# symmetric_linear_bandits_with_hidden_symmetry__7765fa93.pdf Symmetric Linear Bandits with Hidden Symmetry Nam Phuong Tran Department of Computer Science University of Warwick Coventry, United Kingdom nam.p.tran@warwick.ac.uk The Anh Ta CSIRO s Data61 Marsfield, NSW, Australia theanh.ta@csiro.au Debmalya Mandal Department of Computer Science University of Warwick Coventry, United Kingdom debmalya.mandal@warwick.ac.uk Long Tran-Thanh Department of Computer Science University of Warwick Coventry, United Kingdom long.tran-thanh@warwick.ac.uk High-dimensional linear bandits with low-dimensional structure have received considerable attention in recent studies due to their practical significance. The most common structure in the literature is sparsity. However, it may not be available in practice. Symmetry, where the reward is invariant under certain groups of transformations on the set of arms, is another important inductive bias in the highdimensional case that covers many standard structures, including sparsity. In this work, we study high-dimensional symmetric linear bandits where the symmetry is hidden from the learner, and the correct symmetry needs to be learned in an online setting. We examine the structure of a collection of hidden symmetry and provide a method based on model selection within the collection of low-dimensional subspaces. Our algorithm achieves a regret bound of O(d2/3 0 T 2/3 log(d)), where d is the ambient dimension which is potentially very large, and d0 is the dimension of the true low-dimensional subspace such that d0 d. With an extra assumption on well-separated models, we can further improve the regret to O(d0 p 1 Introduction Stochastic bandit is a sequential decision-making problem in which a player, who aims to maximize her reward, selects an action at each step and receives a stochastic reward, drawn from an initially unknown distribution of the selected arm, in response. Linear stochastic bandit (LSB) [1] is an important variant in which the expected value of the reward is a linear function of the action. It is one of the most studied bandit variants and has many practical applications [27]. Actions in LSB are specified as feature vectors in Rd for very large feature dimension d, with performance i.e. the resulting regret scaling with d. Many works have addressed this curse of dimensionality by leveraging different low-dimensional structures as inductive biases for the learner. For example, sparsity, which assumes that the reward is a sparse linear function, has been used extensively in LSB to design bandit algorithms with better performance [2, 36, 23]. However, when the reward function lacks the structure for sparsity (which may occur in many real-world situations), a question arises: Are there different structures in the features of LSB that we can exploit to overcome the curse of dimensionality and design bandit algorithms with better performance? In this paper, we study the inductive bias induced by symmetry structures in LSB, which is a more general model inductive bias than sparsity, and can facilitate efficient and effective learning [9]. Symmetry describes how, under certain transformations of the input of the problem, the outcome 38th Conference on Neural Information Processing Systems (Neur IPS 2024). should either remain unchanged (invariance) or shift predictably (equivariance). In supervised learning, it has been empirically observed [17, 12] and theoretically proven [16, 6] that explicitly integrating symmetry into models leads to improved generalization error. However, in the literature on sequential decision-making, unlike sparsity, symmetry is rarely considered to date. This leads us to the following research question: Can one leverage symmetry in sequential decision-making tasks to enable effective exploration, and eventually break the curse of dimensionality? In the machine learning literature, especially in supervised learning, most studies on symmetry assume prior knowledge of symmetry structures of the tasks under consideration [31, 16]. However, in numerous practical scenarios, the learner can only access to partial knowledge of the symmetry, necessitating the incorporation of symmetry learning mechanisms into the algorithms to achieve better performance. Examples of hidden symmetry can be found in multi-agent learning with cooperative behavior. As a motivating example, consider a company undertaking a large project that consists of several subtasks. The company must hire subcontractors with the goal of maximizing project quality while staying within budget constraints. Symmetry may arise in this situation when coalitions form among subcontractors, where members of a coalition work together to complete their allocated tasks using shared resources. In particular, the allocation of tasks within a coalition can be swapped without affecting overall team performance, inducing symmetry (i.e., performance remains invariant under permutation) in the task assignments. Coalitions among subcontractors often arise since sharing labor and resources reduces operational costs, making their work more efficient and cost-effective. However, these coalitions are typically hidden from the hiring company. One reason is that if the hiring company were aware of these collaborations, they could use this information to negotiate lower prices, knowing that the subcontractors are benefiting from shared resources. Another reason is that coalitions may raise concerns about collusion. In particular, in a competitive market, such as when subcontractors are hired through a bidding platform, coalition members can collaborate to manipulate the bidding process, which is considered unfair and could undermine the integrity of the bidding. For more practical examples of hidden symmetry in multi-agent reinforcement learning [34] and robotics [3], we refer the reader to Appendix D.2. Motivated by these examples, we believe that hidden symmetry is much more relevant in the context of sequential decision-making because the environment and its symmetry structure may not be readily available to the learner, as opposed to supervised learning and offline settings where data are provided during the training phase. As the learner has the power to freely collect data, it is expected that they will learn the hidden symmetry structures as they explore the environment. Against this background, we ask the question of whether learner can leverage symmetry to enable effective exploration, and break the curse of dimensionality without a prior knowledge of the symmetry structure? Moreover, in the presence of symmetry, when can we design learning algorithms with optimal regret bounds? Towards answering this question, we investigate the setting of symmetric linear stochastic bandit in Rd, where d is potentially very large, and the expected reward function is invariant with respect to the actions of a hidden group G of coordinate permutations. Our contributions are summarised as follows: 1. We first give an impossibility result that no algorithm can get any benefit by solely knowing that G is a subgroup of permutation matrices. We achieve this by formally establishing a relation between the class of subgroup to the partition over the set {1, ..., d}. A direct implication of this impossibility result is that it is necessary to have further information about the structure of the hidden subgroup in order to achieve improved regret bounds. 2. Given this, we establish a cardinality condition on the class of symmetric linear bandits with hidden G, in which the learner can learn the true symmetry structure and overcome the curse of dimensionality. Notably, this class includes a sparsity as a special case, and therefore inherits all the computational and statistical complexities of sparsity. Apart from sparsity, this class includes many other practically relevant classes, such as partitions that respect underlying hierarchies, non-crossing partitions, and non-nesting partitions (see Subsection 4.2.1 and Appendix D.2). 3. We cast our problem of learning with hidden subgroup G into model selection with collection of low-dimensional subspaces [25, 35]. To address the polynomial scaling of regret bounds with respect to the number of models and arms in previous works, we depart from model aggregation, which is typically used in LSB model selection, and introduce a new framework inspired by Gaussian model selection [7] 1 and compressed sensing [8]. Based on this framework, we introduce a new algorithm, called EMC (for Explore Models then Commit). Under the assumption that the set of arm is exploratory, we prove that the regret bound of the EMC algorithm is O(d2/3 0 T 2/3 log(d)), and O(d0 T log(d)) with an additional assumption on well-separated partitions, where d0 d is the dimension of the low-dimensional subspace associated with group G. To the best of our knowledge, our work is the first in the linear stochastic bandits literature that leverages symmetry in designing provably efficient algorithms. To save space, all proofs in this paper are deferred to the Appendix. 1.1 Related Work We now briefly outline related work and compare them with our results. We refer the reader to Appendix F for a more in-depth literature review. Sparse linear bandits. As we will explain in Section 4.2, sparsity is equivalent to a subset symmetry structures, and thus, can be seen as a special case of our setting. As such, we first review the literature of sparsity. Sparse linear bandits were first investigated in [2], where the authors achieve a regret of O( ds T), with O disregarding the logarithmic factor, and s representing the sparsity level, and T is the time horizon. This matches the regret lower bound for sparse bandits, which is Ω( ds T) [27]. More recently, the contextual version of linear bandits has gained popularity, where additional assumptions are made regarding the context distribution and set of arms [26, 36, 28, 11, 23] to avoid polynomial dependence on d. Notably, with the assumption on exploratory set of arms, [23] propose an Explore then Commit style strategy that achieves O(s 2 3 T 2 3 ), nearly matching the regret lower bound Ω(s 2 3 T 2 3 ) [24] in the data-poor regime. As sparsity is equivalent to a subclass of hidden symmetry, all the lower bounds for sparse problems apply to our setting of learning with hidden symmetry. Model selection. Our problem is also closely related to the problem of model selection in linear bandits, as the learner can collect potential candidates for the hidden symmetry model. Particularly, in model selection, there is a collection of M features, and different linear bandits running with each of these features serve as base algorithms. By exploiting the fact that the data can be shared across all the base algorithms, the dependence of regret in terms of the number of features can be reduced to log(M). In particular, [25] propose a method that concatenates all M features of dimension d into one feature of dimension Md, and uses the Lasso estimation as a aggregation of models. Their algorithm achieves a regret bound of O(T 3 4 p log(M)) under the assumption that the Euclidean norm of the concatenated feature is bounded by a constant. However, in our case, the Euclidean norm of the concatenated feature vector can be as large as M, which leads to a M multiplicative factor in the regret bound. Besides, [35] uses the online aggregation oracle approach, and is able to obtain regret of O( p Kd T log(M)), where K is the number of arms. In contrast, we use algorithmic mechanisms that are different from aggregation of models. In particular, we explicitly exploit the structure of the model class as a collection of subspaces and invoke results from Gaussian model selection [21, 7] and dimension reduction on the union of subspaces [8]. With this technique, we are able to achieve O(T 2 3 log(M)), which is rate-optimal in the data-poor regime, has logarithmic dependence on M without strong assumptions on the norm of concatenated features, and is independent of the number of arms K. We refer the reader to Section 4.2 for a more detailed explanation. Symmetry in online learning. The notion of symmetry in Markov decision process dates back to works such as [22, 40]. Generally, the reward function and probability transition are preserved under an action of a group on the state-action space. Exploiting known symmetry has been shown to help achieve better performance empirically [43, 42] or tighter regret bounds theoretically [41]. However, all these works requires knowledge of symmetry group, while our setting consider hidden symmetry group which may be considerably harder. Hidden symmetry on the context or state space has been studied by few authors, with the term context-lumpable bandits [29], meaning that the set of contexts can be partitioned into classes of similar contexts. It is important to note that the symmetry group acts differently on the context space and the action space. As we shall explain in detail in Section 3, while one can achieve a reduction in terms of regret in the case of hidden symmetry acting on context 1We note that Gaussian model selection is a technique in statistics, similar to model aggregation (see [21] s chapter 2 and 4), which should not be confused with model selection in the bandit literature. spaces [29], this is not the case when the symmetry group acts on the action space. The work closest to ours is [39], where the authors consider the setting of a K-armed bandit, where the set of arms can be partitioned into groups with similar mean rewards, such that each group has at least q > 2 arms. With the constrained partition, the instance-dependent regret bounds are shown asymptotically to be of order O K q log T . Comparing to [39], we study the setting of stochastic linear bandits with similar arms, in which the (hidden) symmetry and linearity structure may intertwine, making the problem more sophisticated. We also impose different constraints on the way one partitions the set of arms, which is more natural in the setting of linear bandits with infinite arms. 2 Problem Setting For any k N+, denote [k] = {1, . . . , k}. For X Rd, let (X) denote the set of all probability measures supported on X. Given a set S Rk, for some k > 1, denote ΠS(x) as the Euclidean projection of x Rk on S, and conv(S) as the convex hull of S. We denote by T the number of rounds, which is assumed to be known in advance. Each round t [T], the agent chooses an arm xt X Rd, and nature returns a stochastic reward yt = xt, θ + ηt, where ηt is an i.i.d. σ-Gaussian random variable. Now, denote f(xt) = E[yt | xt]. A bandit strategy is a decision rule for choosing an arm xt in round t [T], given past observations up to round t 1. Formally, a bandit strategy is a mapping A : (X R)T (X). Let x = arg maxx X f(x), and let RT = E h PT t=1 x xt, θ i denote the expected cumulative regret. In this paper, we investigate the question whether one can get any reduction in term of regret, if the reward function is invariant under the action of a hidden group of transformations on the set of arms. We define the notion of group of symmetry as follows: Group and group action. Given d N+, let Sd denote the symmetry group of [d], that is, Sd := {h : [d] [d] | h is bijective} the collection of all bijective mappings from [d] to itself. We also define the group action ˆϕ of Sd on the vector space Rd as ϕ : Sd Rd Rd g, (xi)i [d] 7 (xg(i))i [d] (1) In other words, a group element g acts on an arm x Rd by permuting the coordinates of x. In the setting of linear bandit, the permutation group action also acts on the set of parameters via coordinate permutation. For brevity, we simply denote g θ and g x as ϕ(g, θ) and ϕ(g, x), respectively. Denote by Ag the permutation matrix corresponding to g. We write G Sd to denote that G is a subgroup of Sd. Given any point θ Rd, we write G θ = {g θ | g G} to denote the orbit of θ under G. It is well known that the orbit induced by the induced action of a subgroup G Sd corresponds to a set partition of [d]. We denote this partition as πG. Let G be a subgroup of Sd that acts on Rd via the action ϕ. In a symmetric linear bandit, the expected reward is invariant under the group action of G on X, that is, f(g x) = f(x). Due to the linear structure of f, this is equivalent to g θ = θ for all g G. We assume that, while the group action ϕ is known to the learner, the specific subgroup G is hidden and must be learned in an online manner. 3 Impossibility Result of Learning with General Hidden Subgroups We now show how to frame the learning problem with hidden symmetry group as the problem of model selection. We further analyse the structure of the collection of models, and show that no algorithm can benefit by solely knowing that G Sd, which implies that further assumptions are required to achieve significant improvement in term of regret. 3.1 Fixed Point Subspace and Partition The analysis of learning with hidden subgroup requires a group-theoretic notion which is referred to as fixed-point subspaces [10]. As we shall explain promptly, there is a tight connection between the collection of fixed-point subspaces and set partitions. Fixed-point subspaces. For a subset X Rd, denote Fix G(X) := {x X | g x = x, g G} as the fixed-point subspace of G; and FSd(X) := {Fix G(X) | G Sd} as the collection of all fixedpoint subspaces of all subgroups of Sd. We simply write FSd = FSd(Rd) and Fix G = Fix G(Rd) for brevity. Set partition. Given d N+, we denote Pd as the set of all partitions of [d]. Let Pd,k as the set of all partitions of [d] with exactly k classes, and Pd, k be the set of all partitions of [d] with at most k classes. The number of set partitions with k classes |Pd,k| is known as the Stirling number of the second kind, and |Pd| is known as Bell number. 3.2 Impossibility Result Problem with known symmetry. Before discussing the problem of hidden symmetry, let us explain why the learner with an exact knowledge of G can trivially achieve smaller regret. The reason is that θ Fix G by the assumption that θ is invariant w.r.t the action of group G. If G is known in advance, the learner can restrict the support of θ in Fix G, and immediately obtains that the regret scales with dim(Fix G) instead of d, which can be significantly smaller (e.g., if G = Sd, then dim(Fix G) = 1). For any subgroup, there exists a fixed point subspace, and some subgroups may share the same fixed point subspace. Therefore, instead of constructing a collection of subgroups, one can create a smaller collection of models using the collection of fixed point subspaces. As G is hidden, one must learn Fix G within the set of candidates FSd, leading to the formulation of the model selection. From the setting with hidden subgroup to the setting with hidden set partition. Now, we discuss the structure of the collection of models FSd. First, we show the equivalent structure between the collection of fixed point subspaces and the set partitions as follows. Proposition 1. There is a bijection H between Pd and FSd. As there is a bijection between Pd and FSd, we can count the number of subspaces of each dimension k explicitly using the following. Proposition 2 ([10] s Theorem 14). Given a subgroup Γ Sd and its fixed-point subspace FixΓ, suppose that πΓ partitions [d] into k classes, then dim(FixΓ) = k. By Proposition 2, we have that the number of subspaces of dimension k in FSd is exactly the number of set partitions with k-classes. Suppose that the learner knows that the orbit under action of G partitions the index of θ into 2 equivalent classes that is, dim(Fix G) = 2. The learner cannot get any reduction in terms of regret. Proposition 3. Assume that the action set is the unit cube X = {x Rd | x 1}, and f is invariant w.r.t. action of subgroup G Sd, such that dim(Fix G) = 2. Then, the regret of any bandit algorithm is lower bounded by RT = Ω(d The implication of Proposition 3 is that even if the learner knows θ lies in an extremely lowdimensional subspace within the finite pools of candidates, they still suffer a regret that scales linearly with the ambient dimension d. This suggests that further information about the group G must assumed to be known in order to break this polynomial dependence on d in the regret bound. 4 The Case of Hidden Subgroups with Subexponential Size As indicated by Proposition 3, there is no improvement in terms of regret, despite the learner having access to a collection of extremely low-dimensional fixed point subspaces. Therefore, we assume that the learner can access only a reasonably small subset of the collection of low-dimensional fixed point subspaces. Let d0 be the upper bound for the dimension of fixed point subspaces; that is, we know that the orbit of G partitions [d] into at most d0 classes. Now, let us assume that the learner knows that G does not partition [d] freely, but must satisfy certain constraints, that is, πG Qd, d0 Pd, d0. Here, Qd, d0 is a small collection of partitions with at most d0 classes, which encodes the constraints on the way G partitions [d]. We introduce an assumption regarding the cardinality of Qd, d0, which is formally stated in Section 4.2. Using the Proposition 1, we can define the collection of fixed point subspaces associated with the collection of partition Qd, d0 via the bijection H as M := H (Qd, d0) and M := |M|. In addition, let us define the extension of the collection M as M := {conv (m m ) | m, m M} , where conv(S) is the convex hull of the set S Rn. We have that M is a collection of subspaces, that is, conv (m m ) is indeed a subspace [8]. Denote M := |M|, then we have M = (M 2 M)/2. Moreover, if dimension of subspace in M is at most d0, then the dimension of subspace in M is at most 2d0. 4.1 The Explore-Models-then-Commit Algorithm Given some n [T], we define Y = Xθ + η, (2) where Y Rn, X = [x1, . . . , xn] Rn d is the design matrix, θ Rd is the true model; η = [η1, ..., ηn]. We have the information that θ must be contained in some (not necessarily unique) subspace m M. Denote by dm the dimension of m, we have dm d0 for any m M. Let Xm = [Πm(xt)] t [n], and Sm be the column space of Xm, one has dim(Sm) dm. For any m M, and given Y , let ΠSm( ) be the projection onto Sm. Define b fm := ΠSm(Y ); bθm := arg min θ m Y Xθ 2 2 . (3) Now, given n data points, we can choose the model bm M that minimises the least square error bm arg min m M Y b fm 2 2. (4) Based on the framework of model selection, we now introduce our Algorithm 1, Explore-Models-then-Commit (EMC). Our algorithm falls into the class explore-then-commit bandit algorithms. The exploration phase consists of t1 rounds. During this phase, one samples data independently and identically distributed (i.i.d.) from an exploratory distribution ν. After the exploration phase, one computes the solution to the model selection problem and then commits to the best arm corresponding to the chosen model. Remark 4. The key step of Algorithm 1 that may incur significant costs is solving equation (4) (line 6). Without additional information about M, one might need to enumerate all models in M and optimize among them, which would induce a time complexity of O(ndcd0). However, if we have more information about the partitions, e.g., if they are non-crossing or non-nesting partitions, their lattice structures can be exploited to speed up the optimization process of solving equation (4). Due to space limitations, we refer readers to Appendix D.3 for a detailed explanation of a subroutine that leverages these lattice structures for more efficient computation. Additionally, Section 6 demonstrates that our Algorithm 1, when using the lattice search algorithm for non-crossing partitions and nonnesting partitions as a subroutine, achieves polynomial computational complexity of O(nd5) and guarantees low regret. 4.2 Regret Analysis Algorithm 1 Explore Models then Commit 1: Input: T, ν, t1 2: for t = 1, . . . , t1 do 3: Independently pull arm xt according to ν and receive a reward yt. 4: end for 5: X [x1, ..., xt1] , Y [yt]t [t1]. 6: Compute bm as (4). 7: Compute bθt1 as (3) corresponding to bm. 8: for t = t1 + 1 to T do 9: Take greedy actions: xt = arg min x X D bθt1, x E . 10: end for The regret analysis of Algorithm 1 uses results from the Gaussian model selection literature [7, 21] as a basis. As such, we first state the assumptions that are common in the Gaussian model selection literature on the collection of models M and the set of arms X (Section 4.2.1). We then provide our main analysis in Section 4.2.2, highlighting the key technical novelties of our approach. 4.2.1 Assumptions Recall that due to our lower bound in Proposition 3, further assumptions are required on the collection of fixed-point subspaces to achieve a reduction in terms of regret. As suggested by the model selection literature [25, 35], one can achieve regret in terms of log(M) for a collection of M models. Adopting this idea, we make the following assumption regarding the number of potential fixed-point subspaces and the set of arms. Assumption 5 (Sub-exponential number of partitions). The partition corresponding to G belongs to a small subclass of partitions Qd, d0 Pd, d0. In particular, πG Qd,d , for some d d0, and for each k [d0], there exists a constant c > 0, such that |Qd,k| O(dck). Assumption 6 (Bounded set of arms). There are positive numbers Kx, Rmax, such that, for all x X and m M, Πm(x) 2 2 Kx, and | x, θ | Rmax. As a consequence of Assumption 5, the cardinality of the collection of fixed point subspaces is not too large, particularly, M = O(dcd0). First, we note that this class includes interval partitions, a structure equivalent to sparsity as a strict subset, as explained below. Remark 7 (Equivalence between sparsity and interval partition). A set partition of [d] is an interval partition or partition of interval if its parts are interval. We denote Id as the collection of all interval partition of d. Id admits a Boolean lattice of order 2d 1, making it equivalent to the sparsity structure in d 1 dimensions. Specifically, consider the set of entries of parameters φ Rd with a linear order, that is, φ1 φ2 φd. Then define the variable θ Rd 1 such that θi = (φi φi+1). Each interval partition on the entries of φ will determine a unique sparse pattern of θ. Therefore, it is clear that the cardinality of the set of interval partition with d0 classes is bounded as |Id, d0| = O(dd0). Moreover, as a result, symmetric linear bandit is strictly harder than sparse bandit and inherits all the computational complexity challenges of sparse linear bandit, including the NP-hardness of computational complexity. Figure 1: Partition that respects the underlying ordered tree. Apart from sparsity, class of partitions with subexponential size also naturally appears when there is a hierarchical structure on the set [d], and the partitioning needs to respect this hierarchical structure. A partition that respects an ordered tree groups the children of the same node into a single equivalence class, for example, see Figure 1. It is shown in [15] and the cardinality of the set of partitions that respect ordered trees is sub-exponential. Furthermore, as shown in [15] there is a bijection between partitions that respect ordered trees and the set of noncrossing partitions. A real-life example that meets these assumptions is the subcontractor example in the introduction: A hierarchical structure may exist, where a hired subcontractor can further subcontract parts of the work to others. A tree represents the hierarchical order among subcontractors, where subcontractors hired by another contractor can be grouped into one class. Further real-life examples of non-crossing partitions and other structured partitions that satisfy sub-exponential cardinality, such as non-nesting partitions and pattern-avoidance partitions [33], can be found in Appendix D.2. Next, similar as [23], we define the exploratory distribution as follows. Definition 8 (Exploratory distribution). The exploratory distribution ν (X) is solution of the following optimisation problem ν = arg max ω (X) λmin Ex ω[xx ] , V := Ex ν[xx ], Cmin(X) := λmin(V ). (5) Since our setting includes sparsity as a special case, the regret lower bound in [24] applies to our setting as well. In particular, we have: Proposition 9 (Regret lower bound). There exist symmetric linear bandit instances in which Assumption 5, 6 hold with Kx = 8d0, such that, any bandit algorithm must suffer regret RT = Ω min Cmin(X) 1 2 3 0 T 2 3 , We note that the lower bound can be relaxed if we have a stronger assumption on the group G, which allows algorithm to go beyond the lower bound in Proposition 9 of the sparsity class. For example, if we consider a collection of fixed-point subspaces with a nested structure, similar to those discussed in [20], the algorithm may achieve a O(d0 T) rate. The key takeaway is that symmetry exhibits significant flexibility in structure. Depending on the specific class of symmetry, one may achieve either no reduction at all or significant reduction in terms of regret bound. 4.2.2 Main Regret Upper Bound Result We now state the main result of the regret upper bound for Algorithm 1. Theorem 10 (Regret upper bound). Suppose the Assumptions 5, 6 hold. With the choice of t1 = R 2 3 maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 , then the regret of Algorithm 1 is upper bounded as 1 3maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 (6) Remark 11. We note that when Kx = O(d0), as in the lower bound instances, our upper bound is O C 1 2 3 0 T 2 3 , which matches the lower bound in Proposition 9. The main idea is to bound the risk error after exploration rounds, as stated in the following lemma which implies the regret bound after standard manipulations. Lemma 12. Suppose the Assumptions 5, 6 hold. For t1 = Ω(K2 xd0C 2 min(X) log(d/δ)), with probability at least 1 δ, one has the estimate θ bθt1 2 = O σ2d0 log(d/δ) Remark 13 (Non-triviality of Lemma 12). At the first glance, it seems that we can cast the problem of learning with a collection of M subspaces into a model selection problem in linear bandit with M features. This leads to a question: Can we apply the model selection framework based on model aggregation in [35, 25] to our case? First, let us explain how to cast our problem into a model selection problem in linear bandit. For each subspace m, let Φm : Rd Rd0 be the feature map that computes the image of the projection Πm with respect to the orthogonal basis of subspace m. Thus, we then have a collection of M features {Φm}m M. Consider the algorithm introduced in [25], which concatenates the feature maps into Φ(x) = [Φ1(x), . . . , ΦM(x)] RMd0, and the regret bound depends on Φ(x) 2. However, in our case where Φm(x) 2 < 1, we can only bound Φ(x) 2 M, which leads to a M dependence on regret, if we use their algorithm. Regarding [35], their algorithm aggregates the predictions among models for each arm, and based on that, they compute the distribution for choosing each arm. This leads to the regret scaling with the number of arms K, which is not feasible in our case when K = . We note that the similarity with the model selection technique in [25, 35] is that they use model ag- gregation among M to bound the prediction error PT t=1 D xt, bθ θ E2 , but this does not necessarily guarantee the risk error bθ θ 2. The reason is that, although model aggregation can guarantee a small prediction error, it imposes no restriction on the estimator bθ, which limits its ability to leverage the further benign property of designed matrix X. Instead of model aggregation, our algorithm explicitly picks the best model from the pool M, ensuring the following two properties: (1) The prediction error is small, similar to model aggregation; and (2) We can guarantee that bθ lies in one of the subspaces of M. The second property gives us control over (bθ θ ) by ensuring it lies in at most M 2 subspaces. Then, exploiting the restricted isometry property (see Definition 15) of designed matrix X, we can guarantee that with O(log(M)) exploratory samples, we can bound the risk error bθ θ 2. This is crucial for eliminating polynomial dependence on M and the number of arms K. Proof sketch of Lemma 12. We provide a proof sketch here and defer their full proof to Appendix B.1. Our proof borrows techniques from Gaussian model selection [21] and the compressed sensing literature [8]. There are two steps to bound the risk error as in Lemma 12: Step 1 - Bounding the prediction error. We can bound the prediction error Xθ Xbθt1 2 using the Gaussian model selection technique [21] as follows. Proposition 14. Let f = Xθ . For the choice of b f b m as in Eqn. (3) & Eqn. (4), with probability at least 1 δ, there exists a constant C > 1 such that b f b m f 2 2 Cσ2 log Mδ 1 . (8) Step 2 - Bounding the risk error from prediction error. To bound the risk error from the prediction error, we invoke the restricted isometry property on the union of subspaces of a sub Gaussian random matrix as in [8]. Note that bθ and θ can belong to two different subspaces of M, and bθ θ may not lie in any subspace of M, but in M. An important property of the design matrix X, which allows one to recover θ with the knowledge that θ is in a subspace m M, can be captured by the following notion of restricted isometry property (RIP): Definition 15 (Restricted isometry property). For any matrix X, any collection of subspaces M and any θ m M, we define M-restricted isometry constant δM(X) to be the smallest quantity such that (1 δM(X)) θ 2 2 Xθ 2 2 (1 + δM(X)) θ 2 2. (9) We have the minimum number of samples required so that a random matrix X satisfies RIP for a given constant with high probability as Proposition 16 below. Then, Lemma 12 is followed by combining Proposition 16 and Proposition 14. Proposition 16. Let X = [xt]t [n], where xt s are is i.i.d. drawn from ν, and let n = Ω C 2 min(X)K2 x log(2Md0δ 1) . Then, with probability at least 1 δ, and for any θ1, θ2 in subspaces of M, one has that θ1 θ2 2 2 2C 1 min(X)n 1 X(θ1 θ2) 2 2 . (10) 5 Improved Regret Upper Bound with Well-Separated Partitions We now show that by adding more structure (i.e., well-separatedness) to the setting, we can further improve the regret upper bound to O( T). In particular, we will introduce the notion of wellseparatedness in this section, and show that this notion can lead to improved (i.e., O( T)) regret bounds. Algorithm 2 Exploring Model then Commit with well-separated partition 1: Input: T, ν, t2 2: for t = 1, , t2 do 3: Independently pull arm xt according to ν and receive a reward yt. 4: end for 5: X [x1, ..., xt1] , Y [yt]t [t1]. 6: Compute bm as (4). 7: for t = t2 + 1 to T do 8: Playing OFUL algorithm [1] on bm. 9: end for For each partition p Pd, there is a unique equivalence relation on [d] corresponding to p. Denote by p the equivalence relation corresponding to p. Next, we define well-separatedness. Assumption 17 (Well-separated partitioning). Given the true subgroup G, and the corresponding partition πG. For all (i, j) such that i πG j, it holds that |θ ,i θ ,j| ε0, for some ε0 > 0. The implication of Assumption 17 is that the projection of θ to any subspace m M not containing θ will cause some bias in the estimation error. In particular, one can show that for any m M such that θ / m, it holds that θ Πm(θ ) 2 2 ε2 0/2. (11) We now show that under the Assumption 17, after the exploring phase, the algorithm returns a true fixed-point subspace bm θ with high probability. Theorem 18. Suppose the Assumptions 5, 6, 17 hold. Let t2 = Ω σ2K2 xd0 log(d T ) C2 min(X)ε2 0 . Then, Algorithm 2 returns bm θ with probability at least 1 1/T, and its regret is upper bounded as RT = O Rmaxσ2K2 xd0 log(d T) C2 min(X)ε2 0 + σd0 p T log(Kx T) . (12) That is, if the separating constant ε0 is known in advance and ε0 T 1/4, then we can achieve O(d0 T log(Kx T)) regret upper bound. Remark 19. A weakness of Algorithm 2 is that without knowing that ε0 T 1/4 is true a priori, there may be possible mis-specification error, which leads to linear regret if one applies the algorithm naively. On the other hand, Algorithm 1 can always achieve regret O(T 2/3) in the worst case. As such, the following question arises: Does there exist an algorithm that, without the knowledge of ε0, can achieve regret O( T) whenever ε0 T 1/4, but guarantees the worst-case regret of O(T 2/3)? Toward answering this question, we propose a simple method which has O( T) regret whenever the separating constant is large, and enjoys a worst-case regret guarantee of O(T 3/4) (slightly worse than O(T 2/3)). We refer the reader to Appendix C.2 for a detailed description of the algorithm, its regret bound and further discussion. 6 Experiment To illustrate the performance of our algorithm, we conduct simulations where the entries of θ satisfy three cases: sparsity, non-crossing partitions and non-nesting partitions. We refer readers to Appendix D.1 for a more formal description of non-crossing partitions, non-nesting partitions, and why the interval partition (i.e., the partition structure equivalent to sparsity) is a strict subset of both non-crossing and non-nesting partitions. Since sparsity is equivalent to a strict subset of non-crossing and non-nesting partitions, we compare our Algorithm 1 with the sparse-bandit ESTC algorithm proposed in [23] as a benchmark in all environments. The set of arms X is d Sd 1, σ = 0.1, and d = 100, d0 = 15. The ground-truth sparse patterns, partitions and θ are randomized before each simulation. The regret of both algorithms is shown in Figure 2, which indicates that our algorithm performs competitively in the sparsity case and significantly outperforms the sparse-bandit algorithm in cases of non-crossing and non-nesting partitions. Due to space limitations, we refer the reader to Appendix E for a detailed description of the experiments, including how we applied the sparse-bandit algorithm in the cases of non-crossing and non-nesting partitions, and how we ran Algorithm 1 in the case of sparsity. Additionally, we explain how we exploited the particular structure of non-crossing and non-nesting partitions to enable efficient computation in Appendix E. Figure 2: Regret of EMC (Algorithm 1) and of ESTC proposed in [23], in cases of sparsity, noncrossing partitions, and non-nesting partitions. 7 Conclusion and Future Work In this paper, we study symmetric linear stochastic bandits in high dimensions, where the linear reward function is invariant with respect to some hidden subgroup G Sd. We first prove that no algorithm can gain any advantage solely by knowing G Sd. Given this, we introduce a cardinality condition on the hidden subgroup G, allowing the learner to overcome the curse of dimensionality. Under this condition, we propose novel model selection algorithms that achieve regrets of O(d2/3 0 T 2/3) and O(d0 T) with an additional assumption on the well-separated partition. 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Contents of Appendix A Impossibility Result of Learning with General Hidden Subgroups 14 B The Case of Hidden Subgroups with Subexponential Size 15 B.1 Lower Bound for the Case of Hidden Subgroups with Subexponential Size . . . . . 15 B.2 High Probability Prediction Error of Model Selection . . . . . . . . . . . . . . . . 17 B.3 Regret Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 C Improved Regret Bound 21 C.1 Improve Regret Bound with Well-Separated Partitions . . . . . . . . . . . . . . . . 21 C.2 Adapting to Separating Constant ε0 . . . . . . . . . . . . . . . . . . . . . . . . . 23 D On Collections of Partitions with Subexponential-Size 24 D.1 Important Classes of Partitions with Subexponential-Size . . . . . . . . . . . . . . 24 D.2 Practical Examples of Partitions with Subexponential-Size . . . . . . . . . . . . . 25 D.3 Efficient Greedy Algorithm for Specific Classes of Partitions . . . . . . . . . . . . 25 E Experiment details 26 F Extended Related Work 27 Additional notations For any subset S Rd, and matrix W Rd d, we write W(S) := {Wx | x S}. A Impossibility Result of Learning with General Hidden Subgroups To prove Proposition 1, we need to prove Proposition 20 and use Proposition 21. Proposition 20. For each Γ Sd acting naturally on [d], the orbit of Γ on [d] forms a unique partition of [d]. Moreover, for each partition ρ Pd, there exists at least one Γ Sd such that its orbit on [d] under natural action is exactly ρ. Proof. The first claim is obvious by the property of orbit, that is, orbit consists of non-empty and disjoint subsets of [d], whose union is [d]. We prove the second claim. Let ρ = {ρi}i I, where I is the index of partition; note that ρi is nonempty and mutually disjoint. Define a group as follows Γi = {f : ρi ρi | f is bijective} ; (13) It is clear that Γi is a group under function composition. Now, define the product group and the action ψ : Γ [d] [d] such that ψ ((fi)i I, x) := fj(x), for ρj x. (14) Therefore, it is clear that (fi)i I is a bijection from [d] onto itself, hence, Γ is a subgroup of Sd. Let E = (ei)i [d] be the standard basis. Given that group G partition [d] into k disjoint orbits, G also partition E into k disjoint orbits corresponding to its action ϕ on Rd, that is, Let Vi = Span(Ei), then, one has the following. Proposition 21 ([10] s Theorem 14). Fix G(Rd) = i=1 Fix G(Vi). (15) We now state the proof of Proposition 1 as a corollary of Proposition 20 and Proposition 21. Proposition 1. There is a bijection H between Pd and FSd. Proof. First, we show that for each set partition of [d], there exists a unique H FSd(Rd). This is straightforward due to the fact that, given a set partition P, the partition of basis (Ei)i I is unique by definition, so as Vi = span(Ei). As a result, let H = Lk i=1 FixΓ(Vi), by Proposition 21, H FSd(Rd). As (Vi)i I is unique, H is unique. Second, we show that for each H FSd(Rd), there is a unique set partition/equivalent relation P. Denote P as an equivalent relation under the set partition P. Suppose there are two different set partitions P, Q, there must be two set element p q such that p P q under P, and p Q q under Q. Denote HP , HQ as the subspaces defined by P and Q respectively. As p Q q, there must exist a point x HQ such that xp = xq. However, xp = xq for all x HP . Hence, HP cannot be the same as HQ. Proposition 3. Assume that the action set is the unit cube X = {x Rd | x 1}, and f is invariant w.r.t. action of subgroup G Sd, such that dim(Fix G) = 2. Then, the regret of any bandit algorithm is lower bounded by RT = Ω(d Proof. Suppose there is a collection of model parameter Θ = { ε, ε}d, for some ε > 0. For each θ Θ, it is straightforward that since θ has two classes of indices, by Proposition 1 and 2, there must be a subgroup G Sd such that θ Fix G and dim (Fix G) = 2. Now, Θ can be used as a family of problem instances for minimax lower bound of linear bandit. By Theorem 24.1 in [27], we have that with the choice of ε = T 1/2, one has that B The Case of Hidden Subgroups with Subexponential Size B.1 Lower Bound for the Case of Hidden Subgroups with Subexponential Size Proposition 9 (Regret lower bound). There exist symmetric linear bandit instances in which Assumption 5, 6 hold with Kx = 8d0, such that, any bandit algorithm must suffer regret RT = Ω min Cmin(X) 1 2 3 0 T 2 3 , Proof sketch. First, we show a change of variable between the sparsity and interval partition cases, including the parameter space and set of arms. Next, we demonstrate that mapping the problem instances in the lower bound of sparse bandits in [23, 24] satisfies the constraints in Assumption 6. After that, the proof follows immediately by carrying out step-by-step calculations similar to those in [23, 24]. First, let us define the sparse bandit instances. Let Z be the set of arm of sparse bandit problem, such that z Z, z 1. Let φ Rd be a d0-sparse parameters. Now, let us define the mapping that corresponds to change of variable between θ and φ as follows: (1) For i [d 1], φi = θi θi+1, (2) φd = θd. Therefore, one can write 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 1 1 0 0 0 0 0 1 | {z } W Rd d It is easy to verify that entries of θ has k equivalent classes if and only if φ is k sparse. Now, suppose expected reward for a arm x in interval-partition instance is equal to expected reward for an arm z in sparsity instance, that is, x, θ = (W ) 1x, Wθ = z, φ . (18) Therefore, to guarantee the bijection between sparsity and interval partition instance, we define the set of arm X = W (Z) := W z | z Z . We need to prove that X satisfies Assumption 6. Therefore, x = W z, and X = W (Z). Our job is to verify that, for all z [ 1, 1]d, the projection Πm(V z) 2 8d0, for any m such that dim(m) 2d0. Let S1, ..., Sdim(m) be the classes according to the interval partition corresponding to m, such that Πm(x) 2 2 = j 1 such that b f b m f 2 2 Cσ2 log Mδ 1 . (8) Proof. Define E1 as the event such that for all m M, ΠSm(η) 2 σ p By Proposition 22 and union bound, E1 occurs with probability 1 Mδ0. For the rest of the proof, we assume E1 occurs . Let us denote b f := b f b m. By the model selection procedure (4), one has that Y b f 2 Y b fm 2. (24) Also, as Y = f + η, one has that, for some K > 1 b f f 2 2 + 2 D η, f b fm E + Kσ2d b m 2 D η, f b f E Kσ2d b m. (25) First, consider the term f b fm 2 2. Note that, ΠSm (f ) = f , as f Sm . We have that, 2 = f ΠSm (f + η) 2 2 = f f 2 2 2 ΠSm (η), f f + ΠSm (η) 2 2 2σ2d0 + 4σ2 log 1 Second, consider the term D η, f b fm E . We have that, D η, f b fm E = η, f ΠSm (f + η) = η, f f ΠSm (η) 2 2 0. (27) Third, we need to control the magnitude of the term 2 D η, b f f E Kσ2d b m. We show that this term can be bounded as 2 D η, b f f E Kσ2d b m a 1 b f f 2 2 + O(log(1/δ0)), with probability at least 1 Mδ0, for any constant a > 0. Denote f be the line that is spanned by f . For each m M, let us define the subspace Sm = Sm + f . Define Sm Sm as the subspace that is orthogonal to f , that is, one can write Sm = Sm L f . By AM-GM inequality, for some a > 0, we have that 2 D η, b f f E = 2 D Π Sc m(η), b f f E = 2 a Π Sc m(η), 1 a a Π Sc m(η) 2 2 + a 1 b f f 2 = aσ2V + aσ2U ˆm + a 1 b f f 2 where V = σ 2 Π f (η) 2 2 and U ˆm = σ 2 Π S ˆ m(η) 2 2. Note that, as dim( f ) = 1, with probability at least 1 δ0, one has that V 2 + 4 log(δ 1 0 ). (29) Define the event E2 as the event where the above inequality holds. Therefore, our final task is to control the quantity a U ˆm Kd b m. Choose a = (K + 1)/2 > 1, one has that a U ˆm Kd b m = K + 1 U ˆm 2K K + 1d b m Um 2K K + 1dm Now, directly control the magnitude of U ˆm 2K K+1d b m is difficult, as bm depends on η, and their distribution might be complicated. Instead, we will control the maximum of the above quantity Um 2K K+1dm over all m M. Since the dimension of Sm is at most dm, similar as the event E1, we define the event E3 as for all m M, we have that Π Sm(η) 2 σ p By Proposition 22 and the union bound, E3 occurs with probability at least 1 Mδ0. We assume that E3 occurs, then for all m M, 2K K + 1dm + 4K K 1 log 1 Therefore, one has that for all m M, U ˆm 2K K + 1d b m Um 2K K + 1dm Putting things together, let δ = (2M +1)δ0, we assure that event E1 E2 E3 occurs with probability at least 1 δ. Combining (26), (27), (29), (32), with (25), one has that 2 2σ2d0 + 4σ2 log 2M + 1 + σ2(K + 1) + 2σ2(K + 1) log 2M + 1 + σ2 2K(K + 1) K 1 log 2M + 1 Note that, the dominating term in the above equation is log M δ , we have that, there is some constant C > 0. b f f 2 2 Cσ2 log M Proposition 16. Let X = [xt]t [n], where xt s are is i.i.d. drawn from ν, and let n = Ω C 2 min(X)K2 x log(2Md0δ 1) . Then, with probability at least 1 δ, and for any θ1, θ2 in subspaces of M, one has that θ1 θ2 2 2 2C 1 min(X)n 1 X(θ1 θ2) 2 2 . (10) Proof. Our proof strategy is inspired by [8]. First, we compute the subgaussian norm for the normalised distribution along fews directions ν. For any x X, we have that max v 2=1, v V 1/2(m), m M D V 1/2x, v E2 = max V 1/2ω 2=1, D V 1/2x, V 1/2ω E2 = max V 1/2ω 2=1, m M Πm(x) 2 2 C 1 min(X) Kx Cmin(X). This means that, the distribution V 1/2x, v 2 has bounded support [0, Kx Cmin ] for any above direction of v. Next, for any θ m, denote θV := V 1/2θ, note that the direction v corresponding to θV satisfies (35). Using Hoeffding s inequality for bounded random variable, there is an absolute constant c1 > 0 such that D V 1/2xi, θV E2 θV 2 2 2 exp nc1C2 min(X)ϵ2 2 exp nc1C2 min(X)ϵ2 Let Z := 1 n XV 1/2. From Lemma 5.1 in [5], we know that if the above inequality holds, then (1 δM (Z) θV 2) ZθV 2 (1 + δM (Z) θV 2) , (37) holds with probability more than 1 2 12 δM (Z) c1C2 min(X)nδ2 for any θV V 1/2(m) in a subspace m M. Take union bound for M subspaces, and let δ = 2M 2 δM (Z) c1C2 min(X)nδ2 M (Z) C2 min(X) log(2M) + d0 log 1 δM (Z) + log(δ 1) ! M (Z) = 1/2. Then, Given n = O K2 x C2 min(X) log(2M) + d0 + log(δ 1) , for probability at least 1 δ, we have that XV 1/2V 1/2θ 2 2Cmin(X) θ 2 2 1 Let θ := θ1 θ2, for any θ1 m, θ2 m , for any m, m M. We conclude the proof. B.3 Regret Upper bound Lemma 12. Suppose the Assumptions 5, 6 hold. For t1 = Ω(K2 xd0C 2 min(X) log(d/δ)), with probability at least 1 δ, one has the estimate θ bθt1 2 = O σ2d0 log(d/δ) Proof. First, let consider Proposition 16, and recall that log(M) and log(M) are both O(d0 log(d)). Therefore, for the choice of t1 = Ω(K2 xd0C 2 min(X) log(d/δ)), with probability at least 1 δ/2, one has that Cmin(X) θ bθt1 2 X(θ bθt1) 2 Second, by Proposition 14, one has that with probability at least 1 δ/2. X(θ bθt1) 2 2 c1σ2d0 log d for some constant c1 > 0. Therefore, putting thing together, we have that with probability at least 1 δ, one has that θ bθt1 2 c2 Cmin(X)t1 , (40) for some constant c2 > 0. Theorem 10 (Regret upper bound). Suppose the Assumptions 5, 6 hold. With the choice of t1 = R 2 3 maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 , then the regret of Algorithm 1 is upper bounded as 1 3maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 (6) Proof. Let define the pesudo regret as b RT = PT t=1 x xt, θ . Denote m M be the subspace contains θ bθt1. We start by simple regret decomposition as follows. t=1 θ , x xt = t=1 θ , x xt + t=t1+1 θ , x xt D θ bθt1, x xt E + D bθt1, x xt E D θ bθt1, x xt E D θ bθt1, Πm(x xt) E h As θ bθt1 m, i θ bθt1 2 Πm(x xt) 2 Kx θ bθt1 2 ; h Since Πm(x xt) 2 2 p Now, we invoke Lemma 12. Let E is the event in that the exploration error is bounded as in Lemma 12, then there is an absolute constant c1 > 0 such that, RT Rmaxt1 + E Kx θ bθt1 2 + TPr(E)Rmax Rmaxt1 + c1 p σ2d0 log(d/δ) Cmin(X)t1 T + TδRmax. Let δ = 1/T, and t1 = R 2 3 maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 , one has that 1 3maxσ 2 3 C 1 1 3 0 T 2 3 (log(d T)) 1 3 (43) C Improved Regret Bound C.1 Improve Regret Bound with Well-Separated Partitions Theorem 18. Suppose the Assumptions 5, 6, 17 hold. Let t2 = Ω σ2K2 xd0 log(d T ) C2 min(X)ε2 0 . Then, Algorithm 2 returns bm θ with probability at least 1 1/T, and its regret is upper bounded as RT = O Rmaxσ2K2 xd0 log(d T) C2 min(X)ε2 0 + σd0 p T log(Kx T) . (12) Proof. Define the event E as E = n Y ΠSc m(Y ) 2 2 Y ΠSm(Y ) 2 2 | θ bm, θ / m o , (44) The event E is equivalent as Y ΠSc m(Y ) 2 2 Y ΠSm(Y ) 2 2 f + η b f b m 2 2 f + η b fm 2 f b f b m 2 2 + 2 D η, f b f b m E f b fm 2 2 + 2 D η, f b fm E . First, we upper the LHS of (45) with high probability. f f ΠSc m(η) 2 2 + 2 η, ΠSc m(η) = ΠSc m(η) 2 2 2 ΠSc m(η) 2 2 0 [as θ bm] (46) with probability at least 1 δ. The above inequality uses the union bound for all m M. Second, we lower bound the RHS of (45), f b fm 2 2 + 2 D η, f b fm E , with high probability. Let Sm = Sm L f , then D η, f b fm E = D ΠSm(η), f b fm E . Therefore, we have that, with probability at least 1 δ/3. f b fm 2 2 + 2 D ΠSm(η), f b fm E f b fm 2 2 2 ΠSm(η) 2 2 4 σ2(d0 + 1) + σ2(log(3Mδ 1)) 2 5 σ2d0 + σ2(log(3Mδ 1)) , (47) as dm d0. Also, we have that, with probability at least 1 δ/3 f b fm 2 2 = f ΠSm(f ) ΠSm(η) 2 2 f ΠSm(f ) 2 2 Where the inequalities holds because ΠSm(f ) is the projection of f to Sm. Therefore, we have that f b fm 2 2 + 2 D η, f b fm E 1 2 f ΠSm(f ) 2 2 5 σ2d0 + σ2(log(Mδ 1)) . (48) Now, by choosing t2 = Ω C 2 min(X)K2 x log(Mδ 1) , by Proposition 16, we have that with probability at least 1 δ/3, f ΠSm(f ) 2 2 = X(θ ΠSm(θ )) 2 2 t2 2Cmin(X) θ ΠSm(θ ) 2 2 t2ε2 0 4Cmin(X) (49) Therefore, the RHS of (45) is lower bounded as follows f b fm 2 2 + 2 D η, f b fm E t2ε2 0 8Cmin(X) 5 σ2d0 + σ2(log(3Mδ 1)) . (50) Therefore, the sufficient condition for event E holds with probability at least 1 δ is that t2ε2 0 8Cmin(X) 5 σ2d0 + σ2(log(Mδ 1)) 0; (51) Note that as M = O(d0 log(d)), for E holds with probability at least 1 δ, it suffice to choose t2 = Ω σ2K2 xd0 log(dδ 1) C2 min(X)ε2 0 The regret upper bound is an immediate consequence of the fact that Algorithm 2 return the true subspace ˆm θ , combined with the regret bound of OFUL in the exploitation phase and δ = 1/T. C.2 Adapting to Separating Constant ε0 As stated in Theorem 18, if one knows in advance that the separating constant ε0 T 1/4, then using Algorithm 2 leads to T regret. The reason is that the learner can learn the true subspace m after T steps with no mis-specification errors. However, without knowing ε0 T 1/4 a priori, one cannot guarantee to recover the true subspace. Hence, naively using Algorithm 2, which is not aware of potential mis-specification errors, leads to linear regret. On the other hand, using Algorithm 1 can achieve regret T 2/3 in the worst case (without the knowledge of ε0 T 1/4). A question arises: does there exist an algorithm that, without the knowledge of ε0, can achieve regret T whenever ε0 T 1/4, but guarantee the worst-case regret as T 2/3? We note that the role of ε0 is similar to the minimum signal in sparsity, and it is somewhat surprising that the question of adapting to unknown minimum signal has not been resolved in the literature of sparse linear bandits. Towards answering the question, we propose a simple method using adaptation to misspecified error in linear bandit [19], which has a T regret whenever the separating constant is large, and enjoys a worst-case regret guarantee of slightly worse T 3/4 regret. The algorithm described in 4 is a direct application of the algorithm proposed in [19], designed for adapting to misspecification errors in linear bandits. Particularly, the algorithm in [19] can adapt to unknown misspecification errors and achieve a regret bound of O(d0 T + ϵmis T), where ϵmis is the misspecification error. At a high level, our algorithm exploring T rounds using exploratory distribution, which ensures that the misspecification error ϵmis of the chosen subspace ˆm is at most T 1/4. Therefore, we can run multiple linear bandit algorithms using different levels of misspecification error. Particularly, we use a collection of K = log(T) base algorithms, where a base algorithm k [K] is a linear bandit algorithm with misspecified level εk = 2 k. Note that the base algorithm K has the same order of regret T as a well-specified model. Therefore, in exploitation phase, one can guarantee that in the case of well-separated partitions where ϵmis = 0, the algorithm can achieve a regret of d0 T, while in the general case, the regret caused by misspecification error is at most T 3/4 d0. Algorithm 3 Adaptive algorithm 1: Input T, ν, t3. 2: for t = 1, , t3 do 3: Independently pull arm xt according to ν and receive a reward yt. 4: end for 5: X [x1, ..., xt1] , Y [yt]t [t1]. 6: Compute bm as (4). 7: Let K = log(T 1/4) , E = εk := 2 k, k [K] . 8: for t = t2 + 1 to T do 9: Corralling K base misspecified linear bandit algorithms Square CB.Lin+(εk) [19] on bm. 10: end for Corollary 23. Suppose the Assumptions 5, 6 hold. Then, there exists an algorithm which achieves regret bound as follows: (i) [Well-separated partitions] If ε0 T 1/4, then RT = O(d0 (ii) [Non-well-separated partitions] If ε0 < T 1/4, then RT = O(d0 T + T 3 4 d0). Proof sketch. Denote ϵmis = θ Π b m(θ ) 2 as the misspecification error. With t3 = Ω σ2d0 log(dδ 1) We can guarantee that, with probability at least 1 δ, if: (i) If ε0 > T 1/4, then by Theorem 18, θ bm, that is, ϵmis = 0; (ii) If ε0 > T 1/4, then by Lemma 12, we can bound ϵmis T 1/4. The regret of adaptive algorithm in [19] is of the form O(d0 T +ϵmis T d0). Let δ = 1/T. Consider case (i) where ϵmis = 0, we have RT = O(d0 T). Consider case (ii), where ϵmis = T 1/4, we have RT = O(d0 T + T 3/4 d0). The result in Corollary 23 is still sub-optimal in the worst case, as it can only achieve O(T 3/4) regret bound instead of O(T 2/3). We conjecture that new techniques are required to achieve order-optimal regret in both cases, and will continue to investigate this question in future works. D On Collections of Partitions with Subexponential-Size D.1 Important Classes of Partitions with Subexponential-Size In this section, we discuss several important classes of partitions which satisfy Assumption 5. Pattern-avoidance partitions is arguable the most important class of studied partition [33], in which, non-crossing partition is one the most studied. Definition 24 (Non-Crossing Partition). Let [d] admits a cylic order as 1 < 2 < ... < d, and d < 1. A non-crossing partition of [d] is a partition such that for if i, j in one block and p, q in one block, then they are not arranged in the order i < p < j < q. Similarly, we denote NCd, NCd,k, NCd, k as the set of all non-crossing partition of [d], the set of all partition of [d] with k classes, and the set of all partition of [d] with at most k classes. We have the following fact ([33] section 3.2). |NCd| = 1 d + 1 , |NCd,k| = 1 which is the Catalan number and the Narayana number. Note that, Therefore, non-crossing partition satisfied the cardinality restriction as Assumption 5, that is, NCd, d0 Qd, d0. Another important class of pattern-avoidence partitions is nonnesting partition [14]. Definition 25 (Non-Nesting Partition). Let [d] admits a cylic order as 1 < 2 < ... < d, and d < 1. A non-crossing partition of [d] is a partition such that for if i, j in one block and p, q in one block, then they are not arranged in the order i < p < q < j. Similarly, we denote NN d, NN d,k, NN d, k as the set of all non-crossing partition of [d], the set of all partition of [d] with k classes, and the set of all partition of [d] with at most k classes. There is bijections between class of NN d and NCd, non-nesting partitions also satisfy the sub-exponential constraint (Assumption 5). One special case of both non-crossing partitions and non-nesting partitions is interval partition, which has identical structure as sparsity. Definition 26 (Interval Partition). A set partition of [d] is an interval partition or partition of interval if its parts are interval. We denote Id as the collection of all interval partition of d, we have that Id NCd Pd, and Id NN d Pd. Remark 27. Id admits a Boolean lattice of order 2d 1, making it equivalent to the sparsity structure in d 1 dimensions. Specifically, consider the set of entries of parameters φ Rd with a linear order, that is, φ1 < φ2 < < φd. Then define the variable θ Rd 1 such that θi = (φi+1 φi). Each interval partition on the entries of φ will determine a unique sparse pattern of θ. In other words, symmetric linear bandit is strictly harder than sparse bandit and inherits all the computational complexity challenges of sparse linear bandit, including the NP-hardness of computational complexity. Inspired by the literature on sparse linear regression, where one can relax solving exact sparse linear regression by using norm-1 minimization, also known as LASSO methods, we ask whether there is a convex relaxation for the case of non-crossing partitions or pattern-avoidance partitions in general. D.2 Practical Examples of Partitions with Subexponential-Size General hidden symmetries. Examples of hidden symmetry in reinforcement learning tasks can be found in robotic control [32, 3], where robot is initially designed symmetrical, but part of symmetry is destroyed by mechanical imperfection. Further examples of hidden symmetry can be also found in the literature on multi-agent reinforcement learning with a large number of agents. To avoid the curse of dimensionality, researchers often rely on the assumption of the existence of homogeneous agents [13, 34]. In the extreme case where all agents are homogeneous, such as in mean-field games, sample complexity becomes independent of the number of agents [13]. However, in practice, agents can be clustered into different types [34], and this information may not be known in advance to the learner (here symmetry occurs between different agents from the same type). Non-crossing partitions. Sub-exponential size naturally appears when there is a hierarchical structure on the set [d], and the partitioning needs to respect this hierarchical structure. Particularly, let T(d, d0) be the set of ordered trees with (d + 1) nodes and d0 internal nodes (i.e., nodes that are not the leaves). A partition that respects an ordered tree groups the children of the same node into a single equivalence class (for example, see Figure 1). It is shown in [15] that the cardinality of the set of partitions that respect ordered trees in T(d, d0) is sub-exponential. More precisely, it is O(d2d0). Furthermore, there is a natural bijection between partitions that respect ordered trees in T(d, d0) and the set of non-crossing partitions NCd,d0 [15]. Recall the subcontractor example in the introduction. Here, after the company hires subcontractors {1, 4, 6} to do the job, these subcontractors further break down the tasks into smaller subtasks and hire additional subcontractors {2, 3}, {5}, and {7, 8, 9}, respectively, to execute the subtasks. Non-nesting partitions. Besides non-crossing partitions, another sub-exponential-size class of partitions with practical relevance is non-nesting partitions. Consider the resource allocation task where there are d upcoming tasks and d0 machines. The job of the designer is to allocate these tasks to each machine. Now, assume each task will appear in time t1 < t2 < < td, but the exact time (the value of ti) is unknown to the designer. Moreover, the cost of machine k [d0], given a subset of tasks Ak (ordered according to execution time), is ck = tmax(Ak) tmin(Ak). The goal of the designer is to minimize the maximum cost of all machines: min max k [d0] ck. To achieve this, the designer should avoid nesting allocations (i.e., searching among non-nesting partitions). In particular, assuming that if tasks at times ti and tj are assigned to machine k, (where ti < tj), and tasks at times tp and tq are assigned to machine k (where tp < tq), then it should not be the case that ti < tp < tq < tj. This is because the cost of machine k would be significantly higher than that of machine k , and the cost could be reduced by swapping task tq for machine k with task tj for machine k . D.3 Efficient Greedy Algorithm for Specific Classes of Partitions The model selection procedure in the exploration phase of Algorithms 1 and 2 requires finding the best subspace in the pool m M with respect to least square errors. In the worst case, the algorithm needs to solve M linear regression approaches. While the exact computation of the best subspace ˆm as in (4) is an NP-hard problem in general (since it contains interval partition as a subclass), we argue that an greedy algorithm can find the ground-truth subspace m in O(nd5) time complexity, given sufficient large number of samples. The pseudo code of the greedy algorithm is given in Figure 4. Given that the set of partitions Qd is equipped with a lattice structure, in which the finest partition is (1|2| . . . |d) and the coarsest partition is (1, 2, . . . , d). The algorithm starts with the finest partition ˆπ = (1|2| . . . |d). In each iteration, the algorithm finds the finest coarsening of the current partition ˆπ. In graph-theoretic terms, it finds all neighbors of ˆπ in the lattice that are coarsenings of ˆπ. The operator for finding the finest coarsening of ˆπ is denoted as Coarsen(ˆπ), and it returns a collection of the finest coarsened partitions. Next, the algorithm finds the partition that minimizes the prediction error Y ΠSm(Y ) 2 2 among the current coarsening collection. At the end of each iteration, as the number of classes in ˆπ is reduced by one, the dimension variable is also reduced by 1. The while loop stops when the dimension equals d0. Since the algorithm only optimizes locally within the current coarsening collection, it exhibits a behavior similar to a greedy algorithm. We note that for non-crossing and non-nesting partitions, the cardinality of the coarsening collection at any level of the lattice is at most d2. Therefore, assuming that creating a finest coarsening partition take O(d) operator, and solving least square takes O(nd2), the algorithms time complexity is O(nd5). Algorithm 4 Greedily Search within Lattice 1: Input: Compact representation of M, design matrix X, reward vector Y . 2: Initialise ˆπ = (1|2|3|4...|d), dimension = d. 3: while dimension > d0 do 4: Collection = Coarsen(ˆπ). 5: ˆm = arg minm H(Collection) Y ΠSm(Y ) 2 2. 6: ˆπ = H 1( ˆm). 7: dimension = dimension 1. 8: end while 9: ˆθ = arg minθ ˆm Y Xθ 2 2. 10: Return ˆm, ˆθ. E Experiment details We conduct simulations where the entries of θ satisfy non-crossing partition constraints. The set of arms X is d Sd 1, σ = 0.1, and (d, d0) {(40, 4), (80, 10), (100, 15)}. We let exploratory distribution ν be the uniform distribution on the unit sphere. The ground-truth partition πG and θ are randomized before each simulation. To run ESTC-Lasso algorithm [23], we introduce an auxiliary sparse vector φ corresponding to θ , whose entries are defined as φi = θi+1 θi, and φd = θd. We apply Lasso regression for φ , get the estimate ˆφ, then convert back to ˆθ using the map that transforms sparse vector to interval-partition vector (inversion of the map we defined above). Regarding implementing our algorithm, we use greedy Algorithm 4 as introduced in Appendix D.3, to solve the optimisation in equation (3), (4), and its complexity is O(t1d5). It is shown in the simulation result (Figure 2, 3, 4), the greedy algorithm achieves small risk error and consequently leads to small regret. Code is available at: https://github.com/Nam Tran Kek L/Symmetric-Linear-Bandit-with-Hidden-Symmetry. git. Figure 3: Regret of EMC (Algorithm 1) and of ESTC proposed in [23], in cases of sparsity, noncrossing partitions, and non-nesting partitions, with d = 40, d0 = 4. Figure 4: Regret of EMC (Algorithm 1) and of ESTC proposed in [23], in cases of sparsity, noncrossing partitions, and non-nesting partitions, with d = 80, d0 = 10. F Extended Related Work Sparse linear bandits. As we will explain in Section 4.2, sparsity is equivalent to a subset symmetry structures, and thus, can be seen as a special case of our setting. As such, we first review the literature of sparsity. Sparse linear bandits were first investigated in [2], where the authors achieve a regret of O( ds T), with O disregarding the logarithmic factor, and s representing the sparsity level, and T is the time horizon. Without additional assumptions on the arm set and feature distribution, the lower bound for regret in the sparsity case is Ω( ds T) [27]. Consequently, the contextual setting has recently gained popularity in the sparsity literature, where additional assumptions are made regarding the context distribution and set of arms. With this assumption, it can be shown that one can achieve regret of the form O(τs T), where τ is a problem-dependent constant that may have a complex form and varies from paper to paper [26, 36]. Apart from the contextual assumption, to avoid polynomial dependence on T in the regret bound, assumptions are required for the set of arms [28, 11, 23]. Recently, [23] offers a unified perspective on the assumption regarding the set of arms by assuming the existence of an exploratory distribution on the set of arms. With this assumption, the authors propose an Explore then Commit style strategy that achieves O(s 2 3 T 2 3 ), nearly matching the lower bound Ω(s 2 3 T 2 3 ) in the poor data regime [24]. As the sparsity structure can be reduced to a subset of the symmetry structure, all the lower bounds for sparse problems apply to (unknown) symmetric problems. Model selection. Our problem is also closely related to model selection in linear bandits, as the learner can collect potential candidates for the unknown symmetry. Bandit model selection involves the problem where there is a collection of M base algorithms (with unknown performance guarantees) and a master algorithm, aiming to perform as well as the best base algorithm. The majority of the literature assumes the black-box collection of models M base algorithms and employs a variant of online mirror descent to select the recommendations of the base agent [4, 37, 38]. Due to the black-box nature, the regret guarantee bound depends on poly(M). There is a growing literature on model selection in stochastic linear bandits, where there is a collection of M features, and linear bandits running with these features serve as base algorithms. By exploiting the fact that the data can be shared across all the base algorithms, the dependence of regret in terms of the number of models can be reduced to log(M). In particular, [25] propose a method that concatenates all M features of dimension d into one feature of dimension Md, and then runs a group-Lasso bandit algorithm on top of this concatenated feature space, using the Lasso estimation as a aggregation of models. Their algorithm achieves a regret bound of O(T 3 4 p log(M)) under the assumption that the Euclidean norm of the concatenated feature is bounded by a constant. However, in our case, the Euclidean norm of concatenated feature can be as large as M, which leads to M multiplicative factor in regret. Besides, [35] uses the online aggregation oracle approach, and able to obtain regret as O( p Kd T log(M)), where K is the number of arms. In contrast, we use different algorithmic mechanism than aggregation of models. In particular, we explicitly exploiting the structure of the model class as a collection of subspaces and invoking results from Gaussian model selection [21] and dimension reduction on the union of subspaces [8]. With this technique, we are able to achieve O(T 2 3 log(M)), which is rate-optimal in the data-poor regime, has logarithmic dependence on M without strong assumptions on the norm of concatenated features, and is independent of the number of arms K. A special case of feature selection where one can achieve a very tight regret compared to the best model is the nested feature class [18, 20]. In particular, in the nested feature class where dimensions range from {1, . . . , d}, and dm < d represents the realizable feature of the smallest dimension, the regret bound can be O( p Tdm ) as shown in [20]. While the regret bound nearly matches the regret of the best model in the nested feature class, the assumption on nested features cannot be applied in our setting. Symmetry in online learning. The notion of symmetry in Markov Decision Making dates back to works such as [22, 40]. Generally, the reward function and probability transition are preserved under an action of a group on the state-action space. Exploiting known symmetry has been shown to help achieve better performance empirically [43, 42] or tighter regret bounds theoretically [41]. However, all these works requires knowledge of symmetry group, while setting consider unknown subgroup which may be considerably harder. Unknown symmetry on the context or state space has been studied by few authors, with the term context-lumpable bandit [29], meaning that the set of contexts can be partitioned into classes of similar contexts. It is important to note that the symmetry group acts differently on the context space and the action space. As we shall explain in detail in Section 3, while one can achieve a reduction in terms of regret in the case of unknown symmetry acting on context spaces [29], this is not the case when the symmetry group acts on the action space. Particularly, we show that without any extra information on the partition structure of similar classes of arms, no algorithm can achieve any reduction in terms of regret. The work closest to ours is [39], where the authors consider the setting of a K-armed bandit, where the set of arms can be partitioned into groups with similar mean rewards, such that each group has at least q > 2 arms. With the constrained partition, the instance-dependent regret bounds are shown asymptotically to be of order O ((K/q) log T). Comparing to [39], we study the setting of stochastic linear bandits with similar arms, in which the (unknown) symmetry and linearity structure may intertwine, making the problem more sophisticated. We also impose different constraints on the way one partitions the set of arms, which is more natural in the setting of linear bandits with infinite arms. As a result, we argue that the technique used in [39] cannot be applied in our setting. To clarify this, let us first review the algorithmic technique from [39]: The algorithm in [39] assumes there is an equivalence among the parameters θ, and that the set of arms X is a simplex. At each round t, given an estimation ˆθt, the algorithm maintains a sorted list of indices in [d] that follows the ascending order of the magnitude of ˆθi. The algorithm then uses the sorted list (ˆθi)i [d] to choose arm x accordingly. The key assumption here is that, since the set of arms X is a simplex, we can estimate each θi independently. This implies that the list (ˆθi)i [d] should respect the true order of the list (θi)i [d] when there are a sufficiently large number of samples. Unfortunately, this is typically not the case in linear bandits where X has a more general shape. In linear bandits, there can be correlations between the estimates ˆθi and ˆθj for any i, j [d]. Hence, one should not expect that (ˆθi)i [d] will maintain the same order as (θi)i [d]. In other words, the correlations among the estimates {ˆθ1, . . . , ˆθd} may destroy the original order in {θ1, . . . , θd}. In fact, we can only guarantee the risk error of estimation θ, i.e., ˆθ θ is small, but not necessarily the order of the indices in θ. Therefore, the technique used in [39] cannot be directly applied to linear bandits in its current form. Neur IPS Paper Checklist Question: Do the main claims made in the abstract and introduction accurately reflect the paper s contributions and scope? Answer: [Yes] Justification: Our claims made in the abstract and introduction reflect the paper s contributions and scope. 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. 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Guidelines: The answer NA means that paper does not include experiments requiring code. Please see the Neur IPS code and data submission guidelines (https://nips.cc/ public/guides/Code Submission Policy) 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 Neur IPS code and data submission guidelines (https: //nips.cc/public/guides/Code Submission Policy) 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: We described our simulations and provided a link to the code available online. 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: We described our simulations and statistical significance of the experiments. 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: We discussed the time complexity of our algorithm. 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 Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines? Answer: [Yes] Justification: We have reviewed the Neur IPS Code of Ethics. Guidelines: The answer NA means that the authors have not reviewed the Neur IPS 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: [NA] Justification: This paper is primarily theoretical and may have limited direct impact or implications. 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: This paper poses no such risks. 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: [NA] Justification: We does not use existing assets. 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: This paper does not release new assets 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: This paper does not involve crowdsourcing nor research with human subjects. 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 Neur IPS 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: This paper does not involve crowdsourcing nor research with human subjects. 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 Neur IPS 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.