) the embeddings and hidden states encode information relevant to the problem. We also analyze the training dynamics of individual model components to understand the sudden drop in loss.
+
+# 1 Introduction
+
+Large Language Models (LLMs) have revolutionized the field of natural language processing (NLP). However, there are still gaps in our understanding of these models, leading to challenges in controlling their behavior. As a pertinent example, the training of these models appears to demonstrate sudden improvements in metrics correlated with various capabilities [8], prompting questions about whether learning of a given capability can be predicted by tracking predefined progress measures and why such sudden changes occur. If undesirable capabilities can suddenly 'emerge' (despite any explicit supervision for them) [16], such sudden changes can be a challenge for AI regulation [21].
+
+To better understand such sudden changes during model training, this work investigates training BERT [12] on the classical mathematical task of low-rank matrix completion (LRMC) [6]. Making an analogy with masked language modeling (MLM), where sudden learning of syntactical structures was recently demonstrated [8], we argue matrix completion captures the core aspect of this learning problem (Fig. 1): given some relevant context (observed tokens), fill the missing elements (masked tokens). Specifically, we assume access to a matrix with some fraction of its entries missing, and would like to complete the missing entries of this matrix assuming the ground truth matrix is low-rank. We find that despite being a simplified abstraction of MLM, this setting already demonstrates a sharp decrease in loss as the model undergoes training (Fig. 1 (B)), preceded by a loss plateau for a significant number of training steps (akin to Chen et al. [8]). The simplicity of our setting further affords us interpretability, as we find that the point of sudden drop coincides with a precise change in how the model solves the task—we call this change an algorithmic transition. Specifically, we show that the pre-transition model simply copies the input (predicting 0 at masked positions), while the post-transition model accurately predicts missing values at masked positions. To perform the latter,
+
+![](images/e3c024e76b16045bd0b434087e0235c241e887a9bdca438a08453d6454b7f4d3.jpg)  
+Figure 1: (A) Matrix completion using BERT. Similar to completing missing words in an English sentence in MLM, we complete missing entries in a masked low-rank matrix. (B) Sudden drop in loss. During training, the model undergoes an algorithmic shift marked by a sharp decrease in mean-squared-error (MSE) loss. Here, the model shifts from simply copying the input (copying phase) to computing missing entries accurately (completion phase).
+
+![](images/f0a619b63012aec96c484b52dea2c10fcf50314be8a300321c23f7b37cd6bd15.jpg)
+
+distinctive changes occur in the model's attention heads during the period of sudden drop, wherein the model learns to identify relevant positional information to combine various elements in the input matrix and compute missing entries for matrix completion. We perform a range of interventions on the input, model (before and after the transition), and training process to further understand this phenomenon, leading to the following observations.
+
+- Pre-transition: Copying the Input Matrix Before the transition, the model is simply copying the input matrix both at observed entries as well as missing entries, predicted value for missing entries being nearly 0. The attention maps at this stage do not correspond to a particularly interpretable structure, and contribute little to the model output.  
+- Post-Transition: Computing Missing Entries After the transition, the model accurately completes the missing entries, while still copying observed entries. The attention maps at this stage clearly demonstrate that the model 'attends' to relevant tokens in the input, and the attention layers are crucial for accurate matrix completion. Interestingly, the post-transition model can outperform the classical nuclear norm minimization algorithm for matrix completion, suggesting that it does not simply recover this algorithm.  
+- Model Components and Sudden Drop We analyze the training dynamics of individual components, keeping other components fixed to their final values. We find that different components converge to their optimal values at quite different points during this training.
+
+# 2 Preliminaries
+
+# 2.1 Problem Setup
+
+MLM and LRMC In masked language modeling (MLM), a fraction of tokens in the input sequence are masked out and the model is required to predict the correct token for those masked entries. In this setup, the model has access to both the tokens before and after the current token for computing the prediction. Low-rank matrix completion has a similar structure: given a matrix (assumed low-rank) with a fraction of its elements available, the goal is to predict missing entries. For a matrix  $X \in \mathbb{R}^{n \times n}$ , denote its observed entries by the set  $\Omega \subset [n] \times [n]$ , and the set of missing entries  $\Omega^c = [n] \times [n] \setminus \Omega$ . Formally, the problem is
+
+$$
+\min  _ {U} \operatorname {r a n k} (U) \quad \text {s . t .} U _ {i j} = X _ {i j} \forall (i, j) \in \Omega .
+$$
+
+Importantly, both problems (MLM and LRMC) have the same goal—predict the missing entries in the input, i.e., either the language tokens (MLM) or matrix elements (LRMC).
+
+Matrix Completion using BERT BERT [12] is an encoder-only Transformer architecture used widely for MLM. For an input sequence of tokens  $[t_1, t_2, \ldots, t_L]$ , the output is a sequence of
+
+$D$ -dimensional 'hidden states'  $[e_1, \ldots, e_L]^\top \in \mathbb{R}^{L \times D}$ , that is used for prediction. We train a BERT model  $\mathsf{T}\mathsf{F}_{\theta}$  to predict missing entries in a low-rank masked matrix  $\tilde{X}$ . For model output  $\hat{X} := \mathsf{T}\mathsf{F}_{\theta}(\tilde{X}) \in \mathbb{R}^{n \times n}$ , the training objective  $L := L(\theta)$  is the mean-squared-error (MSE) loss over all entries,
+
+$$
+L (\theta) = \frac {1}{n ^ {2}} \sum_ {i, j = 1} ^ {n} (X _ {i j} - \hat {X} _ {i j}) ^ {2}.
+$$
+
+In our experiments, data for matrix completion is generated as
+
+$$
+X = U V ^ {\top}; \quad U, V \in \mathbb {R} ^ {n \times r}, \quad U _ {i j}, V _ {i j} \stackrel {{\mathrm {i i d}}} {{\sim}} \mathrm {U n i f} [ - 1, 1 ] \forall i, j \in [ n ] \times [ r ]
+$$
+
+so that  $X$  has rank at most  $r$ . To mask entries at random, we sample binary matrices  $M \in \{0,1\}^{n \times n}$  such that  $M_{ij} = 0$  with probability  $p_{\mathrm{mask}}$ , and 1 otherwise; that is,  $\Omega = \{(i,j) \mid M_{ij} = 1\}$ .
+
+Nuclear norm minimization Nuclear norm minimization [6] is a widely used convex optimization approach to LRMC; for completeness, we compare our trained models to this approach. Since rank is not a convex function of the matrix, one modifies the low rank completion problem by defining the nuclear norm  $\| U\| _*$ , i.e., sum of singular values of a matrix  $U$ . The overall optimization problem is as follows.
+
+$$
+\min  _ {U} \| U \| _ {*} \quad \text {s . t .} U _ {i j} = X _ {i j} \forall (i, j) \in \Omega . \tag {1}
+$$
+
+# 2.2 Experiments
+
+Training We use a 4-layer, 8-head BERT model [40] for  $7 \times 7$  (rank-2) matrices, with 'absolute' positional embeddings, no token-type embeddings, and no dropout. We fix  $p_{\mathrm{mask}} = 0.3$  for training, and 256 matrices are sampled as training data at each step (in an 'online' training setup). We use Adam optimizer with constant step size 1e-4 for 50000 steps, without weight decay or warmup. In addition to  $L$ , we track MSE over observed and masked entries,
+
+$$
+L _ {o b s} = \frac {1}{| \Omega |} \sum_ {(i, j) \in \Omega} (X _ {i j} - \hat {X} _ {i j}) ^ {2}, \quad \text {a n d} \quad L _ {m a s k} = \frac {1}{| \Omega^ {\mathfrak {c}} |} \sum_ {(i, j) \in \Omega^ {\mathfrak {c}}} (X _ {i j} - \hat {X} _ {i j}) ^ {2}.
+$$
+
+Please see Appendix D for details on tokenizing matrices and other experimental details. Code is available at this https://github.com/pulkitgopalani/tf-matcomp.
+
+Compute Resources For  $7 \times 7$  matrices (training and testing), we used a single  $\{\mathrm{V100 / A100 / L40S}\}$  GPU. A single  $\{\mathrm{A40 / A100 / L40S}\}$  GPU was used for matrices of order 10, 12, 15.
+
+# 3 Sudden Drop in Loss
+
+In our training setup, the model converges to a final MSE of approximately  $4\mathrm{e} - 3$  that is, it can solve matrix completion well (as in Fig. 3, this MSE is lower than nuclear norm minimization). Fig. 2 demonstrates the loss dynamics over the course of training the model on this task.
+
+Interestingly, we observe a sudden drop in training loss at approximately step 15000. This sudden drop in loss is reminiscent of phase transitions in physical systems, that are characterized by sudden observable changes in the system on continuous variation of some parameter (here equivalent to the number of training steps). Motivated by this similarity, we analyse the 'pre-shift' model at step 4000, and 'post-shift' model
+
+![](images/fe5d421f1dcfe472630f414925797517a95b5e630baabd4139d56770de15d5f0.jpg)  
+Figure 2: Sharp reduction in training loss.
+
+at the end of training, i.e., step 50000 to understand model properties and sudden drop in loss.
+
+# 3.1 Before the Algorithmic Shift - Copying Phase
+
+Since the value of  $L_{obs}$  remains quite low in the first phase of model training (Fig. 2), we ask: what algorithm does the model use for predicting matrix entries in this phase?
+
+We find that the model learns to copy the input verbatim in the first phase (with output 0 for missing entries), verified through token interventions (Sec. 3.1.1) and by investigating the contribution of attention heads (Sec. 3.1.2) towards the output.
+
+# 3.1.1 Verifying Copying via Token Intervention
+
+To rigorously verify that the pre-shift model indeed copies the input, we replace the masked elements in the  $7 \times 7$ , rank-2 input by the token corresponding to some  $m \in \mathbb{R}$ . For such input, we would like to see whether the model implements copying and outputs  $m$  at the masked positions. In this setup for model output  $\hat{X}$ , MSE at observed positions is  $L_{obs}$ , and for masked positions the MSE is defined as
+
+$$
+L _ {m a s k} ^ {\prime} = \frac {1}{| \Omega^ {c} |} \sum_ {(i, j) \in \Omega^ {c}} (\hat {X} _ {i j} - m) ^ {2}.
+$$
+
+$L_{obs}$  and  $L_{mask}^{\prime}$  for this experiment averaged over 512 samples are compiled in Table 1 (Appendix A). The small loss values confirm that model output matches the ground truth at observed positions, while at masked positions it outputs a value nearly equal to  $m$ . When the mask token is MASK (i.e., no replacement), we set  $m = 0$ , indicating that the model outputs 0 at the masked locations.
+
+To generalize this observation to OOD matrices, we sample uniform random  $7 \times 7$  matrices for input; i.e., all entries in the matrix are i.i.d. uniformly in  $[-1, 1]$ . Importantly, these matrices do not necessarily have a low-rank structure. With these matrices as input to the same pre-shift model as before, we find that model still copies the input (Table 1). This confirms that the model is indeed not 'computing' any entries in the sense of low-rank matrix completion, and simply copies all entries, masked or observed.
+
+# 3.1.2 Attention Heads - Mostly Inconsequential
+
+Attention heads at this stage (Fig. 22a) do not appear to attend to tokens in an interpretable manner. Since the model is copying the input, and does not need to combine different tokens, Attention heads should not affect the model output at this stage. To confirm that this is indeed the case, we do the following tests.
+
+Uniform Ablation Uniform ablation entails replacing the softmax probabilities in an  $n \times n$  attention head by  $1 / n^2$  for all elements i.e. 'force' the model to equally attend to all tokens (Sec. 4.6, [22]). On such an intervention in our case, there is negligible change in MSE at both observed and masked positions. Averaged over 256 samples,  $L_{obs} = 3.4\mathrm{e} - 4$  and  $L_{mask} = 0.2236$  when using all attention heads; whereas, on ablating all heads, these values are  $3.2\mathrm{e} - 4$  and 0.2236 respectively. The negligible change in MSE supports the hypothesis that attention does not contribute to the model output at this stage.
+
+Model Switching In the extreme case, what if we replace the model weights for some component to check for changes to the output? In model switching, we 'transplant' the attention key, query and value weights in the pre-shift model to those from the post-shift model. Averaged over 256 samples,  $L_{obs}$  is 5e-3, that is similar to the optimal total MSE ( $L$ ) obtained at the end of training, while  $L_{mask} = 0.2246$ , similar to the values obtained without such replacement. This shows that replacing the pre-shift attention weights by the optimal ones does not significantly affect  $L_{obs}$ ,  $L_{mask}$  highlighting that attention layers have little effect on the model output at this stage.
+
+# 3.2 After the Algorithmic Shift - Matrix Completion Phase
+
+In this section, we focus on the model properties in the post-shift phase (specifically, at the end of training at 50000 steps). Since  $L$  are near-optimal in this setting, we ask: What algorithm is the model using for completing missing entries? For example, is it implementing the classical nuclear-norm minimization algorithm? For the second question, we show below that the BERT model is not implicitly implementing nuclear norm minimization for completing missing entries in the input.
+
+Nuclear Norm Minimization We use CVXPY [13] to solve low-rank matrix completion using nuclear-norm minimization at various levels of  $p_{\mathrm{mask}}$ , comparing it to the output of a BERT model trained on  $p_{\mathrm{mask}} = 0.3$ . We find that BERT performs better than nuclear norm minimization with respect to MSE; at the same time, the nuclear norm of BERT solution is larger (Fig. 3).
+
+To verify if the model implicitly optimizes a different objective for nuclear norm minimization, we also compare to the regularized version of the above problem  $(\lambda >0)$
+
+$$
+\min  _ {U} \left[ \frac {1}{| \Omega |} \sum_ {(i, j) \in \Omega} (U _ {i j} - X _ {i j}) ^ {2} + \lambda \| U \| _ {*} \right]
+$$
+
+We find that this is not the case, as for various values of  $\lambda$ , BERT still outperforms regularized MSE minimization w.r.t. MSE (Appendix B). This confirms that the model is not implementing
+
+nuclear norm minimization as its algorithm for computing missing entries.
+
+![](images/f62d38793d7b95b1dc3ce0c58bbd3c968ece228baaf67042a7843fe1867e61d2.jpg)  
+Figure 3: BERT v. Nuclear Norm Minimization. Comparing our model (trained with  $p_{\mathrm{mask}} = 0.3$ ) and nuclear norm minimization on the matrix completion task at various levels of  $p_{\mathrm{mask}}$ . The difference in MSE and nuclear norm of solutions obtained using these two approaches indicates that BERT is not implicitly doing nuclear norm minimization to complete missing entries.
+
+We now move to an interpretability based analysis of the model behavior, to attempt to extract useful signal about the implemented algorithm, analysing model behavior for observed and missing entries separately in the following sections.
+
+# 3.2.1 Observed Entries
+
+Uniform Ablation As in Sec. 3.1.2, to quantify the effect of attention heads at this stage, we uniformly ablate all attention heads in the post-shift model. Averaged over 256 samples, this leads to  $L_{obs} = 9.2\mathrm{e} - 5$  without ablation, and  $3.7\mathrm{e} - 3$  with ablation (close to the value of  $L$  at the end of training). However,  $L_{mask}$  increases from 0.0128 to 0.2183, approximately the value of  $L_{mask}$  in the loss plateau before sudden drop. This difference in effect of ablating attention heads confirms that they are much more important for predicting missing entries than for observed entries.
+
+Model Switching We repeat the model switching experiments from Sec. 3.1.2 in the reverse direction i.e. 'transplant' attention key, query, value weights from pre-shift model to the post-shift model. Note that this direction of weight switching is stronger, in the sense that the learnt information in attention layers is removed. We find that on this modification,  $L_{obs} = 9.5\mathrm{e} - 4$  averaged over 256 samples; that is, the observed loss is still not too large. This test confirms that the prediction at observed entries is not substantially affected by the attention layers.
+
+Position Sensitivity Finally, since the attention mechanism crucially depends on token positions, we intervene on this component of the model by randomly permuting its positional embeddings. Formally, the embedding originally for position  $i$  in the input now represents position  $\pi(i)$  for some random permutation  $\pi : [n^2] \to [n^2]$ . Averaged over 256 samples,  $L_{obs} = 2.4\mathrm{e}-4$ , whereas  $L_{mask} = 0.5687$ , indicating that the observed positions are negligibly affected compared to masked positions due to this intervention.
+
+These results support our 'sub-algorithm' hypothesis; (a) since positional information is intuitively not required for the copying sub-algorithm,  $\bar{L}_{obs}$  remains low; and (b)  $L_{mask}$  increases significantly, demonstrating that removing positional information is detrimental to accurately computing missing entries.
+
+# 3.2.2 Missing Entries
+
+To confirm that attention heads causally affect the model output for missing entries, in addition to uniform ablations, we perform causal interventions (activation patching) [42] on the hidden states just after the attention heads. This involves replacing the hidden state after an attention head for input
+
+![](images/3db443da52d2e15281b1c2e5ec2dd484ebace6c421d458e62a1dfc4647c037c3.jpg)  
+Figure 4: Attention heads in post-shift model attend to specific positions. For example, (Layer 2, Head 1) attends to elements in the same row as the query element, and (Layer 2, Head 2) attends to elements in the same column as the query element. (These attention matrices are an average over multiple independent matrix and mask samples.)
+
+$A$  with the hidden state obtained at the same attention head, but for a different input  $A'$ . Ideally, if that head is causally relevant to the output, then such an intervention should steer the model towards the output for  $A'$ , instead of  $A$ . We find in our case that for  $A = X$  and  $A' = -X$ , such an intervention on all attention heads clearly steers the model output at missing entries towards  $-X$  (more details in Appendix F).
+
+Structure in Attention Heads Denote attention head  $H$  in layer  $L$  by the tuple  $(L,H)$ . We can group the attention heads depending on the specific regions of the input matrix they attend to,
+
+1. [Row Head] same row as the query element - 'block-diagonal' patterns, e.g. (2, 1);  
+2. [Column Head] same column as query element - 'off-diagonal' patterns e.g. (2, 2); and  
+3. [Identity Head] query element itself - 'diagonal' patterns in the last layer, e.g. (4, 3).
+
+There are also some other attention heads that do not neatly fit into either of these 3 categories—for example, all heads in layer 1 except (1,3), (1,4); (3,3); (4,2), (4,5-7). In this context, we note that uniformly abating heads (3,3), (4,2), (4,5-7) gives  $L_{obs} = 9.36\mathrm{e} - 5$ ,  $L_{mask} = 0.01575$  compared to  $L_{obs} = 9.44\mathrm{e} - 5$ ,  $L_{mask} = 0.01428$  without ablation, i.e. these uninterpretable heads do not significantly affect the output.
+
+Attention Heads with 'Structured Masking' Since the maps in Fig. 4 are averages over multiple random masks and input matrices, it is difficult to derive more fine-grained insights into the model computation. To address this, we generate inputs with specific mask structure, see for example Fig. 5. This implies that while averaging the attention probabilities over different input matrices, the mask i.e.  $\Omega^c$  remains the same. This step helps us highlight how an attention head attends to input elements based on the element being masked or observed. From the results in Fig. 5, we find clear evidence that different attention heads focus on specific parts of the input. For instance,
+
+1. [Masked-Row Head] (2, 1), (3,4) and (4,8) are mainly active only at the masked rows, and therein attends to the only observed position in those rows.
+
+![](images/2ac8e8bcfd82b3caae9b95e07cc3647047d00fe44a99c87ddec324fa18a5293a.jpg)  
+Figure 5: Attention heads with specific mask structure in inputs. We can derive fine-grained insights about the functions of individual heads in this setup by using a specific mask structure for all input matrices. (Mask appended below each plot, blue denotes missing entries). For example, multiple attention heads like (Layer 2, Head 2) have negligible attention weight at missing positions in the input matrix, implying that these heads attend only to observed entries in the column of the query element. Further, (Layer 2 Head 1) and similar heads have larger attention weights for the rows with missing entries, and in those rows they attend to the sole observed element.
+
+2. [Observed-Copy Head] (4,3) and (4,4) correspond roughly to an identity map, slightly deviating in the masked rows. In these cases, again the maximal attention score corresponds to the only observed position in these rows.  
+3. [Mask-Ignore Heads] Further, there are multiple 'parallel off-diagonal' heads that completely ignore the masked rows for their computation. These heads include (2,2-4), (2,6); (3,2), (3,3), (3,5). Additionally, there are also attention heads like (3,1), (3,6) that attend to only the observed element of each masked row. Collectively these heads act as 'mask-ignore' heads, attending to only observed entries, and using this information to compute missing entries.  
+4. [Longest-Contiguous-Column Heads] There also exist attention heads that respond systematically to changes in the mask. For example, consider attention heads (2, 5), (2, 7), (2, 8) in Fig. 23. For each row, these heads attend to the element in the 6th and 2nd column respectively for part (a) and (b). On a closer look, the connecting link between these two mask patterns is that, the longest contiguous unmasked column is exactly the column that these heads attend to. We hypothesize that this information is somehow used by the model in its inner computation for masked entries.  
+5. [Input-Processing Heads] Finally, Heads (1,1-2), (1, 5-8) do not fall in any of the categories above. These heads are mostly static across different mask / input variations (for example, comparing Fig 4 and 5), and the patterns suggest that these heads almost exclusively focus on the middle row of the input matrix and some other elements. A possible function of these heads is to process positional and token embeddings (input to the first layer) so that this information can be used appropriately in the subsequent layers.
+
+To quantitatively assess the effect of these attention heads on the model output, we also perform uniform ablations on each sub-group separately (Appendix L), and find that the groups significantly affect the output, to varying degrees depending on the specific group.
+
+Probing We probe for properties of the input matrix in the hidden states of the model, to concretely determine how the model computes the output. We use our 12-layer model in this case, to enhance contrast between probing results in different layers.
+
+Specifically, for every element in the input, we fit a linear probe [3] on its hidden state after a given layer, mapping the hidden state to the  $n$ -dimensional masked row that this element belongs to (missing entries are replaced by 0). That is, element at position  $(i,j)$  maps to the 7-dimensional vector  $\tilde{X}_i$ . The results for this experiment in Fig. 6 demonstrate that the hidden states at layer 3 and 4 in the model correlate quite
+
+strongly with the probe target, compared to other layers. This result suggests that the model tracks input information in its intermediate layers and possibly uses it for computing missing entries.
+
+We also probe for the true matrix element at missing entries, and find that the hidden states at these positions get gradually more correlated with depth (through linear probing). Further, we also attempt to extract information about singular vectors of the ground truth matrix from the hidden states through linear probing, though are unable to conclusively do so. We discuss these results in Appendix K.
+
+![](images/1c674f9d7b7f690f69efc330fe21f2517d627efa3d131d305f08d8463677eec0.jpg)  
+Figure 6: Hidden states encode input information. We probe for the row of each element in the masked matrix input, replacing missing entries by 0. We find that layers 3 and 4 have a much lower MSE and a much larger cosine similarity compared to other layers in the model. Hence, these layers somehow 'store information' about the masked input matrix.
+
+![](images/a3a940b41f7ad6bb746e7226c3772c41502cf45558dbe8b9297959b3d307094c.jpg)
+
+# 3.3 Role of Embeddings
+
+Token Embeddings The  $\ell_2$  norm of token embeddings corresponding to values from  $-1.5$  to  $1.5$  is symmetric w.r.t. 0 as seen in Fig. 7a. Further, the PCA of token embeddings in Fig. 7b shows that the embeddings have a separable structure based on the sign of the real-valued input (y-axis), and continuous variation w.r.t. magnitude of input (x-axis). Importantly, unlike other metrics, token embeddings do not seem to abruptly change only at step 15000; rather, the final structure appears before the sudden drop in loss. Similar to [27], we compute the top-2 principal components of the token embeddings at the final step (50000), and project the token embeddings at intermediate training
+
+![](images/cdb45e115c8ad2f216afbafd46f0036e8c54bb309d4b742a9f2c60f27d96aa95.jpg)  
+(a)
+
+![](images/429e1be5b30d8bd2b06a1b63753320865ca840af1b20ea3cd6c69e589d0bcecf.jpg)  
+(b)
+
+![](images/e478db654a1bf0734a5c43f7e9d43bef37aa42eb7b9e4c2fb9cde67753f97418.jpg)  
+(c)  
+Figure 7: Embeddings demonstrate relevant structure. In the post-shift model, positional and token embeddings exhibit properties demonstrating that the model has learnt relevant information about the matrix completion problem. (a)  $\ell_2$  norm of token embeddings is symmetric around 0. This aligns with the intuition that the norm of token embeddings should depend only on the magnitude of the input, and not on its sign. (b) Top-2 principal components of token embeddings correspond to the magnitude and sign of the real valued input. In our case, the 'y-axis' denotes sign of input, and the 'x-axis' denotes the magnitude of the input value. (c) Positional embeddings of elements in the same column cluster together in the t-SNE projection, showing that the model uses positional information relevant to the matrix completion problem.
+
+steps on these components. The results (Fig. 9, Appendix C) show that the embeddings align very closely to the final arrangement before the actual drop. This is as expected, since the model needs to learn what the tokens actually represent on the real line, before it can use those values for completing missing entries. This also explains to some extent why the model implements copying before the sudden drop, since accurately learning token embedding-unembedding is sufficient for that task.
+
+Positional Embeddings In the t-SNE projection of positional embeddings, positions in the same column tend to cluster together as seen in Fig. 7c. This is non-trivial because we have not used any marker tokens to mark the end of a row or column. Further, note that in contrast to token embeddings, positional embeddings do not have a continual evolution in structure – Fig. 10 (Appendix C) shows that the clustering appears only after the sudden drop (step 20000 and after). This along with the evolution of attention heads (Sec. 3.1, 3.2) aligns with how the pre-shift model copies observed entries with little effect from ablating attention heads or positional embeddings (Sec. 3.2.1).
+
+# 4 Sudden Drop in Loss - Role of Model Components
+
+Is it possible to analyse training dynamics of individual model components to derive insights about the full model training? This is motivated by the findings in the previous section on embeddings, and in Section 3.1; the pre-shift model does not use Attention layers for its computation in that stage, and relies on other components to copy input entries. Hence, in our case, the sudden drop corresponds in large part to learning the right Attention patterns (see Appendix M.1). To analyse training dynamics of different model components, we choose (a set of) components - Attention layers, MLP layers, Positional Embeddings and Token Embeddings, randomly initialize them and freeze the weights of other components to their values at the final step of training (Fig. 8).
+
+We find that (a) MLPs and Token Embeddings converge without any observable plateau or sudden drop in loss; (b) for other components, the dynamics resemble those for the full model training (i.e. plateau and then sudden drop), and (c) Positional embeddings show the longest plateau in loss.
+
+![](images/9fc3ac503582a8c4db194b992fab333b497a9e8440bfe88720fdcd1a023a3d6c.jpg)  
+Figure 8: Individual model components have distinct training dynamics. Training individual model components, initializing others to their final value ('All components' indicates normal training). There is no loss plateau for token embeddings and MLP layers, in contrast to positional embeddings, where the sudden drop occurs just before step 40000. In all other cases the sudden drop occurs before the sudden drop in usual training.
+
+Additional Results To further understand the effect of data and model properties on the sudden drop in loss, we train
+
+- a 2-layer, 2-head GPT model on the matrix completion task (Appendix G);  
+- models of different depth (number of layers) and width (hidden state dimension) (App. H);
+
+- our model on (mixture of) matrices of different sizes keeping the rank fixed at 1 (App. I);  
+- a 12-layer, 12-head model on  $10 \times 10$  matrices of multiple ranks (separately) (App. I);  
+- our model on input matrix entries  $(U_{ij}, V_{ij})$  i.i.d.  $\sim \mathcal{N}(0,1)$  instead of Unif[-1, 1] (App. J.1); we also analyze test-time OOD performance of our model in App. J.2.
+
+# 5 Related Work
+
+[27, 28, 32, 41, 38] analyse 'grokking', the sudden emergence of generalization during model training. In the context of training dynamics of MLM, [8] analyses 'breakthroughs' (sudden drop in loss and associated improvement in generalization capabilities of the model), specifically for BERT. They show that the breakthrough marks the transition of the model to a generalizing one. Their work however is focused on language tasks, distinct from our setting which is mathematical (and hence more controllable) in nature. We also note that their work is not in the online training setting; our setup is online in the sense of sampling new data at every step of training.
+
+Mathematical problem solving capabilities of Transformers have been a topic of interest lately [24, 7, 4]. In fact, [24] show that learning addition from samples is equivalent to low-rank matrix completion. Further, [7] show that it is possible to train a transformer based model to solve various linear algebraic tasks e.g. eigendecomposition, matrix inversion, etc.; however, to the best of our knowledge, interpretability studies for such tasks have not been conducted before. For interpretability in simpler math tasks, [18] mechanistically analyse GPT-2 small on predicting whether a number is 'greater-than' a given number, by formulating the problem as a natural language task. [35, 36, 10] analyse BERT from an interpretability perspective. More recently, there has been a line of research works analysing decoder based models to reverse-engineer the mechanisms employed by these models, termed as 'mechanistic interpretability' [14, 31, 32, 39, 11, 33, 25, 26, 23, 34, 20]. We note that our setting is distinct from the recent work on solving mathematical tasks like linear regression through 'in-context' learning in transformers [4, 1, 9, 15, 17, 2, 30, 37]. Whether our model learns to implicitly 'implement' an optimization procedure as shown in some of these is an open question. We discuss related work in more detail in Appendix E.
+
+# 6 Conclusion
+
+We trained a BERT model on matrix completion, and analyzed it before and after the sudden drop in training loss (algorithmic shift) to interpret the algorithm being learnt by the model, and gain insight on why such a sharp drop in loss occurs.
+
+It is evident in our analysis that both before and after the shift, the model does not really compute anything at observed positions, and simply copies these entries. For missing entries, we have shown that the model learns useful abstractions rapidly through the algorithmic shift. Mathematically formulating what algorithm the model employs to implement matrix completion for missing entries is a direction for future work.
+
+Since our work is primarily interpreting model training and mechanism, all experiments are with small scale matrices (largest being  $15 \times 15$ ), and the current method would likely need modifications to scale to larger matrices. Finally, we only intended to study Transformers on matrix completion as a toy task from an interpretability viewpoint, and do not advocate replacing existing efficient solvers for matrix completion with our approach.
+
+Societal Impact We study Transformer based models on their ability to solve a mathematical task (matrix completion) and the associated training dynamics. The work focuses on Transformer interpretability, aiding in improving our understanding of these models and their training and thus we do not foresee any negative societal impact of our work.
+
+Acknowledgements We thank Yu Bai, Andrew Lee, Naomi Saphra and anonymous reviewers for their helpful comments. WH acknowledges support from the Google Research Scholar Program. ESL's time at University of Michigan was supported by NSF under award CNS-2211509 and at CBS, Harvard by the CBS-NTT Physics of Intelligence program.
+
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+
+# A Copying in Pre-Shift Model
+
+<table><tr><td rowspan="2">Step</td><td rowspan="2">Input Samples</td><td colspan="2">Mask = “MASK”</td><td colspan="2">Mask = “0.44”</td><td colspan="2">Mask = “-0.24”</td></tr><tr><td>L′mask</td><td>Lobs</td><td>L′mask</td><td>Lobs</td><td>L′mask</td><td>Lobs</td></tr><tr><td rowspan="2">1000</td><td>Rank-2 matrices</td><td>1.4e-3</td><td>9.3e-4</td><td>7.6e-4</td><td>1e-3</td><td>6.7e-4</td><td>9.6e-4</td></tr><tr><td>Random matrices</td><td>1.5e-3</td><td>8.3e-4</td><td>7.8e-4</td><td>1e-3</td><td>6.8e-4</td><td>9.6e-4</td></tr><tr><td rowspan="2">4000</td><td>Rank-2 matrices</td><td>3e-4</td><td>3.3e-4</td><td>4e-4</td><td>2.8e-4</td><td>3.7e-4</td><td>2.7e-4</td></tr><tr><td>Random matrices</td><td>2.8e-4</td><td>3.5e-4</td><td>3.7e-4</td><td>3e-4</td><td>3.6e-4</td><td>2.8e-4</td></tr><tr><td rowspan="2">14000</td><td>Rank-2 matrices</td><td>1.6e-5</td><td>3.4e-5</td><td>1.8e-5</td><td>4.9e-5</td><td>5.1e-6</td><td>6.7e-5</td></tr><tr><td>Random matrices</td><td>1.1e-5</td><td>3.7e-5</td><td>3.0e-5</td><td>8.5e-5</td><td>4.1e-6</td><td>1.1e-4</td></tr></table>
+
+Table 1: Models at different steps before sudden drop implement copying, predicting the value for mask token at missing entries.
+
+# B Nuclear Norm Minimization
+
+We use the regularized version of the nuclear norm minimization problem as detailed in Sec. 3.2, and obtain the following  $L$ ,  $L_{obs}$ ,  $L_{mask}$  for various values of  $\lambda$ . We average our results over 256 samples generated in the same way as the training data for BERT (including rounding off to 2 decimal places) for the sake of comparison.
+
+<table><tr><td>λ</td><td>Lobs</td><td>Lmask</td><td>L</td></tr><tr><td>0.0005</td><td>1.015e-5</td><td>0.040728</td><td>0.012173</td></tr><tr><td>0.001</td><td>3.686e-5</td><td>0.040456</td><td>0.01211</td></tr><tr><td>0.0015</td><td>7.959e-5</td><td>0.040505</td><td>0.012155</td></tr><tr><td>0.002</td><td>0.00013769</td><td>0.040734</td><td>0.012264</td></tr><tr><td>0.005</td><td>0.00078591</td><td>0.043402</td><td>0.013516</td></tr></table>
+
+# C Embeddings Progress during Training
+
+![](images/8fe5f2fedffff5b28444216b2888004202786718ac86c611a09a50741558b1ba.jpg)  
+Figure 9: Token embedding structure appears before sudden drop. Projection of token embeddings along principal components of embeddings at step 50000. (Labels same as Fig. 7). Embeddings align with the principal components early on in training before the sudden drop in loss.
+
+![](images/22c515b4b37e1ce3b7ed162df6de5f96a7a29c2f4d5027d318c30fba2ce4e547.jpg)  
+Figure 10: Positional embedding structure appears after sudden drop. Evolution of t-SNE projection of positional embeddings with training. (Labels follow the same color coding as Fig. 7). In this case, the positional embeddings show clustering some time after the sudden drop in loss has occurred at step 15000.
+
+# D Experimental Details
+
+Online Training In online training, the data is sampled afresh from the distribution at every step. Since data is not partitioned into fixed train and test sets, we only analyze the training loss in all cases.
+
+Tokenizing Matrices For tokenizing real values, we discretize the range  $[-10, 10]$  in steps of size  $\epsilon = 0.01$ , and assign token IDs starting from 1; the mask token (MASK) is assigned ID 0. Input to the model is the tokenized masked sequence  $X_{mask} = \mathrm{TOK}\left(\operatorname{Vec}(X \odot M)\right)$ , where  $\odot$  denotes the element-wise product,  $\operatorname{Vec}$  denotes vectorizing the  $n \times n$  matrix to a  $n^2$ -dimensional vector, and  $\mathrm{TOK}$  denotes tokenization. Due to this preprocessing, in all cases  $X_{ij}$  is rounded to 2 decimals for computing  $L$ .
+
+# E Related Work
+
+In the online training setup, [5] study the parity learning problem using a variety of model architectures, and show that 'hidden progress measures' can be used to track abrupt changes in model performance during training. [29] study abrupt learning in an autoregressive (GPT), language data setup, connecting learning the grammar to percolation on graphs. [19] discuss abrupt learning dynamics in the context of transformers and claim that the softmax function in Attention leads to longer training loss plateaus – however, reducing the length of plateau does not explain why the drop in loss is sudden and sharp when it occurs. [43] show that initialization of the model affects whether it learns to infer the compositional structure of the task, or simply memorizes the solution.
+
+[4] show that in an in-context learning framework, a transformer based model can learn to select the most optimal statistical method for solving the task in the prompt, without explicitly being provided any information about the optimal method (called 'in-context algorithm selection' in their work). We emphasize that our setup is not in-context learning, and is quite distinct from [4] as far as the task being solved is concerned. However, whether the framework of layer-wise in-context gradient descent can also be used in our setup is a plausible and open direction for future work.
+
+In [7], the author shows empirically that an encoder-decoder transformer can be trained to solve various linear algebraic tasks, such as eigendecomposition, SVD, matrix inverse etc. They support their findings by showing that the model generalizes to matrix distributions outside the training distribution to some extent, and that OOD performance can be improved by training on non-Wigner matrices. While many experiments in [7] also show a sudden jump in accuracy (Fig 1,2), they do not analyze why such a sudden jump occurs during optimization. In our work, we analyze the sudden drop and the model before and after it to derive insights into the sudden drop in loss in our setup.
+
+[24] show that even a small transformer model can be trained to perform arithmetic operations like add, subtract, multiply accurately through appropriate data selection and formatting, and using Chain-of-Thought prompting. They further show that learning addition is connected to rank-2 matrix completion, and that the sudden jump in accuracy with increasing number of observed entries of the matrix is recovered when their model is trained on datasets of different sizes. This is because the size of the dataset for addition can be seen as the number of observed entries of the rank-2 matrix representing the addition table. We point out that while the task in this case is related to matrix completion, ours is a completely different setup, where the sudden drop happens with the number of training steps with each step consisting of 256 low-rank matrices, each with a fixed fraction  $(p_{\mathrm{mask}})$  of observed entries.
+
+# F Causal Intervention on Attention heads
+
+In the uniform ablation setup, it is possible that setting the softmax probabilities to a given value might change the distribution of resultant hidden states, and consequently degrade model performance. A more principled technique to analyze the effect of a specific component is to replace the hidden state just after that component by hidden states on a different input, and analyze how this affects the final output [42]. In our case, we intervene on attention heads by replacing the hidden state after an attention head for input matrix  $X$  by the hidden state for input  $(-X)$ . Importantly, this change does not affect properties like rank of the input, and hence the hidden states obtained are from the same distribution as those for input  $X$ .
+
+Step 1 Extract the hidden states for all attention layers from the model on some input matrix  $X$ ; call these  $h_{+}$ . Concretely, these hidden states are obtained just after the matrix product of the softmax attention probabilities and the value matrix and hence before the output matrix product.  
+Step 2 Change the input to the model to  $-X$ , however, also replace the hidden states just after the attention layers with  $h_{+}$  obtained in Step 1. Call the output of the model in this setup as  $f_{p}(-X,X)$ .
+
+We observe that, the MSE between  $f_{p}(-X,X)$  and  $X$ , averaged over 256 samples at masked positions is approximately 0.014 (this is comparable to optimal  $L_{mask}$ ), compared to the MSE between  $f_{p}(-X,X)$  and  $-X$  being 0.8066. This demonstrates that the attention heads are causally relevant to the model output for missing entries.
+
+# G Autoregressive Setup
+
+![](images/bf888fa822e3295b165cc1ec8a7b17c2695be8b4aab366dfad6ec2026ad83707.jpg)  
+Figure 11: GPT also shows abrupt learning for matrix completion. Training a 2-layer, 2-head GPT model on  $7 \times 7$ , rank-2 matrices in the autoregressive training setup. Here, the model is trained using cross-entropy loss in a next-token prediction setup over full input sequences of the form  $\tilde{X}_{11}, \tilde{X}_{12}, \ldots, \tilde{X}_{77}$ , [SEP],  $X_{11}, X_{12}, \ldots, X_{77}$  where  $\tilde{X}, X$  are partially observed and fully observed matrices, flattened and tokenized as in the BERT experiments. We find that the sudden drop corresponds to the model learning to copy the observed entries in the input matrix. While we could not achieve performance comparable to BERT for missing entries, we believe it should be possible with some modifications to the training setup.
+
+![](images/ee927443596d71da24b672a1145645df556403888890ed50dc188a896365930e.jpg)
+
+![](images/feedf2c9f2a82d0d4491946667a81bd854523d5fb387a4f7ae3c372ec009090e.jpg)  
+Figure 12: Attention heads demonstrate sudden change in matrix completion using GPT. We find that even in the GPT case, the attention heads change from trivial (Layer 2, attending to the [SEP] token for all positions in the output) to those in Layer 1 attending to the corresponding positions in the input ( $\sim$ -identity maps). This corroborates with our finding about the model learning to copy observed entries after the sudden drop, in Fig. 11.
+
+![](images/0f066aa081729f776a4a6b5a83453e1f4533bfbefa673396a549e3eb6f6eb574.jpg)
+
+# H Effect of Model Size
+
+![](images/4ecd2772d84188257c99ca0637b002f5003a7d2134211e3c068b75df4eb3f06b.jpg)  
+Figure 13: Effect of model width. Training with different model widths (hidden state dimensionality) on  $7 \times 7$  rank-2 inputs. The plot demonstrates that  $d = 64$  is too small to obtain accurate matrix completion, and that the performance is sub-optimal for  $d = 128$ . We scale the hidden layer width of the 2-layer MLPs as  $4d$ , as is done in practice.
+
+![](images/70c9d380727c2573920684885e7407a5ab54b7434cbabdcdd32e5815fa5deb69.jpg)  
+Figure 14: Effect of model depth. Training with different model depths (number of layers) on  $7 \times 7$  rank-2 inputs. The plot demonstrates that as depth increases from 4 to 6, the sudden drop occurs earlier, but increasing depth beyond this (8, 12) has little effect. The final MSE obtained also follows the intuitive ordering (largest for  $L = 4$  decreasing with  $L$  upto  $L = 12$ ; though the variation is not significant.
+
+![](images/0e9f8cb87480bbb6fef4f99877e2a8f7e85e4288d3daa316c6439e3a24cd2d3a.jpg)  
+Figure 15: Using 1 attention head per layer. We find that training a 12-layer, 1-head BERT model on matrix completion leads to similar loss (4e-3) and attention heads as the 4-layer, 8-head model.
+
+# I Effect of Matrix Size and Rank on Training
+
+![](images/c05777591aad83bebc1ea60db8bd2d96b0d2ebfbbfeeceda79fe299e97f4003e.jpg)  
+Figure 16: Learning order when trained on mixture of matrices. Training on uniform mixture of  $5 \times 5$ ,  $7 \times 7$  and  $9 \times 9$  rank-1 matrices i.e., at each step, 256 samples of size  $n \times n$ , with  $n$  chosen randomly from  $\{5, 7, 9\}$ . The plots show the test set MSE on separate 256 samples of the 3 different matrix sizes.
+
+![](images/e5dae5799f374fcd96138be8f3ceee85ab75e3a0eea629973e8974c19ac5a4c5.jpg)  
+Figure 17: Effect of problem complexity. Training a 12-layer, 12-head model on  $10 \times 10$  matrices of rank  $r = 1, 2, 3$ . There is a clear progression in terms of the training step where sudden drop occurs, and the final loss values scale roughly as  $L \sim c \cdot 10^r$ ,  $c \approx 2 \times 10^{-6}$ . This also predicts that  $L \sim 0.02$  for  $r = 4$ , i.e. the model does not solve matrix completion to low MSE, which is what we also observe in practice.
+
+# J Effect of Input Distribution
+
+# J.1 Training
+
+![](images/322f2115841df3e01b251316af9a3862c3dd818d1fa83ac7d2e2f106dec35bb0.jpg)  
+Figure 18: Sudden drop is not limited to uniform distribution. Training on i.i.d.  $\mathcal{N}(0,1)$  entries. We find that the sudden drop also occurs in this case, and the final loss value  $\sim 5.6\times 10^{-3}$ , similar to the value obtained for i.i.d. Uniform[-1, 1] entries.
+
+# J.2 Inference
+
+![](images/cd17c59437d85006944ed8b91f3f568efaea634bc9ad1b502138b64f7141507c.jpg)  
+(a) Rank
+
+![](images/4117e721130011d893279d6b30b0c53deb60c6a1cde4d0d97662bf69237a56cc.jpg)  
+(b) No. of Rows  
+Figure 19: Model performs similar to nuclear norm minimization on OOD samples. OOD performance at inference for various values of rank, number of rows and columns of the input matrix. Except (c), the OOD performance of the model is close to the nuclear norm minimization solution for the same inputs. For (c), since we observed that positional embeddings depend on the column of the element, changing the number of columns adversely affects performance.
+
+![](images/2ea4d38bff7ec3551572e5c95afe208e29aea6c88bcb24167ce028947eb0e658.jpg)  
+(c) No. of Columns
+
+Matrix Distribution We also change the input distribution of the matrix entries to Gaussian and Laplace, and measure average MSE over 1024 samples of size  $7 \times 7$  and rank-2, to evaluate the OOD performance of the trained model. We find that
+
+- for entries i.i.d.  $\sim \mathcal{N}(0, 0.25)$ ,  $L \approx 4 \times 10^{-3}$ , and  
+- for entries i.i.d.  $\sim$  Laplace(0, 0.25), (parameterized by mean and scale),  $L \approx 2 \times 10^{-3}$ .
+
+That is, the OOD performance on these distributions is similar to the MSE obtained for the in-training distribution (Uniform[-1, 1]).
+
+# K Probing: Additional Results
+
+![](images/98a729a7cb930ad7c8fdae758e1465e2bc11c9a7efe2c39ffaa0e8deecf46e9d.jpg)  
+Figure 20: Hidden states encode the true value at missing entries. Probing for the corresponding element in the fully observed matrix  $X$ . (Left) comparing the train and test MSE of the linear probe, to confirm that the probe does not overfit. (Right) Test MSE evaluated on the masked elements, that shows an interesting variation: the MSE goes down at a nonlinear rate, hinting that implicit layer-wise optimization could be taking place.
+
+![](images/d6c0fe1df8afa088877eeec5c129d44a40696594fc18bbe996431796351c1b07.jpg)
+
+![](images/afc7d6446f8f6ad4dd8d9099c7f05f86b9389121c04186d9e0ed105b23555716.jpg)  
+Figure 21: Model does not encode singular vectors in hidden states. Probing for the first left singular vector at all positions; that is, for fully observed input matrix  $X \in \mathbb{R}^{n \times n}$  with SVD  $X = U\Sigma V^{\top}$ , we probe for  $u = [U_{11}, U_{21}, \ldots, U_{n1}]^{\top} \in \mathbb{R}^{n}$ . (Left) Train and Test MSE across different layers of the model; (Right) Average Absolute Cosine similarity of the predicted  $u$  with the actual  $u$ . Both evaluations are inconclusive, since the MSE is too large, and the cosine similarity is not much larger than the average absolute cosine similarity ( $\approx 0.3$ ) between 2 i.i.d. vectors  $\sim \mathcal{N}(0, I_{7 \times 7}) \in \mathbb{R}^{7}$
+
+![](images/e797af44bb805d57bdbfd4a425e2f5014fabe2fa1204a32d23f257acadc17df9.jpg)
+
+# L Ablating Groups of Structured Attention Heads
+
+<table><tr><td rowspan="2">Group of Attention Heads Ablated</td><td colspan="2">L</td><td colspan="2">Lobs</td><td colspan="2">Lmask</td><td rowspan="2">Ratio of L (w/ to w/o)</td></tr><tr><td>w/ ablation</td><td>w/o ablation</td><td>w/ ablation</td><td>w/o ablation</td><td>w/ ablation</td><td>w/o ablation</td></tr><tr><td>(2,1), (3,4), (4,8)</td><td>0.0073</td><td>0.0035</td><td>0.0001</td><td>9e-5</td><td>0.0245</td><td>0.0117</td><td>2.09</td></tr><tr><td>(4,3), (4,4)</td><td>0.0079</td><td>0.0045</td><td>3e-4</td><td>8.8e-5</td><td>0.026</td><td>0.015</td><td>1.76</td></tr><tr><td>(2,2-4), (2,6), (3,2), (3,3), (3,5), (3,1), (3,6)</td><td>0.057</td><td>0.0043</td><td>2.1e-4</td><td>9e-5</td><td>0.1885</td><td>0.0142</td><td>13.26</td></tr><tr><td>(2,5), (2,7), (2,8)</td><td>0.0112</td><td>0.0049</td><td>8.5e-5</td><td>9.2e-5</td><td>0.037</td><td>0.016</td><td>2.29</td></tr><tr><td>(1,1-2), (1,5-8)</td><td>0.0314</td><td>0.0048</td><td>1.7e-4</td><td>8.8e-5</td><td>0.1038</td><td>0.0157</td><td>6.54</td></tr></table>
+
+# M Attention Heads
+
+# M.1 Variation along training
+
+![](images/c599f3a6d53bc078bc275ec25900f5c4daffc89b75ea03932af45693d83f513c.jpg)  
+(a) Step 4000
+
+![](images/11f5c0c4a289875ef9b8461be92832d23c34a764004334ecaf6f7993262761bb.jpg)  
+(b) Step 14000
+
+![](images/4c312b0a0b6de91f95a68c3d5481ddd4c73aa5a8dd2b91c6b1602522a47fe81d.jpg)  
+(c) Step 16000
+
+![](images/9a527c50eb543db047dacee92bd533a2db5fac928887cc35dce76fcbae2ec2ee.jpg)  
+(d) Step 20000  
+Figure 22: Attention heads across various training steps.
+
+# M.2 Effect of changing mask structure
+
+![](images/86225258a051f9eae0baa74a0eb7d41f3a9fc55ea2c3b9777ba4b8c4a94fd5d6.jpg)  
+(a)
+
+![](images/d4348f9dc6103f3370985fe8edd9e882ffc10ddcd01717a97d73341282ea2359.jpg)  
+(b)  
+Figure 23: Attention heads and corresponding masks; blue denotes masked position in the input matrix.
+
+# N Attention Heads for larger inputs
+
+![](images/21da95c1df69e9fc472cddfbd20983f5ee70b56d2bdb41c0611ff7cdaab27e9f.jpg)  
+Figure 24: Attention heads in 12 layers, 12-heads model on  $7 \times 7$  rank-2 input
+
+![](images/7b74c488667fabed6e9a9a704caa26ecd3f3edca840b202e498f55d51b19e2df.jpg)  
+Figure 25: Attention heads in 12 layers, 12-heads model on  $12 \times 12$  rank-3 input
+
+![](images/f1e2b33dfb644a72f14b34346bbaea4be82f2057b45e6136de712c5e3dafdfcd.jpg)  
+Figure 26: Attention heads in 12 layers, 12-heads model on  $15 \times 15$  rank-4 input
+
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+# Absorb & Escape: Overcoming Single Model Limitations in Generating Genomic Sequences
+
+Zehui Li $^{1,*}$ , Yuhao Ni $^{1}$ , Guoxuan Xia $^{1}$ , William Beardall $^{1}$ , Akashaditya Das $^{1}$ , Guy-Bart Stan $^{1,*}$ , Yiren Zhao $^{1,*}$
+
+<sup>1</sup>Imperial College London, {zehui.li22, harry.ni21, g.xia21, william.beardall15, akashaditya.das13, g.stan, a.zhao}@imperial.ac.uk
+
+*Correspondence: {zehui.li22, g.stan, a.zhao}@imperial.ac.uk
+
+# Abstract
+
+Recent advances in immunology and synthetic biology have accelerated the development of deep generative methods for DNA sequence design. Two dominant approaches in this field are AutoRegressive (AR) models and Diffusion Models (DMs). However, genomic sequences are functionally heterogeneous, consisting of multiple connected regions (e.g., Promoter Regions, Exons, and Introns) where elements within each region come from the same probability distribution, but the overall sequence is non-homogeneous. This heterogeneous nature presents challenges for a single model to accurately generate genomic sequences. In this paper, we analyze the properties of AR models and DMs in heterogeneous genomic sequence generation, pointing out crucial limitations in both methods: (i) AR models capture the underlying distribution of data by factorizing and learning the transition probability but fail to capture the global property of DNA sequences. (ii) DMs learn to recover the global distribution but tend to produce errors at the base pair level. To overcome the limitations of both approaches, we propose a post-training sampling method, termed Absorb & Escape (A&E) to perform compositional generation from AR models and DMs. This approach starts with samples generated by DMs and refines the sample quality using an AR model through the alternation of the Absorb and Escape steps. To assess the quality of generated sequences, we conduct extensive experiments on 15 species for conditional and unconditional DNA generation. The experiment results from motif distribution, diversity checks, and genome integration tests unequivocally show that A&E outperforms state-of-the-art AR models and DMs in genomic sequence generation. A&E does not suffer from the slowness of traditional MCMC to sample from composed distributions with Energy-Based Models whilst it obtains higher quality samples than single models. Our research sheds light on the limitations of current single-model approaches in DNA generation and provides a simple but effective solution for heterogeneous sequence generation. Code is available at the Github Repo<sup>1</sup>.
+
+# 1 Introduction
+
+DNA sequences, as the blueprint of life, encode proteins and RNAs and directly interact with these molecules to regulate biological activities within cells. The success of deep generative models in image [28], text [27], and protein design [35] has drawn the attention of deep learning researchers to the problem of DNA design, i.e. applying these models to genomic sequence generation [4, 34, 38]. However, one rarely explored issue is how well existing methods can handle the unique property of DNA sequences: heterogeneity. DNA sequences are highly heterogeneous, consisting of multiple
+
+![](images/9c246773eef81896aa34e3e964bcf5d0ce68f1d62a9ed36c3c98cdac7988a2df.jpg)  
+(a) A DNA generated by A&E, interacting with TATA-binding protein. The DNA sequences highlighted in magenta are the TATA-box motif. The confirmation is predicted by AlphaFold 3 [1]: the DNA bends at the TATA-box position.
+
+![](images/d50301461c819e1ec602d55bb22486901fa2dabf0bcdb3ccc6b3cac94c1c20f3.jpg)  
+(b) The proposed framework Fast Absorb & Escape (Fast A&E): The DM and AR models jointly optimize a given sequence by alternating between the A-step and the E-step. See Section 4 for a detailed explanation.  
+Figure 1: (a) Generated DNA interacting with TATA-binding protein. (b) Proposed A&E framework.
+
+connected functional regions (e.g. Promoter Regions, Exons, and Introns) in sequential order. While elements within each functional region might be homogeneous (coming from the same distribution), the overall sequence is non-homogeneous. This heterogeneity, along with the discrete nature of genomic sequences, poses challenges to popular deep generative methods.
+
+Limitations of Existing Single-Model Approaches in Generating Genomic Sequences AutoRegressive Models (AR) [8, 17, 24] are one of the most dominant approaches for discrete sequence generation. To model the data distribution of a sequence  $x$  of length  $T$ , the probability of  $x$  is factorized as:
+
+$$
+p ^ {A R} (\mathbf {x}) = \prod_ {i = 1} ^ {T} p _ {\theta} \left(x _ {i} \mid x _ {1}, x _ {2}, \dots , x _ {i - 1}\right). \tag {1}
+$$
+
+An issue arises when modeling heterogeneous data, where the value of  $\theta$  may vary significantly from one segment to another. Additionally, AR models assume a dependency between the current element and previous elements; this assumption may not hold true for heterogeneous sequences, potentially hindering the learning process (see Section 3 for details).
+
+On the other hand, Diffusion Models (DMs), initially proposed by [30], have been dominant in image generation. In the probabilistic denoising view [16], DMs gradually add noise to the input data  $x_0$ , and a reverse diffusion (generative) process is trained to gradually remove the noise from the perturbed data  $x_t$ . DMs directly model the data distribution without AutoRegressive factorization, thereby avoiding the issues associated with AR models. However, it has been shown that DMs are less competent than AR models for discrete data generation [21, 36]. When it comes to modeling heterogeneous genomic sequences, it remains unclear how the performance of DMs compares to AR models within each homogeneous segment.
+
+Model Composition As a Solution Balancing the ability of generative algorithms to capture both local and global properties of the data distribution is central to the problem. An obvious solution could be to combine these two types of models and perform generation using the composed models. However, this typically requires converting these two models into an Energy-Based Model and then sampling using Markov Chain Monte Carlo (MCMC), which can be inherently slow due to the sampling nature of the algorithm [10], and the potential long inference time of individual models. With the goal of accurate and efficient DNA generation, we aim to investigate two key questions in this work: (i) How well does a single AR model or DM perform in DNA generation, given the heterogeneous nature of genomic sequences? (ii) Is there an efficient algorithm to combine the benefits of AR models and DMs, outperforming a single model? In answering these two questions, our contribution is three-fold:
+
+(a) We study the properties of AR models and DMs in heterogeneous sequence generation through theoretical and empirical analysis (Section 3).
+
+(b) We design the theoretical framework Absorb & Escape (A&E) to sample from the compositional distribution of an AR model and a DM (Section 4.1). Furthermore, as shown in Figure 1b, we propose an efficient post-training sampling algorithm termed Fast A&E to sample from the composed model, requiring at most one forward pass through the pretrained DM and AR model (Section 4.2).  
+(c) We design a comprehensive evaluation workflow for DNA generation, assessing the sequence composition, diversity, and functional properties of generated genomic sequences. Extensive experiments (15 species, 6 recent DMs, 1 AR model, and 3 types of evaluations) reveal: 1) the limitations of existing models in DNA generation (Section 5.3), and 2) that the proposed algorithm Fast A&E consistently outperforms state-of-the-art models as measured by motif distribution and functional property similarity to natural DNA sequences (Sections 4.2 and 5.3).
+
+# 2 Preliminaries and Related Work
+
+# 2.1 Problem Formulation: DNA Generation
+
+DNA generation aims to produce synthetic sequences that functionally approximate real DNA sequences. Formally, let  $\mathbb{N}_4 = \{1,2,3,4\}$ , where each element represents one of the four nucleotides: adenine (A), thymine (T), guanine (G), and cytosine (C). A DNA sequence of length  $L$  can be represented as  $\mathbf{x} \in \mathbb{N}_4^L$ , with each element/nucleotide denoted by  $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_L$ .
+
+Unconditional Generation: Given a dataset of real-world DNA sequences  $\mathcal{X} = \{\mathbf{x}^{(n)}\}_{n=1}^{N}$  collected from some distribution  $p(\mathbf{x})$ , where each sequence  $\mathbf{x}^{(n)} \in \mathbb{N}_4^L$  represents a chain of nucleotides, the objective is to develop a generative model  $p_{\theta}(\mathbf{x})$  of the data distribution  $p(\mathbf{x})$  from which we can sample novel sequences  $\tilde{\mathbf{x}} \sim p_{\theta}(\mathbf{x})$ . These sequences should be structured arrangements of A, T, G, and C, reflecting the complex patterns found in actual DNA. Earlier works applying Generative Adversarial Networks (GANs) [13] to generate protein-encoding sequences [14] and functional elements [33, 34, 38] fall into this category.
+
+Conditional Generation: In this task, the dataset of DNA sequences  $\mathcal{X} = \{\mathbf{x}^{(n)},c^{(n)}\}_{n = 1}^{N}$  is sampled from the joint distribution  $p(\mathbf{x},c)$ , where  $c$  represents the condition associated with each sequence. The objective is to develop a model  $p_{\theta}(\mathbf{x}|c)$  that generates new DNA sequences  $\tilde{\mathbf{x}}$  given condition  $c$ . Recently, a discrete diffusion model DDSM [4] and an AutoRegressive model RegML [20] have used expression level as the condition, while DNADiffusion [26] has used cell type as the condition.
+
+# 2.2 Homogeneous vs. Heterogeneous Sequences
+
+Homogeneous Generation Process In the context of sequence generation, a homogeneous Markov Chain is characterized by constant probabilistic rules for generating the sequence at each time step  $t$ . More generally, a process is defined as homogeneous if the transition probabilities are independent of time  $t$ . This means there exists a constant  $P_{\mathrm{c},j}$  such that:
+
+$$
+P _ {\mathbf {c}, j} = \Pr [ \mathbf {x} _ {t} = j \mid \mathbf {x} _ {1: t - 1} = \mathbf {c} ] \tag {2}
+$$
+
+holds for all times  $t$ , where  $P_{\mathbf{c},j}$  is a constant, and  $\mathbf{c}$  represents a specific sequence of past values, i.e.,  $\mathbf{c} = (c_1, c_2, \ldots, c_{t-1})$ .
+
+Heterogeneous Generation Process Assuming homogeneous properties simplifies modeling but can be overly restrictive for certain modalities, leading to inaccuracies. For example, DNA sequences consist of various functionally distinct regions, such as promoters, enhancers, regulatory regions, and protein-coding regions scattered across the genome [11]. Each region may be assumed to be homogeneous, but the overall sequence is non-homogeneous due to the differing properties of these elements.
+
+For sequences like DNA, which consist of locally homogeneous segments, we define them as heterogeneous sequences. Formally, a heterogeneous sequence is defined as follows: Suppose a sequence  $\mathbf{x}$  is divided into segments  $S_{1}, S_{2}, \ldots, S_{m}$ , where each segment  $S_{i}$  is homogeneous. For each segment  $S_{i} = (\mathbf{x}_{t}, \mathbf{x}_{t-1}, \mathbf{x}_{t-2}, \ldots, \mathbf{x}_{t-k})$ , there exists a constant  $P_{\mathbf{c}_{i,j}}$  such that:
+
+$$
+P _ {\mathbf {c} _ {i}, j} = \Pr [ \mathbf {x} _ {t} = j \mid \mathbf {x} _ {t - k: t - 1} = \mathbf {c} _ {i} ] \tag {3}
+$$
+
+holds for all times  $t$  within segment  $\mathcal{S}_i$ , where  $P_{c_{i,j}}$  is a constant, and  $\mathbf{c}_i = (c_{i,t - 1}, c_{i,t - 2}, \dots, c_{i,t - k})$  represents values characterizing the history within that segment. While segment  $\mathcal{S}_i$  is homogeneous, the entire sequence is non-homogeneous due to the varying properties across different segments.
+
+![](images/6c4e4a12bb8e47eed763300d04afda181bb136191084db964542c7dc5e86548f.jpg)  
+Figure 2: A toy example with heterogeneous sequences: (a) The overall training set consists of  $N = 50,000$  heterogeneous sequences, where each sequence further consists of 16 homogeneous segments. We apply an autoregressive and a diffusion model to learn the underlying distribution. (b) Within each segment, the sequences are generated with a simple Hidden Markov Model (HMM), with deterministic transition probability and emission probability.
+
+# 3 Single Model Limitations in Heterogeneous Sequence Generation
+
+How powerful are AR models and DMs in modelling heterogeneous sequences? We first provide a theoretical analysis, and then perform experiments on synthetic sequences to validate our assumption.
+
+AutoRegressive (AR) Models Suppose a heterogeneous sequence  $\mathbf{x}$  consist of two homogeneous segments of length  $k$ , then  $\mathbf{x} = \{\{x_1, x_2, \dots, x_k\}, \{x_{k+1}, x_{k+2}, \dots, x_{2k}\}\}$ . AR models factorize  $p(\mathbf{x})$  into conditional probability in eq. (4); consider the case where the true factorisation of  $p(x)$  follows eq. (5).
+
+$$
+p ^ {A R} (\mathbf {x}) = p _ {\theta} \left(x _ {1}\right) p _ {\theta} \left(x _ {2} \mid x _ {1}\right) \dots p _ {\theta} \left(x _ {k} \mid \mathbf {x} _ {1: k - 1}\right) \cdot p _ {\theta} \left(x _ {k + 1} \mid \mathbf {x} _ {1: k}\right) p _ {\theta} \left(x _ {k + 2} \mid \mathbf {x} _ {1: k + 1}\right) \dots p _ {\theta} \left(x _ {2 k} \mid \mathbf {x} _ {1: 2 k - 1}\right) \tag {4}
+$$
+
+$$
+p ^ {\text {d a t a}} (\mathbf {x}) = \underbrace {p _ {1} \left(x _ {1}\right) p _ {1} \left(x _ {2} \mid x _ {1}\right) \cdots p _ {1} \left(x _ {k} \mid \mathbf {x} _ {1 : k - 1}\right)} _ {\text {S e g m e n t 1}} \cdot \underbrace {p _ {2} \left(x _ {k + 1}\right) p _ {2} \left(x _ {k + 2} \mid \mathbf {x} _ {k + 1}\right) \cdots p _ {2} \left(x _ {2 k} \mid \mathbf {x} _ {k + 1 : 2 k - 1}\right)} _ {\text {S e g m e n t 2}} \tag {5}
+$$
+
+AR factorisation allows the accurate modelling of the first homogeneous segment; however, it may struggle to disassociate the elements of the second segment from the first segment. More precisely, sufficient data is needed for AR model to learn that  $p_{\theta}(x_{k + 1}), p_{\theta}(x_{k + 2}), \dots, p_{\theta}(x_{2k})$  should be independent to the elements  $\{x_1, x_2, \dots, x_k\}$  in the first segments. Secondly, when the context length of the AR model is shorter than the sequence length  $2k$ , it could struggle to capture the difference between  $p_1$  and  $p_2$  with a single set of parameters  $\theta$ .
+
+Diffusion Models (DMs) On the other hand, DMs estimate the overall probability distribution  $p(\mathbf{x})$  without factorization. The elements of  $\mathbf{x}$  are usually generated in parallel. Thus, they do not suffer from the conditional dependence assumption. However, the removal of the conditional dependence assumption may also decrease the accuracy of generation within each homogeneous segment compared to AR models, as DMs do not explicitly consider previous elements.
+
+# 3.1 A Toy Example
+
+To evaluate the performance of Autoregressive (AR) models and Diffusion Models (DMs) in generating heterogeneous sequences, we consider a toy example with 50,000 heterogeneous sequences  $\mathcal{X} = \{\mathbf{x}^{(n)}\}_{n=1}^{50000}$ . Each sequence contains 16 segments, as illustrated in Figure 2(a), and each segment comprises 16 elements, resulting in a total sequence length of 256 ( $\mathbf{x} \in \mathbb{N}_4^{256}$ ). A simple Hidden Markov Model (HMM) is used to generate each segment, as shown in Figure 2(b), with deterministic transition and emission probabilities that ensure homogeneity within each segment. The emitted tokens differ from one segment to another, mimicking the properties of real DNA
+
+sequences. Whilst it is possible to use more complex distributions for each segment, doing so could complicate the evaluation of the generated sequences.
+
+Evaluation Under our toy HMM setup, a generative model could make two types of mistakes within each segment: 1) Illegal Start Token: The generated sequence starts with a token which has zero emission probability. E.g. in Figure 2(b), the starting token could only be  $\{A,T\}$ .  $\{G,C\}$  at the beginning of the sequence are classified as illegal start tokens. 2) Incorrect Transition: The generated sequence contains tokens with zero transition probability. E.g. in Figure 2(b) given the start of the sequence is  $(A,T,A,T)$ , the next token must be  $A$ , any other tokens such as  $\{T,G,C\}$  are classified as incorrect transitions. We use the number of incorrect tokens as the metric for evaluation.
+
+Experiment We use HyenaDNA [20, 24] as the representative autoregressive (AR) model. For the diffusion model, we develop a simple latent Discrete Diffusion model termed DiscDiff. It resembles the design of StableDiffusion [28], a latent diffusion model for image generation. DiscDiff consists of
+
+Table 1: Number of Incorrect Tokens on Synthetic Dataset. The performance metrics used are the number of Illegal Start (IS) Tokens and Incorrect Transition (IT) Tokens. Note that there are a total of  $4,000 \times 256 = 1024,000$  tokens.
+
+<table><tr><td></td><td>HYENADNA</td><td>DISCDIFF</td></tr><tr><td># IS TOKENS ↓</td><td>812</td><td>0</td></tr><tr><td># IT TOKENS ↓</td><td>3,586</td><td>110,192</td></tr></table>
+
+a CNN-based Variational Encoder-decoder, trained with cross entropy, to map the discrete DNA data into a latent space, and a standard 2-D UNet as the denoising network (detailed in appendix A). The training dataset consists of  $\mathcal{X} = \{\mathbf{x}^{(n)}\}_{n=1}^{50,000}$ . For detailed training procedures see Appendix B. We generate 4,000 sequences from each model and present the evaluation results in Table 1. As expected, the diffusion model DiscDiff makes fewer errors regarding Illegal Start (IS) tokens but tends to generate more Incorrect Transition (IT) tokens. Conversely, while the AR model HyenaDNA generates some IS token errors, it produces significantly fewer IT token errors. This motivates the question: can we combine the strengths of both algorithms to achieve better sequence generation?
+
+# 4 Method
+
+# Algorithm 1 Absorb & Escape Algorithm
+
+Require: Pretrained AutoRegressive model  $p_{\theta}^{AR}(\mathbf{x})$  and pretrained Diffusion Model  $p_{\beta}^{DM}(\mathbf{x})$
+
+1: Initialize  $\tilde{\mathbf{x}}^0\sim p_\beta^{DM}(\mathbf{x})$  
+2: Set  $t = 0$  
+3: Assume  $\mathbf{x} = \{\mathbf{s}_1, \mathbf{s}_2, \dots, \mathbf{s}_n\}$ , where each  $\mathbf{s}_k = \{\mathbf{x}_i, \mathbf{x}_{i+1}, \dots, \mathbf{x}_j\}$  is a segment  
+4: while not converged do  
+5: Sample a segment  $\mathcal{S} \in \{\mathbf{s}_1, \mathbf{s}_2, \dots, \mathbf{s}_n\}$  
+6: Set  $i =$  start index of  $\mathbf{s}_k$ ,  $j =$  end index of  $\mathbf{s}_k$  
+7: Absorb step:  
+8:  $\begin{array}{rlr}\overline{\mathbf{x}}^{\prime}_{i:j} & \sim & p(\mathbf{x}_{i:j}|\mathbf{x}_{0:i - 1},\mathbf{x}_{j + 1:L})\\ & \approx & \\ p_{\theta}^{AR}(\mathbf{x}_{i:j}|\mathbf{x}_{0:i - 1}) / / Refine~segment~\mathbf{s}_k~us-ing~the~AR~model \end{array}$  
+9: Escape step:  
+10:  $\tilde{\mathbf{x}}_{i:j}^{t} = \tilde{\mathbf{x}}_{i:j}^{\prime} / / U p d a t e\tilde{\mathbf{x}}^{t}$  
+11: Increment  $t = t + 1$  
+12: end while  
+13: Output: Final sample  $\tilde{\mathbf{x}}^t$  with improved quality
+
+# Algorithm 2 Fast Absorb & Escape Algorithm
+
+Require: Absorb Threshold  $T_{Absorb}$ , Pretrained AutoRegressive model  $p_{\theta}^{AR}(\mathbf{x})$  and pretrained Diffusion Model  $p_{\beta}^{DM}(\mathbf{x})$
+
+1: Initialize  $\tilde{\mathbf{x}}^0\sim p_\beta^{DM}(\mathbf{x})$  
+2: for  $i$  in len(x) do  
+3: if  $p^{DM} < T_{Absorb}$  then  
+4: Absorb step:  
+5:  $\mathrm{j} = \mathrm{i} + 1$  
+6:  $\tilde{\mathbf{x}}_j^{\prime}\sim p_{\theta}^{AR}(\mathbf{x}_j|\mathbf{x}_{0:i})$  
+7: while  $p^{AR}(\tilde{\mathbf{x}}_j^{\prime}) > p^{DM}(\tilde{\mathbf{x}}_j)$  do  
+8: Increment  $j = j + 1$  
+9:  $\tilde{\mathbf{x}}_j^{\prime}\sim p_{\theta}^{AR}(\mathbf{x}_j|\mathbf{x}_{0:i},\mathbf{x}_{i:j - 1})$  //Refine Inaccurate region of the sequence token by token  
+0: end while  
+1: Escape step:  
+12:  $\tilde{\mathbf{x}}_{i:j} = \tilde{\mathbf{x}}_{i:j}^{\prime}$  //Update  $\tilde{\mathbf{x}}$  
+13: Increment  $\mathrm{i} = \mathrm{i} + \mathrm{j}$  
+14: end if  
+15: end for  
+16: Output:  $\tilde{\mathbf{x}}$  with improved quality
+
+# 4.1 The Absorb & Escape Framework
+
+Given a pretrained AutoRegressive model  $p_{\theta}^{AR}(\mathbf{x})$  and a Diffusion Model  $p_{\beta}^{DM}(\mathbf{x})$ , we aim to generate a higher quality example  $\tilde{\mathbf{x}}$  from the composed distribution  $p_{\theta, \beta}^{C}(\mathbf{x}) = p_{\theta}^{AR}(\mathbf{x}) \circ p_{\beta}^{DM}(\mathbf{x})$ . However, directly computing  $p_{\theta, \beta}^{C}(\mathbf{x})$  is generally intractable, as both the autoregressive factorizations from  $p^{AR}$  and score functions from  $p^{DM}$  are not directly composable [9, 10]. We propose the Absorb & Escape (A&E) framework, as shown in Algorithm 1, to efficiently sample from  $p_{\theta, \beta}^{C}(\mathbf{x})$ .
+
+Absorb-Step Inspired by Gibbs sampling [12], which iteratively refines each dimension of a single sample and moves to higher density areas, our algorithm starts with a sequence  $\mathbf{x}^0\sim p^{DM}(\mathbf{x})$  generated by the diffusion model and then refines the samples through the Absorb step and Escape step. By exploiting the heterogeneous nature of the sequence, we assume that  $\mathbf{x}^0$  can be factorized into multiple segments  $\{\mathbf{s}_1,\mathbf{s}_2,\dots ,\mathbf{s}_n\}$ . For each segment  $\mathbf{s}_k$ , we set  $i$  and  $j$  as the start and end indices, respectively. During the Absorb step, we sample a subset of segments  $\mathcal{S}\subseteq \{\mathbf{s}_1,\mathbf{s}_2,\dots ,\mathbf{s}_n\}$  and refine each segment  $\mathbf{s}_k$  by sampling  $\tilde{\mathbf{x}}_{i:j}^{\prime}\sim p(\mathbf{x}_{i:j}|\mathbf{x}_{0:i - 1},\mathbf{x}_{j + 1:L})\approx p_{\theta}^{AR}(\mathbf{x}_{i:j}|\mathbf{x}_{0:i - 1})$ , using the autoregressive model to approximate the conditional probability.
+
+Escape-Step After refining the segment in the Absorb step, we proceed with the Escape step where we update the refined segment  $\tilde{\mathbf{x}}_{i:j}^{t}$  to  $\tilde{\mathbf{x}}_{i:j}^{\prime}$ . This iterative process continues for each selected segment  $\mathbf{s}_k$ , with  $t$  incrementing after each update. By leveraging the ability of the diffusion model to capture the overall data distribution and the autoregressive model to refine homogeneous sequences within each segment, our algorithm efficiently improves the quality of the generated samples. The final output  $\tilde{\mathbf{x}}^t$  is hereby closer to the true data distribution  $p(\mathbf{x})$  compared to the initial sample  $\tilde{\mathbf{x}}^0$ . A proof for the convergence in Proposition 1 is provided in Appendix C.
+
+Proposition 1. The Absorb & Escape (A&E) algorithm converges to the target distribution  $p_{\theta, \beta}^{C}(\mathbf{x}) = p_{\theta}^{AR}(\mathbf{x}) \circ p_{\beta}^{DM}(\mathbf{x})$ , under the assumptions that both models are properly trained, the segments of  $\mathbf{x}$  are homogeneous, the subset of segments is chosen randomly, and the conditional distribution  $p(\mathbf{x}_{i:j} | \mathbf{x}_{0:i-1}, \mathbf{x}_{j+1:L})$  is accurately approximated by  $p_{\theta}^{AR}(\mathbf{x}_{i:j} | \mathbf{x}_{0:i-1})$ .
+
+# 4.2 Practical Implementation: Fast A&E
+
+While A&E offers a method to sample from a compositional distribution, two practical issues remain unresolved. Firstly, the algorithm may take a considerable amount of time to converge. Secondly, a crucial step in Line 3 of Algorithm 1 involves splitting  $\mathbf{x}$  into homogeneous segments  $\{s_1, s_2, \dots, s_n\}$  and then sampling a subset of these segments. Segmentation is straightforward when the boundaries of functional regions of the DNA sequence are known, such as protein-coding regions, exons, or introns, where each region naturally forms a homogeneous segment. However, this information is often unavailable in practice.
+
+To address these challenges, we propose a practical implementation termed Fast A&E. For generating a sequence  $\mathbf{x} \in \mathbb{N}_4^L$ , it requires at most  $L$  forward passes through the AR model. As shown in Algorithm 2 and Figure 1b, Fast A&E adopts a heuristic-based approach to select segments for refinement. It scans the sequence from left to right, identifying low-quality segments through a thresholding mechanism. Tokens with predicted probabilities smaller than the  $T_{absorb}$  threshold trigger the absorb action, while the autoregressive process terminates once the probability of a token generated by the AR model  $p^{DM}(\tilde{\mathbf{x}}_j')$  is smaller than that of the diffusion model  $p^{AR}(\tilde{\mathbf{x}}_j)$ . In this manner, Fast A&E corrects errors made by the diffusion model with a maximum running time of  $O(T_{DM} + T_{AR})$ , where  $T_{DM}$  and  $T_{AR}$  are the times required for generating a single sequence from the diffusion model and autoregressive model, respectively.
+
+# 5 Experiment
+
+# 5.1 Transcription Profile (TP) conditioned Promoter Design
+
+We first evaluate Fast A&E in the task of TP-conditioned promoter design, following the same evaluation procedures and metrics as used by DDSM [4] and Dirichlet Flow Matching (DFM) [32].
+
+Data Format & Evaluation Metric: Each data point in this task is represented as a (DNA, signal) pair, where signal corresponds to the CAGE values for a given DNA sequence, providing a quantitative measure of gene expression around the transcription start site (TSS). Both the DNA sequence and the signal have a length of 1024. The goal of this task is to generate DNA sequences conditioned on specified signals. During evaluation, for a given test set data point  $(x,c)$ , the generated sequence  $\tilde{\mathbf{x}}$  and the ground truth sequence are processed through the genomic neural network Sei [37]. The performance metric is the mean squared error (MSE) between  $Sei(x)$  and  $Sei(\tilde{\mathbf{x}})$ .
+
+Table 2: Evaluation of transcription profile conditioned promoter sequence design. A&E achieves the smallest MSE with Language Model and DFM distilled being the AR and DM components.  
+
+<table><tr><td>Method</td><td>MSE↓</td></tr><tr><td>Bit Diffusion (bit-encoding)*</td><td>.0414</td></tr><tr><td>Bit Diffusion (one-hot encoding)*</td><td>.0395</td></tr><tr><td>D3PM-uniform*</td><td>.0375</td></tr><tr><td>DDSM*</td><td>.0334</td></tr><tr><td>Language Model*</td><td>.0333</td></tr><tr><td>Linear FM*</td><td>.0281</td></tr><tr><td>Dirichlet FM (DFM)*</td><td>.0269</td></tr><tr><td>Dirichlet FM distilled (DFM distilled)*</td><td>.0278</td></tr><tr><td>A&amp;E (Language Model+Dirichlet FM distilled)</td><td>.0262</td></tr></table>
+
+Results: As shown in Table 2, we ran Fast A&E on this task with a default threshold of  $T_{\mathrm{absorb}} = 0.85$ , using the pretrained model as both the autoregressive (AR) model and the denoising model (DM) component. The evaluation followed the same procedure as described in the DFM repository. The Fast A&E model, comprising AR and DM components (i.e., the language model and the distilled DFM checkpoints provided in the DFM repository), achieved state-of-the-art results with an MSE of 0.0262 on the test split.
+
+# 5.2 Multi-species Promoter Generation
+
+# 5.2.1 Experimental Setup
+
+Dataset Construction Prior efforts in DNA generation have been constrained by small, single-species datasets [19, 34]. To better evaluate the capability of various generative algorithms in DNA generation, we construct a dataset with 15 species from the Eukaryotic Promoter Database (EPDnew)[23]. Table 3 compares our EPD dataset with those used in previous studies, including DDSM[4], ExpGAN [38], and EnhancerDesign [33]. The key advantage of EPD is its diversity in both species types and DNA sequence types. Additionally, although the number of sequences in EPD is on a similar scale to that of DDSM, EPD offers greater uniqueness: each sequence corresponds to a unique promoter-gene combination, a guarantee not provided by the other datasets.
+
+Baseline Model We evaluate the state-of-the-art diffusion models for DNA sequence generation:  $DDSM$  [4], DNADiffusion [26],  $DDPM$  [2, 3], and a AR model Hyena [20, 24]. In addition, we implement a VAE with a CNN-based encoder-decoder architecture. Adding UNet as the denoising network to VAE results in another baseline latent diffusion model termed  $DiscDiff$ . For a fair evaluation, we maximally scale up the denoising networks of each diffusion model to fit
+
+Table 3: A comparison of DNA generation datasets. EPD used in this work is significantly larger in size and contains fifteen species. Reg. represents the regulatory regions, and Prot. represents the protein encoding region.  
+
+<table><tr><td>Dataset</td><td># DNA</td><td>Multi Species</td><td>DNA Regions</td></tr><tr><td>EPD (Ours)</td><td>160,000</td><td>✓</td><td>Reg. &amp; Prot.</td></tr><tr><td>DDSM [4]</td><td>100,000</td><td>×</td><td>Reg. &amp; Prot.</td></tr><tr><td>ExpGAN [38]</td><td>4238</td><td>×</td><td>Reg.</td></tr><tr><td>EnhancerDesign [33]</td><td>7770</td><td>×</td><td>Reg.</td></tr></table>
+
+into an 40GB NVIDIA A100. Additionally, we adapt four pretrained Hyena models from Hugging-Face for comprehensive fine-tuning. The additional details of the network architectures are shown in Appendix D.
+
+Model Training All the models are implemented in Pytorch and trained on a NVIDIA A100-PCIE-40GB with a maximum wall time of 48 GPU hours per model; most of the models converged within the given time. Adam optimizer [7] is used together with the CosineAnnealingLR [22] scheduler. The learning rate of each model are detailed in Appendix D. For the evaluation of various diffusion models in unconditional generation (see Section 5.2.2), we sample 50,000 sequences from each model. For
+
+the conditional generation across 15 species (see Section 5.3), we generate 4,000 sequences. In both cases, we use the DDPM sampler [31] with 1,000 sequential denoising steps.
+
+# 5.2.2 Evaluating Diffusion Models on Mammalian Model Organisms
+
+Table 4: Comparison of diffusion models on unconditional generation evaluated on EPD (256 bp) and EPD (2048 bp). Metrics include S-FID,  $\mathrm{Cor}_{\mathrm{TATA}}$ , and  $\mathrm{MSE}_{\mathrm{TATA}}$ . The best and second-best scores are highlighted in bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">EPD (256 bp)</td><td colspan="3">EPD (2048 bp)</td></tr><tr><td>S-FID ↓</td><td>CorTATA ↑</td><td>MSE TATA ↓</td><td>S-FID ↓</td><td>CorTATA ↑</td><td>MSE TATA ↓</td></tr><tr><td>Random (Reference)</td><td>119.0</td><td>-0.241</td><td>8.21</td><td>106.0</td><td>0.030</td><td>1.86</td></tr><tr><td>Sample from Training Set</td><td>0.509</td><td>1.0</td><td>0</td><td>0.100</td><td>0.999</td><td>0</td></tr><tr><td>VAE</td><td>295.0</td><td>-0.167</td><td>26.5</td><td>250.0</td><td>0.007</td><td>9.40</td></tr><tr><td>BitDiffusion</td><td>405</td><td>0.058</td><td>5.29</td><td>100.0</td><td>0.066</td><td>5.91</td></tr><tr><td>D3PM (small)</td><td>97.4</td><td>0.0964</td><td>4.97</td><td>94.5</td><td>0.363</td><td>1.50</td></tr><tr><td>D3PM (large)</td><td>161.0</td><td>-0.208</td><td>4.75</td><td>224.0</td><td>0.307</td><td>8.49</td></tr><tr><td>DDSM (Time Dilation)</td><td>504.0</td><td>0.897</td><td>13.4</td><td>1113.0</td><td>0.839</td><td>2673.7</td></tr><tr><td>DiscDiff (Ours)</td><td>57.4</td><td>0.973</td><td>0.669</td><td>45.2</td><td>0.858</td><td>1.74</td></tr></table>
+
+One prerequisite of Fast A&E (Algorithm 2) is that the diffusion model  $P_{\beta}^{DM}$  should be properly trained and provide accurate approximations of underlying data distribution. We first evaluate existing Diffusion Models on a subset of EPD datasets. This subset includes sequences from four mammalians H. Sapiens (human), Rattus Norvegicus (rat), Macaca mulatta, and Mus musculus (mouse), which collectively represent  $50\%$  of the total dataset. Training on this subset allows for a more precise assessment of the generative algorithm's accuracy in a unconditional generation setting.
+
+# Metrics
+
+1. Motif Distribution Correlation  $(\mathrm{Cor}_{\mathbf{M}})$  and Mean Square Error  $(\mathbf{MSE}_{\mathbf{M}})$ :  $\mathrm{Cor}_{\mathbf{M}}$  is the Pearson correlation between the motif distributions of generated and natural DNA sequences for motifs like TATA-box, GC-box, Initiator, and CCAAT-box.  $\mathrm{MSE}_{\mathbf{M}}$  is the average squared differences between these motif distributions.  
+2. S-FID (Sei Fréchet Inception Distance): Measures the distance between distributions of generated and natural DNA sequences in latent space similar to the FID metric [15] for images, replacing the encoder with the pre-trained genomic neural network, Sei [37].
+
+The results are presented in Table 4, indicating that most existing models perform worse than the simple baseline DiscDiff proposed here, as measured by S-FID,  $\mathrm{Cor}_{\mathrm{TATA}}$ , and  $\mathrm{MSE}_{\mathrm{TATA}}$ . TATA is one of the most fundamental motifs for gene transcription – a special type of protein called transcription factors binds to TATA tokens on the DNA as shown in Figure 1a. It changes the shape of DNA and then enables the gene transcription. The failure of existing diffusion models to capture the TATA-box distribution indicates a potential gap in existing research. In the following, we hereby use DiscDiff as the  $p_{\beta}^{DM}$  to initialize the A&E algorithm.
+
+# 5.3 Multi-species DNA Sequences Generation
+
+We compare our model Fast A&E with Hyena [24] and the best-performing diffusion model from Section 5.2.2, DiscDiff, on the task of generating species-specific DNA sequences.
+
+Motif-centric Evaluation We consider four types of motifs closely related to promoter activities  $\{\mathrm{TATA}$  -box, GC-box, Initiator, CCAAT-box\}. We calculate 4 types of motif distributions for 15 species across 3 models, resulting in 180 frequency distributions.
+
+Figure 3 shows the average MSE and Correlation between generated and natural DNA distributions for each model and motif type across 15 species. Fast A&E improves upon Hyena and DiscDiff, generating the most realistic sequences across almost all species and motif types. It achieves the lowest MSE and highest Correlation across all four motifs. This pattern is consistent across all 15 species. The motif plots for all 15 species are provided in Appendix F. As an example, Figure 4 shows the motif distributions of sequences generated by the three models versus real DNA sequences
+
+![](images/cc7c793397613d1f672463db24edfea450d03b512be589f8159507e50a84f26a.jpg)  
+Figure 3: The average MSE and Correlation between generated and natural DNA distributions for each model and motif type across 15 species. Fast A&E outperforms Hyena and DiscDiff, generating the most realistic sequences with the lowest MSE and highest Correlation across four motif types. This pattern is consistent across all 15 species.
+
+for macaque. Fast A&E closely resembles the natural motif distribution, especially for the TATA and GC box motifs, while Hyena and DiscDiff fail to capture the values or trends accurately.
+
+Sequence Diversity To assess the diversity of the generated sequences, we applied BLASTN [18] to check (1) the similarity between the training set (Natural DNA) and generated sequences from three models, and (2) the similarity within the generated sequences. BLASTN takes a query DNA sequence and compares it with a database of sequences, returning all the aligned sequences in the database that are similar to the query. For each alignment, an alignment score, the alignment length (AlignLen), and statistical significance (eValue) are provided to indicate the quality of the alignment, where a larger alignment score, a smaller statistical significance (eValue), and a longer alignment sequence (AlignLen) indicate a higher similarity between the query sequence and the database
+
+sequences. Ideally, when using generated sequences to query the training dataset, a good generative sequence should align better than a random sequence, but not replicate the sequences in the training set.
+
+![](images/c156b1bff2e59bcfae2b1236e3052ec0d76f4386a74a0ef7de49dfc77ca72b7b.jpg)  
+Figure 4: Motif distributions in macaque DNA compared across natural DNA, FAST A&E, DiscDiff, and Hyena. FAST A&E closely aligns with natural DNA, especially for the TATA and GC motifs.
+
+Table 5 shows the results of the BLASTN algorithm. From the table, DiscDiff, Hyena, and A&E all satisfied the mentioned criteria. In terms of the diversity within the groups, none of the algorithms generated repetitive sequences. Furthermore, A&E tends to lie between Hyena and DiscDiff in terms of the diversity of the generated sequences, implying that A&E may combine the properties of AR and DM models. One notable fact is that the alignment scores are very high within the natural sequences, potentially indicating that natural sequences
+
+naturally have repetitive properties (conservative motifs), while the generative sequences do not have this characteristic.
+
+Table 5: BLASTN Results  
+
+<table><tr><td>Query vs. Database</td><td>Score↑</td><td>eValue↓</td><td>AlignLen↑</td></tr><tr><td>Random vs. Natural DNA</td><td>17.78</td><td>1.1769</td><td>24.2</td></tr><tr><td>A&amp;E vs. Natural</td><td>21.39</td><td>0.1695</td><td>35.9</td></tr><tr><td>Hyena vs. Natural</td><td>22.89</td><td>0.2895</td><td>40.1</td></tr><tr><td>DiscDiff vs. Natural</td><td>20.25</td><td>0.2098</td><td>31.4</td></tr><tr><td>A&amp;E vs. A&amp;E</td><td>20.14</td><td>0.0968</td><td>33.69</td></tr><tr><td>Hyena vs. Hyena</td><td>19.57</td><td>0.0843</td><td>28.7</td></tr><tr><td>DiscDiff vs. DiscDiff</td><td>20.95</td><td>0.1029</td><td>37.6</td></tr><tr><td>Natural vs. Natural DNA</td><td>57.06</td><td>0.0633</td><td>77.6</td></tr></table>
+
+Genome Integration with Promoter Sequences As shown in Figure 5, to evaluate the functional properties of sequences generated by Hyena, DiscDiff, and  $A\& E$ , we inserted the generated promoter sequences of length 128 bp upstream  $(5^{\prime})$  of three commonly studied genes in oncology: TP53,
+
+![](images/5d85daf56466a6c98e516ba5a0528f9a7526fcea3404e37f9eec550960289c38.jpg)  
+Figure 5: Evaluation of Generated Promoters for gene regulation through Genome Integration EGFR, and AKT1 [6, 25, 29], which are closely related to tumor activities. Our goal was to determine which model generates promoter sequences that produce gene expression levels closest to those of natural promoters when reinserted into the human genome.
+
+We use Enformer [5] to predict transcription profiles. Enformer takes a DNA sequence of 200k bps as input and outputs a matrix  $P \in \mathbb{R}^{896 \times 638}$ , representing a multi-cell type transcription profile. We sampled 300 promoter sequences from each source: Natural DNA promoters, Hyena, DiscDiff, and A&E. For each set, we calculated the average transcription profile across the sequences. The Sum of Squared Errors (SSE) between these average transcription profiles of the generated sequences and those of natural promoters
+
+Table 6: Sum of Squared Errors (SSE) of Transcription Profiles between Real and Generated Sequences  
+
+<table><tr><td></td><td>TP53↓</td><td>EGFR↓</td><td>AKT1↓</td></tr><tr><td>Random</td><td>278.18</td><td>8.09</td><td>65.70</td></tr><tr><td>A&amp;E</td><td>17.21</td><td>0.28</td><td>1.65</td></tr><tr><td>Hyena</td><td>36.25</td><td>0.89</td><td>2.88</td></tr><tr><td>DiscDiff</td><td>124.03</td><td>2.17</td><td>25.50</td></tr></table>
+
+are shown in Table 6. The results indicate that A&E produces the smallest SSE, suggesting it best captures the properties of natural DNA. This finding highlights the potential of generative algorithms to create promoter sequences that effectively regulate gene expression, with applications in bioproduct manufacturing and gene therapy.
+
+# 5.3.1 Sensitivity Analysis of  $T_{\mathrm{Absorb}}$
+
+We perform a sensitivity analysis of A&E algorithm over hyperparameter  $T_{\mathrm{absorb}}$ . As shown in Figure 6, with a small  $T_{\mathrm{absorb}}$ , the sequences generated by A&E are dominated by the diffusion model. As  $T_{\mathrm{absorb}}$  increases, the AR helps to correct the errors made by the DM. Finally, when  $T_{\mathrm{absorb}}$  is larger than 0.7, the correlation flattens and fluctuates. In conclusion, A&E is robust under different values of  $T_{\mathrm{absorb}}$ , and it is best to use the validation dataset to choose the optimal value. However, a wide range of  $T_{\mathrm{absorb}}$  can still be used with improved performance.
+
+![](images/129c849e9e21d6e0756cb011a265a6975084742b68dc5c3ab2155162d6e4a9b7.jpg)  
+Figure 6: Sensitivity of A&E under various  $T_{\mathrm{absorb}}$ . For each value of  $T_{\mathrm{absorb}}$ , 3000 sequences are generated and compared with natural DNA. The correlation between the generated sequences and natural DNA increases initially as  $T_{\mathrm{absorb}}$  increases, and then it flattens. An optimal  $T_{\mathrm{absorb}}$  can be selected based on the validation set, or a default value of 0.85 can be used.
+
+# 6 Conclusion
+
+This paper demonstrates that (i) both the AutoRegressive (AR) model and Diffusion Models (DMs) fail to accurately model DNA sequences due to the heterogeneous nature of DNA sequences when used separately, and (ii) this limitation can be overcome by introducing A&E, a novel sampling algorithm that combines AR models and DMs. Additionally, we developed a fast implementation of the proposed algorithm, Fast A&E, which enables efficient generation of realistic DNA sequences without the repetitive function evaluations required by conventional sampling algorithms. Experimental results across 15 species show that Fast A&E consistently outperforms single models in generating DNA sequences with functional and structural similarities to natural DNA, as evidenced by metrics such as Motif Distribution, Sequence Diversity, and Genome Integration. Regarding the future work, the generated DNA sequences still require validation through wet-lab experiments before they can be directly used in clinical settings.
+
+# Acknowledgments and Disclosure of Funding
+
+This research project has benefitted from the Microsoft Accelerate Foundation Models Research (AFMR) grant program. Zehui Li is grateful to his family—Jinghai Li, Huiqing Wang, and Yiqiu Sun for their support.
+
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+
+![](images/af0d92fd4e5df829119cfb154af2fc30dbaf833b8e13ffec599ff85b0712e2e3.jpg)  
+Figure 7: DiscDiff Model: A two-step process for DNA sequence generation. Step 1: VAE Training: A sequence  $s \in \{A, T, G, C\}^{2048/256}$  is encoded via a 1D-Encoder to a 2D-Encoder. The latent space representation  $Z$  with parameters  $\mu, \epsilon, \sigma$  is then decoded back to  $\tilde{s}$  through a 2D-Decoder and 1D-Decoder. Step 2: Denoising Network Training: The latent representation  $Z$  is processed through a denoising network comprising a ResNet Block, optional Self-Attention, and Cross Attention, with species and time information. The network outputs a Gaussian distribution  $N(z; \mu, \Sigma)$ . A U-Net architecture takes this distribution to produce various  $z_0$  representations, which a Locked Decoder (fronzen parameters) used to generate the final DNA sequences.
+
+# A A simple baseline latent diffusion model for discrete data: DiscDiff
+
+# A.1 The DiscDiff Model
+
+We design DiscDiff, a Latent Discrete Diffusion model for DNA generation tasks. As shown in Figure 7, this model is structured into two main components: a Variational-Auto-Encoder (VAE) and a denoising model. The VAE consists of an encoder  $\mathbf{E}:\underline{\mathbf{s}}\mapsto \underline{\mathbf{z}}$  , which maps discrete input sequence s to a continuous latent variable  $\underline{\mathbf{z}}$  , and a decoder  $\mathbf{D}:\underline{\mathbf{z}}\mapsto \tilde{\underline{\mathbf{s}}}$  , which reverts  $\underline{\mathbf{z}}$  back to  $\tilde{s}$  in the discrete space. The denoising model  $\varepsilon_{\theta}(\mathbf{z}_t,t)$  is trained to predict the added noise  $\varepsilon$  in the latent space.
+
+# A.1.1 VAE Architecture
+
+The choice of VAE architecture in LDMs is critical and often domain-specific. We find that mapping the input data to a higher dimension space can help to learn a better denoising network, generating more realistic DNA sequences. We hereby propose to use a two-stage VAE architecture as shown in Figure 7.
+
+The first stage encoder  $\mathbf{E}_{\phi_1}:\mathbb{N}_4^L\to \mathbb{R}^{K\times M}$  maps  $\underline{\mathbf{s}}\in \mathbb{N}_4^L$  to a 2D latent space  $\underline{\mathbf{z}}_1\in \mathbb{R}^{K\times M}$ , where  $K$  is the number of channels and  $M$  is the length of the latent representation. The second stage encoder  $\mathbf{E}_{\phi_2}:\mathbb{R}^{1\times K\times M}\to \mathbb{R}^{C\times K'\times M'}$  first adds a dummy dimension to  $\underline{\mathbf{z}}_1$  such that  $\mathbf{z}_1\in \mathbb{R}^{1\times K\times M}$  and then maps it to 3d latent space  $\mathbf{z}\in \mathbb{R}^{C\times K'\times M'}$ , where  $C$  is the number of channels,  $K'$  and  $M'$  are the reduced dimensions of  $K$  and  $M$  respectively. The decoder in the first and second stage are  $\mathbf{D}_{\theta 1}$  and  $\mathbf{D}_{\theta 2}$  respectively. Which are symmetric to the encoders. Overall, we have  $\mathbf{z} = \mathbf{E}_{\phi}(\mathbf{s}) = \mathbf{E}_{\phi_2}(\mathbf{E}_{\phi_1}(\mathbf{s}))$ , and the reconstruction is  $\tilde{\mathbf{s}} = \mathbf{D}_{\theta}(\mathbf{z}) = \mathbf{D}_{\theta 1}(\mathbf{D}_{\theta 2}(\mathbf{z}))$ .
+
+# A.1.2 VAE Loss
+
+When training the VAE, we propose to use Cross Entropy (CE) as reconstruction loss. The loss function is given by:
+
+$$
+\mathbf {L} _ {\theta , \phi} = \underbrace {\mathbb {E} _ {p (\mathbf {s})} \left[ \mathbb {E} _ {q _ {\phi} (\mathbf {z} | \mathbf {s})} \left[ - \sum_ {l = 1} ^ {L} \sum_ {i = 1} ^ {4} \delta_ {i s _ {l}} \log p _ {\theta} (\mathbf {s} _ {l} | \mathbf {z}) \right] \right]} _ {\text {R e c o n s t r u c t i o n L o s s}} +
+$$
+
+$$
+\beta \cdot \underbrace {\mathbb {E} _ {p (\mathbf {s})} \left[ \mathrm {K L} \left(q _ {\phi} (\mathbf {z} | \mathbf {s}) \mid \mid \mathcal {N} (\mathbf {z} ; \mu , \boldsymbol {\Sigma})\right) \right]} _ {\text {K L D i v e r g e n c e}}
+$$
+
+where  $\delta_{ij}$  is the Kronecker delta,  $p_{\theta}(\mathbf{s}|\mathbf{z})$  is the probabilistic decoder output from  $\mathbf{D}_{\theta}$ ;  $q_{\phi}(\mathbf{z}|\mathbf{s})$  is the probabilistic output from encoder  $\mathbf{E}_{\phi}$  that represents the approximate posterior of the latent variable  $\mathbf{z}$  given the input  $\mathbf{s}$ ;  $\mathcal{N}(\mathbf{z};\mu,\Sigma)$  is the prior on  $\mathbf{z}$ . Here we use a simple isotropic.  $\beta$  is a mixing hyperparameter.  $\beta$  is set to  $10^{-5}$  in the experiments used in this paper.
+
+# A.1.3 Denoising Network Training
+
+Once  $\mathbf{D}_{\theta}$  and  $\mathbf{E}_{\phi}$  are trained in the first step, we train a noise prediction  $\varepsilon_{\theta}$  in the latent space  $\mathbf{z} = \mathbf{E}_{\phi}(\mathbf{s})$  with Equation (6).
+
+$$
+\mathbb {E} _ {\mathbf {z}, t \sim U [ 1, T ], \varepsilon \sim \mathcal {N} (\mathbf {0}, \mathbf {I})} \left[ \| \varepsilon - \varepsilon_ {\theta} (\mathbf {z} _ {t}, t) \| _ {2} ^ {2} \right] \tag {6}
+$$
+
+# B Toy Experiment Training Details
+
+We train both HyenaDNA and DiscDiff on an NVIDIA A100-PCIE-40GB using the Adam optimizer. For HyenaDNA, the learning rate is set to 0.0001, and we use the model heyenadna-large-1m-seqlen for this task. The maximum number of epochs is set to 100, with early stopping enabled to facilitate early convergence.
+
+For DiscDiff, the VAE is trained with a learning rate of 0.0001, while the UNet is trained with a learning rate of 0.00005. DiscDiff is trained for 600 epochs; during the inference time, we use DDPM [16] sampler with 1000 denoising steps.
+
+# C Convergence Proof of the Absorb & Escape Algorithm
+
+In this section, we provide a proof for the convergence of the Absorb & Escape (A&E) algorithm under certain assumptions. The convergence proof will demonstrate that the sequence generated by the A&E algorithm converges to a sample from the target distribution  $p_{\theta ,\beta}^{C}(\mathbf{x})$ .
+
+# C.1 Assumptions
+
+We make the following assumptions for the convergence proof:
+
+1. The autoregressive model  $p_{\theta}^{AR}(\mathbf{x})$  and the diffusion model  $p_{\beta}^{DM}(\mathbf{x})$  are both properly trained and provide accurate approximations of the underlying data distribution.  
+2. The initial sample  $\mathbf{x}^0\sim p_\beta^{DM}(\mathbf{x})$  is a valid sample from the diffusion model.  
+3. The segments  $\{\mathbf{s}_1, \mathbf{s}_2, \dots, \mathbf{s}_n\}$  of the sequence  $\mathbf{x}$  are chosen such that each segment is homogeneous.  
+4. The subset of segments  $S$  is chosen randomly and includes all segments over multiple iterations.  
+5. The conditional distribution  $p(\mathbf{x}_{i:j}|\mathbf{x}_{0:i-1}, \mathbf{x}_{j+1:L})$  approximated by  $p_{\theta}^{AR}(\mathbf{x}_{i:j}|\mathbf{x}_{0:i-1})$  is accurate.
+
+# C.2 Proof
+
+We aim to show that the A&E algorithm produces samples from the target distribution  $p_{\theta, \beta}^{C}(\mathbf{x})$ . We do this by showing that the Markov chain defined by the A&E algorithm has  $p_{\theta, \beta}^{C}(\mathbf{x})$  as its stationary distribution.
+
+Step 1: Initialization The initial sample  $\mathbf{x}^0\sim p_\beta^{DM}(\mathbf{x})$  is drawn from the diffusion model. This ensures that  $\mathbf{x}^0$  is a valid sample from  $p_{\beta}^{DM}(\mathbf{x})$ .
+
+Step 2: Absorb Step For each segment  $\mathbf{s}_k$  in the subset  $S$ , the Absorb step samples  $\tilde{\mathbf{x}}_{i:j}^{\prime}$  from the conditional distribution  $p(\mathbf{x}_{i:j}|\mathbf{x}_{0:i-1}, \mathbf{x}_{j+1:L})$ . Under the assumption that  $p_{\theta}^{AR}(\mathbf{x}_{i:j}|\mathbf{x}_{0:i-1})$  accurately approximates this conditional distribution, the refined segment  $\tilde{\mathbf{x}}_{i:j}^{\prime}$  is a valid sample from the target conditional distribution.
+
+Step 3: Escape Step The Escape step updates the segment  $\tilde{\mathbf{x}}_{i:j}^{t}$  to  $\tilde{\mathbf{x}}_{i:j}^{\prime}$ . This ensures that the updated sequence  $\tilde{\mathbf{x}}^t$  incorporates the refinement from the autoregressive model.
+
+Step 4: Stationary Distribution To show that the Markov chain defined by the A&E algorithm converges to  $p_{\theta, \beta}^{C}(\mathbf{x})$ , we need to show that  $p_{\theta, \beta}^{C}(\mathbf{x})$  is the stationary distribution of this Markov chain.
+
+The transition probability for the A&E algorithm is given by the product of the probabilities of the Absorb and Escape steps:
+
+$$
+P (\mathbf {x} \rightarrow \mathbf {x} ^ {\prime}) = P _ {\mathrm {A b s o r b}} (\mathbf {x} \rightarrow \mathbf {x} ^ {\prime}) P _ {\mathrm {E s c a p e}} (\mathbf {x} ^ {\prime} \rightarrow \mathbf {x}).
+$$
+
+Given that  $p_{\beta}^{DM}(\mathbf{x})$  captures the overall structure and  $p_{\theta}^{AR}(\mathbf{x})$  refines the segments, the composed distribution  $p_{\theta,\beta}^{C}(\mathbf{x})$  is achieved by iteratively applying the Absorb and Escape steps. We need to show that the stationary distribution satisfies:
+
+$$
+\int p _ {\theta , \beta} ^ {C} (\mathbf {x}) P (\mathbf {x} \rightarrow \mathbf {x} ^ {\prime}) d \mathbf {x} = p _ {\theta , \beta} ^ {C} (\mathbf {x} ^ {\prime}).
+$$
+
+Using the detailed balance condition for the Markov chain:
+
+$$
+p _ {\theta , \beta} ^ {C} (\mathbf {x}) P (\mathbf {x} \rightarrow \mathbf {x} ^ {\prime}) = p _ {\theta , \beta} ^ {C} (\mathbf {x} ^ {\prime}) P (\mathbf {x} ^ {\prime} \rightarrow \mathbf {x}).
+$$
+
+Given that the Markov chain is ergodic, the detailed balance condition implies that  $p_{\theta,\beta}^{C}(\mathbf{x})$  is the stationary distribution.
+
+Step 5: Ergodicity Ergodicity ensures that the Markov chain will visit all possible states given sufficient iterations. The random selection of segments  $S$  and the iterative updates in the A&E algorithm guarantee that all parts of the sequence are refined over time.
+
+Conclusion By satisfying the detailed balance condition and ergodicity, we have shown that the Markov chain defined by the A&E algorithm converges to the target distribution  $p_{\theta,\beta}^{C}(\mathbf{x})$ . Therefore, the A&E algorithm produces samples from the composed distribution  $p_{\theta,\beta}^{C}(\mathbf{x})$  as the number of iterations  $t$  approaches infinity.
+
+# D Experiment Details
+
+Baselines The details about the architecture and implementation of the baseline models are as below:
+
+- DNADiffusion [26]: We enhance the current DNADiffusion implementation for DNA synthesis, originally from the DNA-Diffusion project², by expanding the models to encompass 380 million parameters. This network is composed of Convolutional Neural Networks (CNNs), interspersed with layers of cross-attention and self-attention. The learning rate is set to 0.0001.  
+- DDSM [4]: We scale up the original implementation of the denoising network used for promoter design in  $\mathrm{DDSM}^3$  to what is the corresponding size of the network given 470 million parameters. It is a convolution-based architecture with dilated convolution layers. The learning rate is set to 0.00001.
+
+- D3PM [3]: We take the implementation of D3PM for biological sequence generation from EvoDiff  $[2]^{4}$ , adopting the algorithm for DNA generation. We use the original implementation of the denoising network, which has two versions: with sizes of 38M and 640M. We hereby have D3PM (small) and D3PM (big), respectively. The learning rate for both D3PM (small) and D3PM (large) are set to 0.0001.  
+- Hyena [24]: We modify the RegLM  $[20]^{5}$ , a existing work uses hyena for DNA generation. Four pretrained Hyena models of different sizes (hyenadna-large-1m-seqlen, hyenadnamedium-160k0seqlen, heynadna-small-32k-seqlen, and hyenaana-tiny-16k-seqlen-d128) are downloaded from HuggingFace $^{6}$  and used for full-size fine-tuning, we apply the fine-tuned models for generations on EPD-GenDNA. The learning rate for fine-tuning is set to 0.0001.  
+- DiscDiff: A 2D-Unet of 500 Million parameters are used as the denoising network. See Appendix A for the implementation details. The learning rate is set to 0.00005 for UNet training, 0.0001 for VAE training.
+
+For the Fast A&B algorithm, we set the  $T_{absorb}$  to 0.80.
+
+# E Content List of Supplementary Code and Data
+
+Our code folder includes the following sub-folders, within each folder there is aREADME file, detailing the steps to run the code. The below is the list of sub-folders.
+
+# E.1 Toy Experiment
+
+Include the source code to reproduce Section 3. No external package is required to run the code except for python
+
+# E.2 DiscDiff
+
+A implementation of the discdiff baseline in Appendix A. Please follow theREADME for running the code.
+
+# E.3 AbsorbEscape
+
+This includes an implementation of the proposed algorithm in Section 4.2; however, it depends on the external AR models, please adopt it accordingly to AR models which you want to try.
+
+# E.4 EPD Data
+
+We include the training dataset used for producing the main results, which includes 160K DNA sequences from EPD, each sequence has a length of 256 bp.
+
+# F Motif Distributions for 15 species
+
+We plot TATA-box, GC content, Initiator, and CCAAT-box for 15 species as below.
+
+![](images/5a69f2b2c5b14d89c775ccf46dc57291316d2ff20c63f5007118c770e128ecd7.jpg)  
+Figure 8: Baking Yeast
+
+![](images/7ce6e069786f6dbc55389b4652bd8e54e04bb10a6ab9703163b7a3c06a4a5319.jpg)  
+Figure 9: Chicken
+
+![](images/52e845df488ed7d9933ecd3987d4bca60a8a5460c35fc2ebc2ea61ea03c6110c.jpg)  
+Figure 10: Chicken
+
+![](images/fa9d8b4575088ab0694a8101def8748663338069880d5ea20b62344d2e1a299e.jpg)  
+Figure 11: Corn
+
+![](images/82ff521378ee08b3b1168fc592fd1c620273883cd8775d95f61f7dcca88cd55a.jpg)  
+Figure 12: dog
+
+![](images/abe1b793c934091d3afdfba203d74e31f383185938a83eb0b705045880ebfc9b.jpg)  
+Figure 13: elegans
+
+![](images/b919ca15dba0f25212e80399033aa152bfe64c16d5de79069e1e5de16e255fca.jpg)  
+Figure 14: fission yeast
+
+![](images/b0e560ce0225aaa0e7cd46ac4f98588ff04d0011d4b2b4118531871b5df2e064.jpg)  
+Figure 15: fruit fly
+
+![](images/e09c675c7feb2770c176dd37c5e9da99c4849cd4555d9bd9e7bca93b1d4da284.jpg)  
+Figure 16: honey bee
+
+![](images/ffe7335e8dd0a285946289155a06ffd0b188b52561330467f569b07ac757bf3a.jpg)  
+Figure 17: human
+
+![](images/26777eaeec3e7fa06f347b6959d9e851d1d540a8a4a25ac647a8e404771242be.jpg)  
+Figure 18: plasmodium
+
+![](images/fbc788421880de2c133db917596a0da368d44e81992bce0d30e3d879168c91c2.jpg)  
+Figure 19: rat
+
+![](images/500beb9bbc518d6c7ee6e7f9ad36d0cf14693301e48120f4ece3d08f284521e7.jpg)  
+Figure 20: thale cress
+
+![](images/5ec491f7a2d6b04d9a325d2a09792dbc56d7f8d4a493bbef0dae96f07dab2a8d.jpg)  
+Figure 21: macaque
+
+![](images/35d53623112f27963ac2d7a9308492709825f44eb64e23f3ace2c4b6605b6c0c.jpg)  
+Figure 22: zebrafish
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
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+
+Answer: [Yes]
+
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+
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+
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
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+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: See Section 5.3 and Appendix D.
+
+# Guidelines:
+
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+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
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+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: For the main result which compares the motif distributions across 15 species, we compute the error bar. For some of the other tasks requiring heavy compute, we didn't repeat the experiment.
+
+# Guidelines:
+
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+
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+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
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+
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+# Accelerating Augmentation Invariance Pretraining
+
+Jinhong Lin* Cheng-En Wu* Yibing Wei Pedro Morgado
+
+University of Wisconsin-Madison
+
+{jlin522，cwu356，wei96，pmorgado}@wisc.edu
+
+# Abstract
+
+Our work tackles the computational challenges of contrastive learning methods, particularly for the pretraining of Vision Transformers (ViTs). Despite the effectiveness of contrastive learning, the substantial computational resources required for training often hinder their practical application. To mitigate this issue, we propose an acceleration framework, leveraging ViT's unique ability to generalize across inputs of varying sequence lengths. Our method employs a mix of sequence compression strategies, including randomized token dropout and flexible patch scaling, to reduce the cost of gradient estimation and accelerate convergence. We further provide an in-depth analysis of the gradient estimation error of various acceleration strategies as well as their impact on downstream tasks, offering valuable insights into the trade-offs between acceleration and performance. We also propose a novel procedure to identify an optimal acceleration schedule to adjust the sequence compression ratios to the training progress, ensuring efficient training without sacrificing downstream performance. Our approach significantly reduces computational overhead across various self-supervised learning algorithms on large-scale datasets. In ImageNet, our method achieves speedups of  $4 \times$  in MoCo,  $3.3 \times$  in SimCLR, and  $2.5 \times$  in DINO, demonstrating substantial efficiency gains.
+
+# 1 Introduction
+
+Self-supervised learning (SSL) has emerged as a powerful pre-training paradigm, demonstrating remarkable success across a variety of domains. By designing pretext tasks that leverage unlabeled data, SSL eliminates the need for costly and labor-intensive manual annotations. Among SSL methods, contrastive [5, 16] and distillation-based [4, 24] learning are among the most effective. They learn representations through transformation invariance, meaning that the model learns to create similar representations for different augmentations of the same image, but different across images. These approaches have led to the development of state-of-the-art models for tasks ranging from image recognition [16, 12] to object detection [16] and video object segmentation [4].
+
+Despite these achievements, SSL requires substantial computational resources for pretraining, with optimal performance often necessitating long training schedules, hindering their practical application. The computational demands of SSL are particularly high for Vision Transformers (ViTs), a promising class of neural networks that has recently gained significant attention for visual understanding tasks. ViTs represent images as sequences of patches, processed through self-attention layers. While self-attention enables the model to capture long-range dependencies and complex patterns more effectively than convolutional networks, their enhanced expressiveness (and weaker inductive bias) require even more extensive pre-training to achieve competitive performance. Our work aims to address these computational challenges by proposing an acceleration framework specifically tailored for ViTs.
+
+Existing attempts to accelerate SSL pretraining primarily focus on defining improved learning rates and data augmentation schedules for faster learning [19] or increasing the strength of supervision through multiple targets [8]. While beneficial, these methods are not tailored to the ViT architecture, and can only provide limited acceleration. They also often change the underlying pretraining
+
+![](images/5db9926f35c8c73b79aaf65fa27d86495fa568891b6589831af97779c79eb01e.jpg)  
+(a) ImageNet-100
+
+![](images/e1d06bd64dcefd4a08e9169060aa16a7410add011015f76b02ceb0ebdd4d9eb1.jpg)  
+Figure 1: Our accelerated MoCo-v3 achieves standard MoCo-v3 performance using only 1/5 of the training budget on ImageNet-100 and 1/3 on ImageNet-1k. The training budget (x-axis) is measured as the training time normalized by the forward pass of the base non-accelerated backbone model, in million (M) units. The results for ImageNet-100 are shown in Fig. 1a and for ImageNet-1k in Fig. 1b.
+
+![](images/06e584cc0f9b5d42dcfb8be8366de2eae808241998377bfc69c58e578d56dab6.jpg)  
+(b) ImageNet-1k
+
+![](images/9e5079f823702f4bc377ce6ba89fe71c20607a7c524e046e350fd8bcdbf37aa9.jpg)
+
+algorithm, making use of additional losses or data augmentations. Instead, we investigate acceleration techniques that leverage the ViT's unique ability to generalize across inputs of varying sequence lengths, while faithfully preserving the model's architectural design.
+
+Since the time complexity of a training iteration is proportional to the input sequence length, our method identifies at each moment in time the most cost-effective mix of two simple sequence compression strategies: (1) randomized token dropout and (2) flexible patch scaling. We show that when applied judiciously with an appropriately optimized schedule, these simple strategies can significantly reduce the cost of gradient estimation, leading to faster convergence without compromising the quality of the learned representations (see Fig. 1). Our approach is general and can be applied to a wide range of SSL pre-training algorithms, as it only modifies the input sequence of the ViT. To demonstrate its effectiveness and generality, we apply our method for MoCo-V3 [16], SimCLR [5] and DINO [4], achieving significant training speed-ups on standard pre-training datasets like ImageNet-1K (between 2.5 to 4 times faster than the original methods). Additionally, we conduct a series of experiments on MoCo-V3 to perform a deeper and more general analysis of the proposed acceleration strategies. Through our analysis, we provide insights into the trade-offs between acceleration and performance, and the intricacies of establishing an optimal acceleration schedule. Specifically, we (1) investigate the gradient estimation error of various acceleration strategies alongside their performance on downstream tasks, study the impact of (2) compression rates on query and target sequences for contrastive learning, and (3) varying training budgets (showing that constant compression fails to meet peak model performance), and (4) establish an optimal acceleration schedule that adjusts to the training progress by minimizing the expected error of gradient estimation. Our analysis shows that, while the early phases of training typically benefit from aggressive acceleration strategies with high token dropout rates or large patch sizes, the gradient estimation biases increase as the model converges. Consequently, the optimal strategy should gradually shift towards smaller patches and lower dropout ratios.
+
+# 2 Related Work
+
+Representation Learning Through Transformation Invariance involves training models to produce consistent representations for augmented versions of the same image, primarily via contrastive learning and distillation methods. Contrastive learning achieves this by contrasting positive pairs, generated from augmenting the same image, with negative pairs from different images, as explored in numerous studies [13, 7, 6, 32, 23, 17, 1, 16, 5, 4]. Distillation methods [4, 24] focus on aligning embedding distributions across augmentations of varying scales without relying on negative samples. Both approaches effectively enhance feature quality without the need for labeled data.
+
+Accelerating Augmentation Invariance Pre-Training Despite the high computational requirements, accelerating pre-training of vision transformers has remained underexplored. A few strategies have nevertheless been proposed. While focusing on ResNets, [19] introduced an architecture-agnostic method that dynamically adjusts augmentation intensity and learning rate schedules to hasten the training process. Some works focus on ViTs but modify its architecture or the underlying algorithm for acceleration. For example, [21] progressively merges tokens within the model, and [8] utilizes multiple small crops as positive pairs to increase the strength of supervision and promising convergence with smaller training budgets. In contrast, we propose a novel acceleration procedure for identifying the optimal schedule of simple sequence compression strategies, by ensuring that gradient estimation is cost-effective without introducing significant estimation biases.
+
+![](images/10624b47b7445410f4afdd3750873095029f51912e7b3445dfa2e5a5c457c02a.jpg)  
+Figure 2: Framework overview. We propose a method for accelerating augmentation invariance pre-training of transformer neural networks. Acceleration is achieved by compressing the ViT's input sequence length using two strategies: (1) randomized token dropout and (2) flexible patch scaling. We further introduce a gradient error analysis framework to assess the efficacy of an acceleration strategy, enabling us to define an optimal acceleration schedule that adjusts to the training progress. The acceleration strategy can be applied to a variety of methods. For example, SimCLR optimizes both encoders by gradient descent, while MoCo and DINO use a momentum encoder to compute the representations for the Key view. The loss function also differs across algorithms.
+
+Accelerating Model Training The computational demands of pretraining extend beyond contrastive learning. Various strategies have been proposed to mitigate this, such as random masking, a technique used in Vision Language pretraining to scale up training [22], and masked image modeling [14, 2] for in-context reconstruction. Other acceleration techniques include curriculum learning, starting with simple samples and progressively moving to harder ones [34, 26]; flexible ViT architectures that gradually increase in depth and width [25]; and resolution scaling, using low-resolution inputs at initial stages for faster convergence [20, 27, 30].
+
+While these methods have been successful in accelerating pre-training, they often require changes to the model architecture or the underlying training algorithm. They have also been primarily designed for a variety of tasks beyond contrastive learning, and thus their findings and proposed algorithms may not be directly applicable. Our work covers this gap by proposing a novel acceleration framework tailored for augmentation invariance pre-training of ViTs. Furthermore, while the building blocks (token dropout and patch scaling) of our acceleration framework are closely related to existing methods, our work is the first to systematically analyze their impact on gradient estimation and to leverage this analysis to define an acceleration schedule that adjusts to the training progress.
+
+# 3 [Background] Augmentation Invariance Pre-Training of ViTs
+
+We present an approach to accelerate augmentation invariance pre-training of Vision Transformers. Since the proposed strategy only modifies the input sequence, it is applicable to a wide range of algorithms. For comprehensive empirical analysis, we apply our method to MoCo-V3 [7], SimCLR [5] and DINO [4].
+
+# 3.1 Augmentation Invariance Pre-Training
+
+While contrastive learning and distillation-based methods may differ in their implementation, they generally rely on augmentation invariance as the source of supervision. Given an input image  $x$ , two views are generated using a random data augmentation procedure  $\mathcal{T}$  and often referred to as the query  $x_{q} = \mathcal{T}(x)$  and the key  $x_{k} = \mathcal{T}(x)$ . These are processed by two encoders  $f_{q}$  and  $f_{k}$ , producing query and key  $n$ -dimensional representations  $\pmb{z}_{q} = f_{q}(x_{q}) \in \Re^{n}$  and  $\pmb{z}_{k} = f_{k}(x_{k}) \in \Re^{n}$ . Augmentation invariance is encouraged by aligning the query  $z_{q}$  and key  $z_{k}$  representations of the same image. Two major differences between algorithms are: the choice of the key encoder  $f_{k}$  and the loss function leveraged to impose augmentation invariance.
+
+Key Encoder While SimCLR uses a shared encoder to encode both views (i.e.,  $f_{k} = f_{q} = f$ ), MoCo-V3 and DINO use a momentum encoder [28] for the keys  $x_{k}$ . Momentum encoders  $f_{k}$  share the architecture, but their parameters are computed as the exponential moving average of the online encoder  $f_{q}$ . Momentum encoders have been shown to yield more stable target representations, and consequently improved representation learning.
+
+Loss function Augmentation invariance can be enforced through a variety of loss functions. Among the chose algorithms, both SimCLR and MoCo-V3 leverage the InfoNCE loss [23]
+
+$$
+\mathcal {L} _ {\text {I n f o N C E}} = - \log \frac {\exp \left(\operatorname {s i m} \left(z _ {q} , z _ {k} ^ {+}\right) / \tau\right)}{\exp \left(\operatorname {s i m} \left(z _ {q} , z _ {k} ^ {+}\right) / \tau\right) + \sum_ {z _ {k} ^ {-} \in \mathcal {Z} _ {k} ^ {-}} \exp \left(\operatorname {s i m} \left(z _ {q} , z _ {k} ^ {-}\right) / \tau\right)}, \tag {1}
+$$
+
+where  $\mathrm{sim}(\cdot, \cdot)$  is the cosine similarity,  $\tau$  a temperature parameter,  $z_{q}$  and  $z_{k}^{+}$  the corresponding query and key representations, and  $\mathcal{Z}_k^-$  a set of negative keys obtained from other images in the same batch. Unlike MoCo-V3 and SimCLR, DINO uses a distillation loss instead, which seeks to align query and key representations (also referred to as student and teacher's representations) without the explicit use of negatives samples. To accomplish this, query and key representations are first converted into a probability vector through a softmax operator  $\pmb{p} = \text{SoftMax}(\pmb{z} / \tau) \in \Re^n$ , and the model is trained to minimize the cross-entropy between the two
+
+$$
+\mathcal {L} _ {\text {d i s t}} = - \sum_ {i = 1} ^ {n} \boldsymbol {p} _ {k, i} \log \boldsymbol {p} _ {q, i}. \tag {2}
+$$
+
+The sum is taken over the dimensions of the probability vector.
+
+# 4 Gradient Acceleration Through Sequence Compression
+
+While augmentation invariance can be used to learn representations with any type of neural network, the proposed acceleration procedure leverages properties specific to transformer models. In this work, we focused on Vision Transformers (ViTs) [10], a widely used architecture for vision tasks. ViTs process an image  $x$  of resolution  $H \times W$  by dividing it into a grid of  $(H // p, W // p)$  patches, each of size  $p \times p$ . After embedding each patch into a  $d$ -dimensional vector and adding positional encodings to mark their spatial location, the ViT processes the sequence of patch embeddings through several self-attention transformer blocks [31]. Since transformer blocks' parameters are shared across the input sequence, the gradients of the loss wrt its parameters can be computed regardless of input sequence lenght. In this work, we tackle two key questions: (1) How to reduce the input sequence length with limited impact on the gradient estimation error? and (2) How to characterize the effectiveness of an acceleration strategy? We introduce two strategies, randomized token dropout and dynamic patch scaling, and propose a methodology to determine the effectiveness of gradient acceleration of a given strategy by analyzing its cost-adjusted bias-variance trade-off. This methodology allows us to identify the optimal acceleration strategy at each moment during training, eliminating the need for manual hyper-parameter tuning.
+
+# 4.1 Randomized Token Dropout
+
+Randomized token dropout (TknDrop) is a simple strategy for reducing the sequence length in ViTs, by simply removing a random subset of tokens from the input sequence. This strategy is especially effective in vision, since neighboring pixels are highly correlated and the model can still infer the visual content from a partial view of the image. However, while highly compressed sequences can speed up gradient estimation, too much compression may cause significant biases in the estimated gradients and consequently degraded model performance. Determining the optimal dropout rate is thus crucial for effective acceleration.
+
+TknDrop is inspired by MIM methods [15, 2], which also mask the input sequence. However, while MIM uses masking to establish reconstruction targets for representation learning, we leverage token dropout to generate compressed input sequences to accelerate augmentation invariance pre-training.
+
+# 4.2 Patch Scaling
+
+The second strategy involves splitting the input image into a coarser grid of patches. As the sequence length  $L$  is inversely proportional to the patch size  $p$ , larger patches allow us to reduce  $L$  without removing any pixels from the input. However, since the patch embedding layer  $W_{patch} : \Re^{p^2} \to \Re^n$  depends on a predefined patch size  $p$ , larger patches cannot be directly encoded. To mitigate this issue, we leverage the flexible patch embeddings introduced in [3], where  $W_{patch}$  are dynamically resized to accommodate different patch sizes. Consider the weights  $w_p \in \Re^{p^2}$  of a single output dimension of  $W_{patch}$ . Instead of simple interpolation, the optimal weights  $w_q \in \Re^{q^2}$  at the larger size  $q$  are computed by finding a projection  $w_q = Pw_p$  that minimizes the distance between the embedding of the original patch  $x_p$  and the interpolated larger patch  $x_{p\rightarrow q}$ . Specifically, the optimal projection  $P$  is obtained by solving
+
+$$
+\arg \min  _ {P} \mathbb {E} _ {x _ {p} \in \mathcal {X} _ {p}} \left[\left(\langle x _ {p}, w _ {p} \rangle - \langle x _ {p \rightarrow q}, P w _ {p} \rangle\right) ^ {2} \right], \tag {3}
+$$
+
+where the expectation is taken over a distribution of patches  $\mathcal{X}_p$ .
+
+<table><tr><td>Method</td><td>Sample Cost (C)</td></tr><tr><td>SimCLR</td><td>3Lq+Lk/Lbase</td></tr><tr><td>MoCo-v3</td><td>3Lq+Lk/Lbase</td></tr><tr><td>DINO</td><td>3Lq+3KLqsmall + Lk/Lbase</td></tr></table>
+
+Table 1: Hardware-independent sample cost of different pre-training algorithms. We assume relatively short sequence lengths (typical of pretraining frameworks) where linear operations dominate over the quadratic self-attention operations.  
+
+<table><tr><td rowspan="6">Patch</td><td>48</td><td>1.54</td><td>1.44</td><td>1.35</td><td>1.25</td><td>1.19</td></tr><tr><td>40</td><td>1.71</td><td>1.57</td><td>1.43</td><td>1.29</td><td>1.20</td></tr><tr><td>30</td><td>2.14</td><td>1.89</td><td>1.64</td><td>1.39</td><td>1.25</td></tr><tr><td>24</td><td>2.69</td><td>2.30</td><td>1.92</td><td>1.53</td><td>1.30</td></tr><tr><td>20</td><td>3.36</td><td>2.80</td><td>2.25</td><td>1.70</td><td>1.37</td></tr><tr><td>16</td><td>4.59</td><td>3.73</td><td>2.87</td><td>2.01</td><td>1.49</td></tr><tr><td></td><td></td><td>0.0</td><td>0.25</td><td>0.5</td><td>0.75</td><td>0.9</td></tr><tr><td></td><td></td><td colspan="5">Token Dropout Rate</td></tr></table>
+
+Figure 3: Accelerated MoCo-v3 sample costs for varying dropout rates and patch sizes. We assume uncompressed key sequences.
+
+# 4.3 Combined Sequence Compression
+
+Large patches can be trivially combined with token dropout by applying the two strategies in sequence. The sequence length are thus modulated by the selected patch size  $q$  and token dropout rate  $d$ . Specifically, an image of size  $H \times W$  split into a grid of  $p \times p$  patches yields an uncompressed sequence of lenght  $L = \left\lfloor \frac{HW}{p^2} \right\rfloor$ . After compression, the sequence lenght is lowered to  $L = (1 - d) \times \left\lfloor \frac{HW}{q^2} \right\rfloor$ .
+
+# 4.4 Quantifying acceleration
+
+Linear complexity assumption. Transformer blocks use two types of operations, token-wise transformations and self-attention. Token-wise operations, such as the MLP block or the query/key/value heads in self-attention, process each token in the sequence independently and thus scale linearly  $\mathcal{O}(L)$  with its length  $L$ . Self-attention operations, on the other hand, establish relationships between all pairs of patches and thus scale quadratically  $\mathcal{O}(L^2)$ . However, the sequence length for most model pre-training frameworks is relatively small (typically  $L = 197$ ). Since there are many more linear operations than quadratic ones, the time complexity of linear operations dominates at this scale. Empirically, we observed that quadratic operations only become significant when the sequence length exceeds 400 patches. Thus, for the sake of simplicity, we assumed the time complexity of ViT pre-training to be linear with  $L$ .
+
+Sample costs of various algorithms. Let the time spent per token be denoted as  $t_{tkn}$ . Then, under the linear complexity assumption, a forward pass takes approximately  $t_{fwd} = L \times t_{tkn}$  seconds, and a backward pass twice as long,  $t_{bwd} = 2L \times t_{tkn}$ , as both the partial derivatives wrt the latent representations and the model parameters need to be computed [18]. Since, for SimCLR, the backward pass is performed on both encoders, the total time per sample is  $t_{smp} = 3(L_q + L_k) \times t_{tkn}$ . For MoCo-v3, the backward pass is only performed on the query encoder, and thus  $t_{smp} = (3L_q + L_k) \times t_{tkn}$ . DINO also uses  $K$  smaller augmentations as additional query sequences for its distillation loss, further increasing the sample time to  $t_{smp} = (3L_q + 3KL_q^{small} + L_k) \times t_{tkn}$ . Finally, hardware dependencies (captured through  $t_{tkn}$ ) can be removed by normalizing  $t_{smp}$  by the forward pass of a standard input  $t_{base} = L_{base} \times t_{tkn}$ , where  $L_{base} = 197$  is the sequence length for a regular  $14 \times 14$  grid. Hardware independent sample costs are summarized in Table 1.
+
+As can be seen, regardless of pre-training algorithm, the sample cost is proportional to the sequence lengths,  $L_{q}$  and  $L_{k}$ . To visualize the impact of compression, we show the sample costs with varying token dropout ratios and patch sizes for MoCo-v3 in Fig. 3. These cost assume an uncompressed key sequence, as we empirically found that model pre-training is often more effective when supervision targets are computed without compression<sup>2</sup>. As can be seen, speed ups as large as  $4\times$  can be achieved with  $90\%$  dropout rates and patches of size  $q = 48$ . However, useful acceleration strategies should not only reduce the sample cost, but also minimize their impact on the estimated gradients.
+
+# 5 Gradient Estimation Analysis of Acceleration Strategies
+
+Given the large search space, empirically selecting the most effective strategy at each stage of training is computationally prohibitive. Instead, we posit that the distribution of the accelerated gradients should closely resemble that of the non-accelerated model. This criterion, further expanded below, can be used to inform us of the optimal mix of accelerated strategies at any point throughout training.
+
+# 5.1 Formulation
+
+Gradient Distribution in Mini-Batch Training Model optimization requires the minimization of a loss function  $l(\pmb{x}; \theta)$  wrt the model parameters  $\theta$ . Assuming independence between samples  $\pmb{x}$  in the training dataset  $\mathcal{D}$ , the expected loss is given by
+
+$$
+\mathcal {L} (\theta) = \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {D}} [ l (\boldsymbol {x}; \theta) ] \approx \frac {1}{| \mathcal {B} |} \sum_ {i \in \mathcal {B}} l \left(\boldsymbol {x} _ {i}; \theta\right) \tag {4}
+$$
+
+where  $|\mathcal{B}|$  is the mini-batch size. Optimization algorithms update the model parameters  $\theta$  using the gradient of the loss.
+
+$$
+\nabla_ {\theta} \mathcal {L} (\theta) = \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {D}} \left[ \nabla_ {\theta} l (\boldsymbol {x}; \theta) \right] \approx \frac {1}{| \mathcal {B} |} \sum_ {i \in \mathcal {B}} \nabla_ {\theta} l \left(\boldsymbol {x} _ {i}; \theta\right) \tag {5}
+$$
+
+For simplicity, denote the sample gradient as  $g_{\theta}(\pmb{x}) = \nabla_{\theta}l(\pmb{x}_i;\theta)$ , and the true gradient (computed over the entire dataset) as  $G_{\theta} = \nabla_{\theta}\mathcal{L}(\theta)$ . Since samples are independently drawn from the training dataset  $\mathcal{D}$ , then the sample gradient  $g_{\theta}(\pmb{x})$  is a random variable with mean and covariance given by
+
+$$
+\mathbb {E} \left[ g _ {\theta} (\boldsymbol {x}) \right] = G _ {\theta} \quad \text {a n d} \quad \operatorname {C o v} \left[ g _ {\theta} (\boldsymbol {x}) \right] = \mathbb {E} \left[ \left(g _ {\theta} (\boldsymbol {x}) - G _ {\theta}\right) \left(g _ {\theta} (\boldsymbol {x}) - G _ {\theta}\right) ^ {T} \right]. \tag {6}
+$$
+
+Similarly, batch gradients are also unbiased estimates of the true gradient but with variance reduced by a factor of  $|\mathcal{B}|$ .
+
+$$
+\mathbb {E} \left[ \frac {1}{| \mathcal {B} |} \sum_ {\boldsymbol {x} \in \mathcal {B}} g _ {\theta} (\boldsymbol {x}) \right] = G _ {\theta} \quad \text {a n d} \quad \operatorname {C o v} \left[ \frac {1}{| \mathcal {B} |} \sum_ {\boldsymbol {x} \in \mathcal {B}} g _ {\theta} (\boldsymbol {x}) \right] = \frac {1}{| \mathcal {B} |} \operatorname {C o v} [ g _ {\theta} (\boldsymbol {x}) ]. \tag {7}
+$$
+
+Gradient Estimation Errors When assessing an acceleration strategy, we need to consider both the mean and variance of the gradient estimates. While the bias is an intrinsic property of each strategy, the variance is a function of their computational cost. Strategies that reduce the computational cost significantly can potentially be used to average the gradients over a larger number of samples and thus reduce their variance. Thus, to fairly capture the bias-variance tradeoff of each strategy, we propose to use the Mean Squared Error (MSE) of the gradient estimate obtained with a cost adjusted batch size. From parameter estimation theory, it can be easily shown that the sample MSE decomposes into (squared) bias and variance components
+
+$$
+\begin{array}{l} \operatorname {M S E} \left(g _ {\theta} ^ {c} (\boldsymbol {x}), G _ {\theta}\right) := \mathbb {E} \left[ \left\| g _ {\theta} ^ {c} (\boldsymbol {x}) - G _ {\theta} \right\| _ {2} ^ {2} \right] (8) \\ = \| G _ {\theta} - \bar {g} _ {\theta} ^ {c} \| _ {2} ^ {2} + \mathbb {E} \left[ \| g _ {\theta} ^ {c} (\boldsymbol {x}) - \bar {g} _ {\theta} ^ {c} \| _ {2} ^ {2} \right] = \operatorname {B i a s} ^ {2} \left(g _ {\theta} ^ {c}, G _ {\theta}\right) + \operatorname {V a r} \left(g _ {\theta} ^ {c}\right) (9) \\ \end{array}
+$$
+
+where  $\bar{g}_{\theta}^{c} = \mathbb{E}\left[g_{\theta}^{c}(\boldsymbol{x})\right]$  is the average accelerated gradient using strategy  $c$ . In other words, the MSE accounts for both the average deviation of the accelerated gradients from the ground truth and their variance across samples. Adjusting the MSE score for the cost of each strategy simply requires adjusting the variance by the number of samples within a fixed budget.
+
+$$
+\operatorname {C A - M S E} \left(g _ {\theta} ^ {c}\right) = \operatorname {B i a s} ^ {2} \left(g _ {\theta} ^ {c}, G _ {\theta}\right) + \frac {\operatorname {C o s t} (c)}{\text {B u d g e t}} \operatorname {V a r} \left(g _ {\theta} ^ {c}\right). \tag {10}
+$$
+
+Estimating Bias and Variance Both the bias and variance components can be efficiently computed given a model with parameters  $\theta$  and a dataset  $\mathcal{D}$ .  $G_{\theta}$  and  $\bar{g}_{\theta}^{c}$  can be approximated by the sample average of the non-accelerated and accelerated gradients, respectively. While automatic differentiation libraries only track the aggregated gradient during batch processing, preventing us from directly computing the sample variance, we can estimate it by dividing the batch into smaller sub-batches (of size  $K$ ) and adjusting the variance across sub-batches by  $K$ . Finally, it should be noted that to obtain a reliable estimate of CA-MSE, we need a large enough number of samples (we used 16k samples).
+
+Dynamic Acceleration The cost-adjusted MSE provides a means to compare acceleration strategies without full training and evaluation. This metric could also be used to select the most effective strategy on-the-fly, at different stages during the training process. However, in practice, we conducted our analysis on intermediate checkpoints of a pre-trained model and used the findings to establish the "CA-MSE optimal" acceleration schedule for each method.
+
+# 5.2 Analysis
+
+In Section 4, we introduced two strategies to reduce the input sequence length (randomized token dropout and patch scaling) thereby reducing the cost of estimating the gradients of the model,
+
+![](images/250a940a92e78c5897de01d46d6d6298deb4f710089b1340ce716d5ffbb0371d.jpg)  
+Figure 4: Error profile of accelerated gradients. From top to bottom, the three panels show the CA-MSE, squared bias and cost-adjusted variance of the gradient estimates, using different acceleration strategies and at different stages of training.
+
+potentially at the expense of inaccurate gradients. To investigate their cost-accuracy trade-off, we measured the cost-adjusted MSE at 5 different stages of training progress, at  $0\%$ ,  $25\%$ ,  $50\%$ ,  $75\%$  and  $100\%$  of training progress. We varied the dropout ratio in the set  $\{0, 0.25, 0.5, 0.75, 0.9\}$  and the patch size in  $\{16, 20, 24, 30, 40, 48\}$ . Fig. 4 presents the CA-MSE score, normalized by the magnitude of the ground-truth gradient, across all configurations. Early in training (as seen in the first panel), both non-accelerated gradients (i.e.,  $0\%$  dropout and patch size 16) and gradients derived from highly compressed input sequences (high dropout ratios and large patches) exhibit high MSE. However, non-accelerated gradients show low bias and high variance, while highly compressed gradients show high bias but low variance. More favorable trade-offs are achieved by combining moderate compression using both strategies simultaneously. As training advances and the model converges (remaining panels), the optimal strategy gradually shifts towards smaller patches and lower dropout ratios. This shift occurs because, as the model converges, gradient magnitudes shrink and the MSE becomes more sensitive to estimation biases.
+
+# 6 Experiments
+
+To assess the impact of acceleration on model performance, we conducted extensive exploration and ablation experiments, using the MoCo-V3 pre-training framework. We also assessed the generalizability of our methodology by accelerating other frameworks like DINO and SimCLR.
+
+# 6.1 Experimental Setup
+
+Dataset We conduct all experiments on the ImageNet dataset [9] using a ViT-Base transformer backbone for both the online and target encoders. Ablations and parametric studies are conducted on the ImageNet-100 (IN100) dataset, a randomly chosen subset of 100 classes from ImageNet. We adhere to the class partitioning used in previous studies [33, 29]. With around 125,000 images, IN100 provides a substantial amount of data for conducting statistically meaningful experiments. We also validate our findings on the full ImageNet-1k dataset to ensure the generalizability of our results.
+
+Downstream Evaluation We assess the quality of the learned representations in three ways. In line with standard practices in self-supervised learning, we measure the classification accuracy on the pre-training dataset either using a linear probe (LP) with frozen features or after full model finetuning (FT). We also measure the nearest neighbor accuracy (NN) as an indicator of the effectiveness of the learned representations. All downstream evaluations are conducted without sequence compression.
+
+Pre-training settings To ensure fair reproduction, we followed the official implementations. In the case of MoCo, the only modification was the use of a non-symmetric loss. Originally, the two augmentations  $x_{q}$  and  $x_{k}$  are used both as queries and targets, forming two pairs for each sample. However, this is equivalent to using only one pair, while doubling both the batch size and number
+
+<table><tr><td>Training Dataset</td><td>Batch Size</td><td>Epochs</td><td>Training Budget</td><td>NN</td><td>LP</td><td>FT</td></tr><tr><td rowspan="3">IN-100</td><td rowspan="3">512</td><td>200</td><td>104M</td><td>58.1</td><td>75.8</td><td>88.0</td></tr><tr><td>600</td><td>312M</td><td>68.7</td><td>80.4</td><td>90.4</td></tr><tr><td>1000</td><td>520M</td><td>70.0</td><td>84.4</td><td>90.6</td></tr><tr><td rowspan="3">IN-1k</td><td rowspan="3">1024</td><td>100</td><td>1028M</td><td>50.3</td><td>70.3</td><td>82.0</td></tr><tr><td>300</td><td>3084M</td><td>59.1</td><td>75.0</td><td>82.3</td></tr><tr><td>600</td><td>6168M</td><td>61.9</td><td>76.0</td><td>82.5</td></tr><tr><td>IN-1k</td><td>4096</td><td>300×2</td><td>6168M</td><td>-</td><td>76.7</td><td>83.2</td></tr></table>
+
+Figure 5: Non-accelerated MoCo-v3 across training budgets and datasets, and comparison to the publically released MoCo-V3 model (last row). The effective training epochs for the official MoCo implementation is doubled, as it uses a symmetric loss.
+
+![](images/206f2e78eebbcb92667b98ced88a99e9ef192c9f3f1356ecb7aea0a2e0c7853a.jpg)  
+Figure 6: Cosine vs Polynomial  $(\alpha = 2)$  learning rate decay schedules.
+
+of epochs. The non-symmetric version also produces more diverse batches, which is advantageous given the use of batch normalization in the prediction heads.
+
+To establish an optimized baseline on ImageNet-100, we empirically search for the learning rate, batch size and the required training budget (default values were used for other hyper-parameters). We observed that performance saturated for batch sizes of 512, and training budgets equivalent to 1000 epochs with a 40-epoch warmup phase. The optimal base learning rate was  $5 \times 10^{-4}$ , adjusted by the batch size scaling rule [11].
+
+As for ImageNet-1k experiments, we followed the official training hyperparameters except for batch size, which was set to 1024 due to hardware limitations. MoCo's baseline performance on both ImageNet-100 and ImageNet-1K are shown in Fig. 5. As can be seen, despite the lower batch size, the model achieves comparable performance on ImageNet-1k (only  $0.7\%$  worse on both LP and FT accuracy).
+
+Accelerated MoCo Pre-training As different gradient acceleration settings decrease the computational cost of a single iteration by different factors (see Fig. 3), fair comparisons require controlling the training budget, rather than the number of epochs. We express the training budget in units of the hardware independent sample costs defined in Section 4.4. On the ImageNet-100 dataset, where optimal performance was achieved at a training budget of 520M units (equivalent to 1000 epochs), we experiment with accelerated budgets ranging from 25M to 200M units. On ImageNet-1k, where baseline performance is achieved with a budget of 6168M, we varied the accelerated budget between 300M and 1500M.
+
+Similarly to [19], we observed that the learning rate schedule also impacts the effectiveness of acceleration strategies. Augmentation invariance pretraining commonly employs a cosine decay schedule, which rapidly decreases in the second half of training. However, under constrained training budgets, this rapid decay hinders the model's ability to learn during the late stages. To address this, we use a polynomial decay schedule (see Fig. 6) to maintain a relatively higher learning rate in the later stages of training.
+
+# 6.2 Constant Gradient Acceleration Strategies
+
+We begin by assessing the representations obtained with each gradient acceleration strategy when applied uniformly and independently throughout training, using the MoCo-v3 pre-training framework.
+
+Randomized Token Dropout We studied the impact of token dropout on the learning process. To assess the efficacy of this approach, we trained the model with varying dropout rates for the query and key sequences, adhering to a restricted training budget of 100M units (20% of the budget utilized for the optimal MoCo setup). The results detailed in Table 2a support three noteworthy observations. First, with randomized token dropout, it is beneficial to keep the key (target) sequence uncompressed to preserve maximum information when calculating the targets for the query (online) encoder. We refer to this strategy as asymmetric acceleration. Second, training MoCo-v3 without acceleration under the constrained training budget (as shown in the last row of Table 2a) yields significantly inferior results compared to any of the tested accelerated versions. For example, acceleration via asymmetric token dropout with  $L_{q} = 50$  surpasses the non-accelerated model by 11.5% in NN accuracy and 4.0% in LP accuracy. This finding highlights the effectiveness of token dropout for accelerating the learning process. Finally, it is possible to compress the sequence too much, as evidenced by the performance degradation when using  $L_{q} = 20$  (90% dropout rate). This result is
+
+<table><tr><td>Lq</td><td>Lk</td><td>Cost</td><td>NN</td><td>LP</td></tr><tr><td rowspan="3">20</td><td>20</td><td>0.4</td><td>58.2</td><td>73.3</td></tr><tr><td>40</td><td>0.5</td><td>59.2</td><td>74.9</td></tr><tr><td>197</td><td>1.3</td><td>63.9</td><td>78.5</td></tr><tr><td rowspan="3">50</td><td>50</td><td>1.0</td><td>65.4</td><td>79.6</td></tr><tr><td>100</td><td>1.3</td><td>67.0</td><td>78.7</td></tr><tr><td>197</td><td>1.8</td><td>69.6</td><td>81.1</td></tr><tr><td rowspan="2">100</td><td>100</td><td>2.0</td><td>59.8</td><td>79.0</td></tr><tr><td>197</td><td>2.5</td><td>69.3</td><td>81.4</td></tr><tr><td>197</td><td>197</td><td>4.0</td><td>58.1</td><td>77.1</td></tr></table>
+
+(a) Sym and asym TknDrop.
+
+<table><tr><td>p</td><td>L</td><td>Cost</td><td>NN</td><td>LP</td></tr><tr><td>16</td><td>225</td><td>4.59</td><td>53.5</td><td>73.6</td></tr><tr><td>20</td><td>144</td><td>2.94</td><td>64.1</td><td>79.9</td></tr><tr><td>24</td><td>100</td><td>2.04</td><td>67.1</td><td>81.2</td></tr><tr><td>30</td><td>64</td><td>1.31</td><td>68.1</td><td>80.6</td></tr><tr><td>40</td><td>36</td><td>0.73</td><td>62.0</td><td>76.3</td></tr></table>
+
+(b) Sym patch scaling.  
+
+<table><tr><td>pq</td><td>pk</td><td>Cost</td><td>NN</td><td>LP</td></tr><tr><td>30</td><td>16</td><td>2.1</td><td>45.3</td><td>71.7</td></tr><tr><td>30</td><td>20</td><td>1.7</td><td>53.8</td><td>77.3</td></tr><tr><td>30</td><td>24</td><td>1.5</td><td>64.3</td><td>80.6</td></tr><tr><td>30</td><td>30</td><td>1.3</td><td>68.1</td><td>80.5</td></tr></table>
+
+(c) Asym patch scaling.
+
+Table 2: Ablation studies of symmetric and asymmetric token dropout and patch scaling (training budget:  ${100M}$  ).
+
+![](images/b3ee9cb17e1958b5336c1733359ac6fb3dd3c80b4d7cbcd914a24de0f2c8e38b.jpg)  
+Figure 7: Training curves using constant symmetric patch scaling (training budget:  $100M$ ).
+
+consistent with the observations from the gradient error analysis (Fig. 4), which shows large gradient estimation biases for such high dropout rates.
+
+Patch Scaling We also examined the impact of patch scaling on learned representations, maintaining a training budget of 100M units. We tested symmetric and asymmetric compression strategies for query and key sequences. For this, we used  $240 \times 240$  resolution images to accommodate patches of size 16, 20, 24, 30, and 40, resulting in an uncompressed sequence length of 225 (with  $p = 16$ ). The results, in Tables 2b and 2c, mirror the findings of token dropout. Models trained with larger patch sizes expedite learning, outperforming the non-accelerated model ( $p = 16$ ) within the same budget, as shown in Fig. 7. Too much acceleration, with patches scaled above 30 pixels, can however degrade performance, which is aligned with the increased bias observed in the gradient error analysis. However, unlike token dropout, symmetric patch scaling, where both the query and key sequences are equally compressed, is more advantageous (see Table 2c). This is likely because patch scaling modifies the distribution of the input patches, making it preferable to maintain the same distribution for both the query and key sequences.
+
+Accelerated Training Across Training Budgets The previous experiments showcased the efficacy of the proposed acceleration strategies within a constrained budget. To characterize their performance across an array of training budgets, we employed two representative strategies: asymmetric token dropout with  $L_{q} = 50$  and  $L_{k} = 197$ , and symmetric patch scaling with  $p = 30$ . We trained the model with increasing budgets, from 25M to 200M units, and evaluated their downstream performance. The results, shown in Table 3, unveil a notable limitation of constant acceleration strategies. While these strategies are effective at lower budgets, they can overfit when the budget is increased. This is especially evident in the case of token dropout, as shown in Fig. 8. As a result of this overfitting, although the proposed acceleration strategies can outperform the non-accelerated model at lower budgets, they fail to meet their peak performance. As suggested by the gradient error analysis in Fig. 4, this overfitting can be traced back to the increased biases of the accelerated gradients often witnessed in the final stages of training. This occurs because, as the model converges, the gradient strength diminishes, allowing biases introduced by the compressed sequences to exert a greater influence on the learning process.
+
+# 6.3 Optimized Acceleration Schedules
+
+To circumvent the overfitting issue, we investigate optimized acceleration schedules that favor higher acceleration at the beginning and lower acceleration at the end of training. Although we can manually specify a schedule based on reasonable intuitions, the defined schedule would likely be suboptimal. Instead, we automate this process by leveraging the gradient error analysis of Fig. 4 to establish a "CA-MSE optimal" acceleration schedule.
+
+As expected, at the beginning of training, a high token drop ratio and larger patch sizes have lower cost-adjusted MSE. However, as the training progresses, smaller patch sizes and lower drop ratios become more effective. We used these automatically derived schedules to train a model with three training budgets: 50M, 104M and 150M. As shown by the training curves in Fig. 8, by lowering the
+
+![](images/d33c558bb68e6c931b769456e077e3f837cb0c0cf406884d2753478e805abf87.jpg)  
+Figure 8: Training curve of three acceleration strategies: constant patch size, constant token dropout, and dynamic scheduling of joint patch scaling and token dropout.
+
+<table><tr><td rowspan="2">Training Budget</td><td colspan="2">TknDrop</td><td colspan="2">PatchScale</td></tr><tr><td>NN</td><td>LP</td><td>NN</td><td>LP</td></tr><tr><td>25M</td><td>37.3</td><td>64.7</td><td>37.4</td><td>66.5</td></tr><tr><td>50M</td><td>56.6</td><td>76.0</td><td>50.1</td><td>74.7</td></tr><tr><td>75M</td><td>65.3</td><td>80.2</td><td>61.1</td><td>79.4</td></tr><tr><td>100M</td><td>69.6</td><td>81.2</td><td>66.7</td><td>80.5</td></tr><tr><td>150M</td><td>68.2</td><td>80.8</td><td>68.9</td><td>82.1</td></tr><tr><td>200M</td><td>65.3</td><td>80.6</td><td>70.1</td><td>83.2</td></tr></table>
+
+acceleration towards the end of training, the model no longer overfits and is capable of reproducing the peak performance of the MoCo-v3 baseline in less than  $30\%$  of the time.
+
+# 6.4 Accelerating Pretraining on ImageNet-1k
+
+Finally, to validate the generalizability of our findings, we deploy the proposed optimized acceleration scheduled on the full ImageNet-1k dataset. Due to the unique characteristics of different SSL algorithms, we tailor our method slightly for each algorithm. As observed in 6.2, token dropout should be applied only to the online encoder, while patch scaling should remain consistent across encoders. However, since SimCLR directly optimizes over both encoders, treating them as online models, we found that its better to apply token dropout to both sequences for SimCLR. As for DINO, we empiri
+
+ically found that the additional small crops used as queries are already compressed enough, and applying additional compression to these sequences is not beneficial.
+
+The results, shown in Table 4, demonstrate that the dynamic acceleration strategy is capable of achieving competitive performance with the non-accelerated model, while significantly reducing the computational requirements for training. For instance, our method achieves comparable LP accuracy for MoCo-V3 (75.9% vs 76.0%) with only 25% of the original budget (1542M vs 6168M iterations). We also achieved significant speedups for other pre-training frameworks, namely 2.5x for DINO and 3.3x for SimCLR.
+
+Table 3: Impact of training budgets on gradient acceleration strategies. Asymmetric token dropout ( $L_{q} = 50$ ,  $L_{k} = 197$ ) and symmetric patch scaling ( $p = 30$ ).  
+
+<table><tr><td>Algorithm</td><td>Accel.</td><td>Budget (M)</td><td>NN</td><td>LP</td><td>FT</td></tr><tr><td rowspan="2">MoCo</td><td>✓</td><td>1542</td><td>60.7</td><td>75.9</td><td>81.9</td></tr><tr><td></td><td>6168</td><td>61.9</td><td>76.0</td><td>81.8</td></tr><tr><td rowspan="2">SimCLR</td><td>✓</td><td>922</td><td>50.7</td><td>68.4</td><td>81.5</td></tr><tr><td></td><td>3075</td><td>50.2</td><td>68.3</td><td>81.3</td></tr><tr><td rowspan="2">DINO</td><td>✓</td><td>1138</td><td>66.0</td><td>77.4</td><td>82.0</td></tr><tr><td></td><td>2846</td><td>67.3</td><td>77.4</td><td>81.8</td></tr></table>
+
+Table 4: Acceleration of three augmentation invariance pretraining algorithms on ImageNet-1K. "Accel" indicated the use of the optimized acceleration schedule.
+
+# 7 Conclusion
+
+In this paper, we propose a general acceleration framework for self-supervised learning that leverages simple sequence compression strategies to reduce the cost of gradient estimation. Our method is shown to significantly speed up the convergence of a variety of self-supervised methods, including constrastive (MoCo, SimCLR), and distillation-based frameworks (DINO) thus demonstrating its broad usability.
+
+Given the compute-intensive nature of model pre-training and its implications on reproducibility, energy costs, and carbon footprints, we believe that further research on accelerated training is essential for advancing sustainable AI practices. Our paper aims to inspire continued exploration in this area, promoting the development of more efficient training methodologies that can (1) reduce the environmental impact of machine learning and (2) improve accessibility for researchers with limited resources.
+
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+[25] Pan, X., Jin, X., He, Y., Song, S., Huang, G., et al.: Budgeted training for vision transformer. In: International Conference Learning Representations (ICLR) (2022) 3  
+[26] Qin, Z., Wang, K., Zheng, Z., Gu, J., Peng, X., Xu, Z., Zhou, D., Shang, L., Sun, B., Xie, X., et al.: Infobatch: Lossless training speed up by unbiased dynamic data pruning. arXiv preprint arXiv:2303.04947 (2023) 3  
+[27] Tan, M., Le, Q.: Efficientnetv2: Smaller models and faster training. In: International Conference on Machine Learning (ICML). pp. 10096-10106. PMLR (2021) 3  
+[28] Tarvainen, A., Valpola, H.: Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In: Advances on Neural Information Processing Systems (NeurIPS). vol. 30 (2017) 3  
+[29] Tian, Y., Krishnan, D., Isola, P.: Contrastive multiview coding. In: European Conference on Computer Vision (ECCV). pp. 776-794. Springer (2020) 7  
+[30] Touvron, H., Cord, M., Jégou, H.: Deit iii: Revenge of the vit. In: European Conference on Computer Vision (ECCV). pp. 516-533. Springer (2022) 3  
+[31] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I.: Attention is all you need. Advances on Neural Information Processing Systems (NeurIPS) 30 (2017) 4  
+[32] Wu, Z., Xiong, Y., Yu, S.X., Lin, D.: Unsupervised feature learning via non-parametric instance discrimination. In: IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 3733-3742 (2018) 2  
+[33] Xiao, T., Wang, X., Efros, A.A., Darrell, T.: What should not be contrastive in contrastive learning. In: International Conference Learning Representations (ICLR) (2020) 7  
+[34] Zhou, T., Wang, S., Bilmes, J.: Curriculum learning by dynamic instance hardness. Advances on Neural Information Processing Systems (NeurIPS) 33, 8602-8613 (2020) 3
+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS paper checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Abstract and Introduction.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: 5 Gradient Estimation Analysis of Acceleration Strategies
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: 6 Experiments
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [No]
+
+Justification: The data (ImageNet100 and ImageNet1000) are public. The code will be released after the paper accepted.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: 6 Experiments
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Our experiments are time-consuming. It is impractical to repeat the experiment for error bars.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: 6 Experiments
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
+
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The techniques presented in this paper can motivate more researchers with limited GPU resources to engage in self-supervised learning.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.
+
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: All data and code used in this paper are cited appropriately.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

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