SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We want to run Algorithm 1 as a subroutine within the main algorithm to estimate the gradient. However, in order to run Algorithm 1, we need to be able to query the oracle within the set  $\mathbb{B}_{\delta}^{\mathrm{aff}(\mathcal{K})}(\mathbf{x})$ . Since the oracle can only be queried at points within the feasible set, we need to restrict our attention to a set  $\kappa_{\delta}$  such that  $\mathbb{B}_{\delta}^{\mathrm{aff}(\mathcal{K})}(\mathcal{K}_{\delta}) \subseteq \mathcal{K}$ . On the other hand, we want the optimal point of  $F$  within  $\kappa_{\delta}$  to be close to the optimal point of  $F$  within  $\kappa$ . One way to ensure
Algorithm 1 Black Box Gradient Estimate (BBGE)
1: Input: Point  $\mathbf{z}$ , sampling radius  $\delta$ , constraint linear space  $\mathcal{L}_0$ ,  $k = \dim(\mathcal{L}_0)$ , batch size  $B$
2: Sample  $\mathbf{u}_1, \dots, \mathbf{u}_B$  i.i.d. from  $S^{d-1} \cap \mathcal{L}_0$
3: For  $\dot{\boldsymbol{i}} = 1$  to  $B$ , let  $\mathbf{y}_i^+\gets \mathbf{z} + \delta \mathbf{u}_i, y_i^- \gets \mathbf{z} - \delta \mathbf{u}_i$ , and evaluate  $\hat{F} (\mathbf{y}_i^+)$ ,  $\hat{F} (\mathbf{y}_i^-)$
4:  $\mathbf{g} \gets \frac{1}{B} \sum_{i=1}^{B} \frac{k}{2\delta} \left[ \hat{F}(\mathbf{y}_i^+) - \hat{F}(\mathbf{y}_i^-) \right] \mathbf{u}_i$
5: Output g
that is to have  $\mathcal{K}_{\delta}$  not be too small. More formally, we want that  $\mathbb{B}_{\delta'}^{\mathrm{aff}(\mathcal{K})}(\mathcal{K}_{\delta}) \supseteq \mathcal{K}$ , for some value of  $\delta' \geq \delta$  that is not too large. The constraint boundary could have a complex geometry, and simply maintaining a  $\delta$  sized margin away from the boundary can result in big gaps between the boundary of  $\mathcal{K}$  and  $\mathcal{K}_{\delta}$ . For example, in two dimensions, if  $\mathcal{K}$  is polyhedral and has an acute angle, maintaining a  $\delta$  margin away from both edges adjacent to the acute angle means the closest point in the  $\mathcal{K}_{\delta}$  to the corner may be much more than  $\delta$ . For this construction, we choose a  $\mathbf{c} \in \operatorname{relint}(\mathcal{K})$  and a real number  $r > 0$  such that  $\mathbb{B}_r^{\mathrm{aff}(\mathcal{K})}(\mathbf{c}) \subseteq \mathcal{K}$ . For any  $\delta < r$ , we define
$$
\mathcal {K} _ {\delta} ^ {\mathbf {c}, r} := \left(1 - \frac {\delta}{r}\right) \mathcal {K} + \frac {\delta}{r} \mathbf {c}. \tag {7}
$$
Clearly if  $\mathcal{K}$  is downward-closed, then so is  $\mathcal{K}_{\delta}^{\mathbf{c},r}$ . Lemma 7 shows that for any such choice of  $\mathbf{c}$  and  $r > 0$ , we have  $\frac{\delta'}{\delta} \leq \frac{D}{r}$ . See Appendix G for more details about the choice of  $\mathbf{c}$  and  $r$ . We will drop the superscripts in the rest of the paper when there is no ambiguity.
Remark 2. This construction is similar to the one carried out in [31] which was for  $d$ -dimensional downward-closed sets. Here we impose no restrictions on  $\mathcal{K}$  beyond Assumption 1. A simpler construction of shrunken constraint set was proposed in [8]. However, as we discuss in Appendix D, they require to be able to query outside of the constraint set.
3.4 Generalized DR-Submodular Frank-Wolfe
The pseudocode of our proposed offline algorithm, Generalized DR-Submodular Frank-Wolfe, is shown in Algorithm 2. At a high-level, it follows the basic template of Frank-Wolfe type methods, where over the course of a pre-specified number of iterations, the gradient (or a surrogate thereof) is calculated, an optimization sub-routine with a linear objective is solved to find a feasible point whose difference (with respect to the current solution) has the largest inner product with respect to the gradient, and then the current solution is updated to move in the direction of that feasible point.
However, there are a number of important modifications to handle properties of the objective function, constraint set, and ora
Algorithm 2 Generalized DR-Submodular Frank-Wolfe
1: Input: Constraint set  $\mathcal{K}$ , iteration limit  $N \geq 4$ , sampling radius  $\delta$ , gradient step-size  $\{\rho_n\}_{n=1}^N$
2: Construct  $\kappa_{\delta}$
3: Pick any  $\mathbf{z}_1\in \mathrm{argmin}_{\mathbf{z}\in \mathcal{K}_{\delta}}\| \mathbf{z}\|_{\infty}$
4:  $\bar{\mathbf{g}}_0\gets \mathbf{0}$
5: for  $n = 1$  to  $N$  do
6:  $\mathbf{g}_n\gets$  estimate-grad  $(\mathbf{z}_n,\delta ,\mathcal{L}_0 = \mathrm{aff}(\mathcal{K}) - \mathbf{z}_1)$
7:  $\bar{\mathbf{g}}_n\gets (1 - \rho_n)\bar{\mathbf{g}}_{n - 1} + \rho_n\mathbf{g}_n$
8:  $\mathbf{v}_n\gets$  optimal-direction  $(\bar{\mathbf{g}}_n,\mathbf{z}_n)$
9:  $\mathbf{z}_{n + 1}\gets \mathrm{update}(\mathbf{z}_n,\mathbf{v}_n,\varepsilon)$
10: end for
11: Output  $\mathbf{z}_{N + 1}$
cle type. For the oracle type, for instance, standard Frank-Wolfe methods assume access to a deterministic gradient oracle. Frank-Wolfe methods are known to be sensitive to errors in estimates of the gradient (e.g., see [16]). Thus, when only a stochastic gradient oracle or even more challenging, only a stochastic value oracle is available, the gradient estimators must be carefully designed to balance query complexity on the one hand and output error on the other. The Black Box Gradient Estimate (BBGE) sub-routine, presented in Algorithm 1, utilizes spherical sampling to produce an unbiased gradient estimate. This estimate is then combined with past estimates using momentum, as seen in [23], to control and reduce variance.
Our algorithm design is influenced by state-of-the-art methods that have been developed for specific settings. One of the most closely related works is [8], which also dealt with using value oracle access for optimizing monotone functions. They used momentum and spherical sampling techniques that are similar to the ones we used in our Algorithm 1. However, we modified the sampling procedure and the solution update step. In their work, [8] also considered a shrunken feasible region to avoid sampling close to the boundary. However, they assumed that the value oracle could be queried outside the feasible set (see Appendix D for details).
In Algorithm 3, we consider the following cases for the function and the feasible set.
(A) If  $F$  is monotone DR-submodular and  $\mathbf{0} \in \mathcal{K}$ , we choose
$$
\operatorname {o p t i m a l - d i r e c t i o n} \left(\bar {\mathbf {g}} _ {n}, \mathbf {z} _ {n}\right) = \operatorname {a r g m a x} _ {\mathbf {v} \in \mathcal {K} _ {\delta} - \mathbf {z} _ {1}} \left\langle \mathbf {v}, \bar {\mathbf {g}} _ {n} \right\rangle , \text {u p d a t e} \left(\mathbf {z} _ {n}, \mathbf {v} _ {n}, \varepsilon\right) = \mathbf {z} _ {n} + \varepsilon \mathbf {v} _ {n},
$$
and  $\varepsilon = 1 / N$ . We start at a point near the origin and always move to points that are bigger with respect to the partial order on  $\mathbb{R}^d$ . In this case, since the function is monotone, the optimal direction is a maximal point with respect to the partial order. The choice of  $\varepsilon = 1 / N$  guarantees that after  $N$  steps, we arrive at a convex combination of points in the feasible set and therefore the final point is also in the feasible set. The fact that the origin is also in the feasible set shows that the intermediate points also belong to the feasible set.
(B) If  $F$  is non-monotone DR-submodular and  $\mathcal{K}$  is a downward closed set containing 0, we choose
$$
\text{optimal - direction} (\bar{\mathbf{g}}_{n},\mathbf{z}_{n}) = \operatorname *{argmax}_{\substack{\mathbf{v}\in \mathcal{K}_{\delta} - \mathbf{z}_{1}\\ \mathbf{v}\leq \mathbf{1} - \mathbf{z}_{n}}}\langle \mathbf{v},\bar{\mathbf{g}}_{n}\rangle ,\text{update} (\mathbf{z}_{n},\mathbf{v}_{n},\varepsilon) = \mathbf{z}_{n} + \varepsilon \mathbf{v}_{n},
$$
and  $\varepsilon = 1 / N$ . This case is similar to (A). However, since  $F$  is not monotone, we need to choose the optimal direction more conservatively.
(C) If  $F$  is monotone DR-submodular and  $\mathcal{K}$  is a general convex set, we choose
$$
\operatorname {o p t i m a l - d i r e c t i o n} \left(\overline {{\mathbf {g}}} _ {n}, \mathbf {z} _ {n}\right) = \operatorname * {a r g m a x} _ {\mathbf {v} \in \mathcal {K} _ {\delta}} \left\langle \mathbf {v}, \overline {{\mathbf {g}}} _ {n} \right\rangle , \operatorname {u p d a t e} \left(\mathbf {z} _ {n}, \mathbf {v} _ {n}, \varepsilon\right) = (1 - \varepsilon) \mathbf {z} _ {n} + \varepsilon \mathbf {v} _ {n},
$$
and  $\varepsilon = \log (N) / 2N$ . In this case, if we update like in cases (A) and (B), we do not have any guarantees of ending up in the feasible set, so we choose the update function to be a convex combination. Unlike (B), we do not need to limit ourselves in choosing the optimal direction and we simply choose  $\varepsilon$  to obtain the best approximation coefficient.
(D) If  $F$  is non-monotone DR-submodular and  $\mathcal{K}$  is a general convex set, we choose
$$
\text {o p t i m a l - d i r e c t i o n} (\bar {\mathbf {g}} _ {n}, \mathbf {z} _ {n}) = \operatorname {a r g m a x} _ {\mathbf {v} \in \mathcal {K} _ {\delta}} \left\langle \mathbf {v}, \bar {\mathbf {g}} _ {n} \right\rangle , \quad \text {u p d a t e} (\mathbf {z} _ {n}, \mathbf {v} _ {n}, \varepsilon) = (1 - \varepsilon) \mathbf {z} _ {n} + \varepsilon \mathbf {v} _ {n},
$$
and  $\varepsilon = \log (2) / N$ . This case is similar to (C) and we choose  $\varepsilon$  to obtain the best approximation coefficient.
The choice of subroutine estimate-grad and  $\rho_{n}$  depend on the oracle. If we have access to a gradient oracle  $\hat{G}$ , we set estimate-grad  $(\mathbf{z},\delta ,\mathcal{L}_0)$  to be the average of  $B$  evaluations of  $P_{\mathcal{L}_0}(\hat{G} (\mathbf{z}))$ . Otherwise, we run Algorithm 1 with input  $\mathbf{z}$ ,  $\delta$ ,  $\mathcal{L}_0$ . If we have access to a deterministic gradient oracle, then there is no need to use any momentum and we set  $\rho_{n} = 1$ . In other cases, we choose  $\rho_{n} = \frac{2}{(n + 3)^{2 / 3}}$ .
3.5 Approximation Guarantees for the Proposed Offline Algorithm
Theorem 1. Suppose Assumption 1 holds. Let  $N \geq 4$ ,  $B \geq 1$  and choose  $\mathbf{c} \in \mathcal{K}$  and  $r > 0$  according to Section 3.3. If we have access to a gradient oracle, we choose  $\delta = 0$ , otherwise we choose  $\delta \in (0, r/2)$ . Then the following results hold for the output  $\mathbf{z}_{N+1}$  of Algorithm 2.
(A) If  $F$  is monotone DR-submodular and  $\mathbf{0} \in \mathcal{K}$ , then
$$
(1 - e ^ {- 1}) F \left(\mathbf {z} ^ {*}\right) - \mathbb {E} \left[ F \left(\mathbf {z} _ {N + 1}\right) \right] \leq \frac {3 D Q ^ {1 / 2}}{N ^ {1 / 3}} + \frac {L D ^ {2}}{2 N} + \delta G \left(2 + \frac {\sqrt {d} + D}{r}\right). \tag {8}
$$
(B) If  $F$  is non-monotone DR-submodular and  $\mathcal{K}$  is a downward closed set containing 0, then
$$
e ^ {- 1} F \left(\mathbf {z} ^ {*}\right) - \mathbb {E} \left[ F \left(\mathbf {z} _ {N + 1}\right) \right] \leq \frac {3 D Q ^ {1 / 2}}{N ^ {1 / 3}} + \frac {L D ^ {2}}{2 N} + \delta G \left(2 + \frac {\sqrt {d} + 2 D}{r}\right). \tag {9}
$$
(C) If  $F$  is monotone DR-submodular and  $\kappa$  is a general convex set, then
$$
\frac {1}{2} F \left(\mathbf {z} ^ {*}\right) - \mathbb {E} \left[ F \left(\mathbf {z} _ {N + 1}\right) \right] \leq \frac {3 D Q ^ {1 / 2} \log (N)}{2 N ^ {1 / 3}} + \frac {4 D G + L D ^ {2} \log (N) ^ {2}}{8 N} + \delta G \left(2 + \frac {D}{r}\right). \tag {10}
$$
(D) If  $F$  is non-monotone DR-submodular and  $\mathcal{K}$  is a general convex set, then
$$
\frac {1}{4} \left(1 - \| \mathbf {z} _ {1} \| _ {\infty}\right) F \left(\mathbf {z} ^ {*}\right) - \mathbb {E} \left[ F \left(\mathbf {z} _ {N + 1}\right) \right] \leq \frac {3 D Q ^ {1 / 2}}{N ^ {1 / 3}} + \frac {D G + 2 L D ^ {2}}{4 N} + \delta G \left(2 + \frac {D}{r}\right). \tag {11}
$$
In all these cases, we have
$$
Q = \left\{ \begin{array}{l l} 0 & \text {d e t . g r a d . o r a c l e ,} \\ \max  \{4 ^ {2 / 3} G ^ {2}, 6 L ^ {2} D ^ {2} + \frac {4 \sigma_ {1} ^ {2}}{B} \} & \text {s t o c h . g r a d . o r a c l e w i t h v a r i a n c e \sigma_ {1} ^ {2} > 0 ,} \\ \max  \{4 ^ {2 / 3} G ^ {2}, 6 L ^ {2} D ^ {2} + \frac {4 C k G ^ {2} + 2 k ^ {2} \sigma_ {0} ^ {2} / \delta^ {2}}{B} \} & \text {v a l u e o r a c l e w i t h v a r i a n c e \sigma_ {0} ^ {2} \geq 0 ,} \end{array} \right.
$$
$C$  is a constant,  $k = \dim(\mathcal{K})$ ,  $D = \mathrm{diam}(\mathcal{K})$ , and  $\mathbf{z}^*$  is the global maximizer of  $F$  on  $\mathcal{K}$ .
Theorem 1 characterizes the worst-case approximation ratio  $\alpha$  and additive error bounds for different properties of the function and feasible region, where the additive error bounds depend on selected parameters  $N$  for the number of iterations, batch size  $B$ , and sampling radius  $\delta$ .
The proof of Parts (A)-(D) is provided in Appendix I-L, respectively.
The proof of parts (A), (B) and (D), when we have access to an exact gradient oracle is similar to the proofs presented in [4, 3, 24], respectively. Part (C) is the first analysis of a Frank-Wolfe type algorithm over general convex sets when the oracle can only be queried within the feasible set. When we have access to a stochastic gradient oracle, directly using a gradient sample can result in arbitrary bad performance as shown in Appendix B of [16]. The momentum technique, first used in continuous submodular maximization in [23], is used when we have access to a stochastic gradient oracle. The control on the estimate of the gradient is deferred to Lemma 9. Since the momentum technique is robust to noise in the gradient, when we only have access to a value oracle, we can use Algorithm 1, similar to [8], to obtain an unbiased estimate of the gradient and complete the proof.
Theorem 2 converts those bounds to characterize the oracle complexity for a user-specified additive error tolerance  $\epsilon$  based on oracle properties (deterministic/stochastic gradient/value). The 16 combinations of the problem settings listed in Table 1 are enumerated by four cases (A)-(D) in Theorem 1 of function and feasible region properties (resulting in different approximation ratios) and the four cases 1-4 enumerated in Theorem 2 below of oracle properties. For the oracle properties, we consider the four cases as (Case 1): deterministic gradient oracle, (Case 2): stochastic gradient oracle, (Case 3): deterministic value oracle, and (Case 4): stochastic value oracle.
Theorem 2. The number of oracle calls for different oracles to achieve an  $\alpha$ -approximation error of smaller than  $\epsilon$  using Algorithm 1 is
$$
\text {C a s e 1 :} \tilde {O} (1 / \epsilon), \quad \text {C a s e s 2 , 3 :} \tilde {O} (1 / \epsilon^ {3}), \quad \text {C a s e 4 :} \tilde {O} (1 / \epsilon^ {5}).
$$
Moreover, in all of the cases above, if  $F$  is non-monotone or  $0 \in \mathcal{K}$ , we may replace  $\tilde{O}$  with  $O$ .
See Appendix M for proof.
4 Online DR-submodular optimization under bandit or semi-bandit feedback
In this section, we first describe the Black-box Explore-Then-Commit algorithm that uses the offline algorithm for exploration, and uses the solution of the offline algorithm for exploitation. This is followed by the regret analysis of the proposed algorithm. This is the first algorithm for stochastic continuous DR-submodular maximization under bandit feedback and obtains state-of-the-art for semi-bandit feedback.
4.1 Problem Setup
There are typically two settings considered in online optimization with bandit feedback. The first is the adversarial setting, where the environment chooses a sequence of functions  $F_{1}, \dots, F_{N}$  and in each iteration  $n$ , the agent chooses a point  $\mathbf{z}_{n}$  in the feasible set  $\mathcal{K}$ , observes  $F_{n}(z_{n})$  and receives the reward  $F_{n}(\mathbf{z}_{n})$ . The goal is to choose the sequence of actions that minimize the following notion of expected  $\alpha$ -regret.
$$
\mathcal {R} _ {\mathrm {a d v}} := \alpha \max  _ {\mathbf {z} \in \mathcal {K}} \sum_ {n = 1} ^ {N} F _ {n} (\mathbf {z}) - \mathbb {E} \left[ \sum_ {n = 1} ^ {N} F _ {n} \left(\mathbf {z} _ {n}\right) \right]. \tag {12}
$$
In other words, the agent's cumulative reward is being compared to  $\alpha$  times the reward of the best constant action in hindsight. Note that, in this case, the randomness is over the actions of the policy.
The second is the stochastic setting, where the environment chooses a function  $F: \mathcal{K} \to \mathbb{R}$  and a stochastic value oracle  $\hat{F}$ . In each iteration  $n$ , the agent chooses a point  $\mathbf{z}_n$  in the feasible set  $\mathcal{K}$ , receives the reward  $(\hat{F}(\mathbf{z}_n))_n$  by querying the oracle at  $z_n$  and observes this reward. Here the outer subscript  $n$  indicates that the result of querying the oracle at time  $n$ , since the oracle is stochastic. The goal is to choose the sequence of actions that minimize the following notion of expected  $\alpha$ -regret.
$$
\mathcal {R} _ {\mathrm {s t o c h}} := \alpha N \max  _ {\mathbf {z} \in \mathcal {K}} F (\mathbf {z}) - \mathbb {E} \left[ \sum_ {n = 1} ^ {N} \left(\hat {F} \left(\mathbf {z} _ {n}\right)\right) _ {n} \right] = \alpha N \max  _ {\mathbf {z} \in \mathcal {K}} F (\mathbf {z}) - \mathbb {E} \left[ \sum_ {n = 1} ^ {N} F \left(\mathbf {z} _ {n}\right) \right] \tag {13}
$$
Further, two feedback models are considered - bandit and semi-bandit feedback. In the bandit feedback setting, the agent only observes the value of the function  $F_{n}$  at the point  $\mathbf{z}_n$ . In the semi-bandit setting, the agent has access to a gradient oracle instead of a value oracle and observes  $\hat{G}(\mathbf{z}_n)$  at the point  $\mathbf{z}_n$ , where  $\hat{G}$  is an unbiased estimator of  $\nabla F$ .
In unstructured multi-armed bandit problems, any regret bound for the adversarial setup could be translated into bounds for the stochastic setup. However, having a non-trivial correlation between the actions of different arms complicates the relation between the stochastic and adversarial settings. Even in linear bandits, the relation between adversarial linear bandits and stochastic linear bandits is not trivial. (e.g. see Section 29 in [19]) While it is intuitively reasonable to assume that the optimal regret bounds for the stochastic case are better than that of the adversarial case, such a result is not yet proven for DR-submodular functions. Thus, while the cases of bandit feedback has been studied in the adversarial setup, the results do not reduce to stochastic setup. We also note that in the cases where there are adversarial setup results, this paper finds that the results in the stochastic setup achieve improved regret bounds (See Table 3 in Supplementary for the comparison).
4.2 Algorithm for DR-submodular maximization with Bandit Feedback
The proposed algorithm is described in Algorithm 3. In Algorithm 3, if there is semi-bandit feedback in the form of a stochastic gradient sample for each action  $\mathbf{z}_n$ , we run the offline algorithm (Algorithm 2) with parameters from the proof of case 2 of Theorem 2 for  $T_0 = \lceil T^{3/4} \rceil$  total queries. If only the stochastic reward for each action  $\mathbf{z}_n$  is available (bandit feedback), we run the offline algorithm (Algorithm 2) with parameters from the proof of case 4 of Theorem 2 for  $T_0 = \lceil T^{5/6} \rceil$  total queries. Then, for the remaining time (exploitation phase), we run the last action in the exploration phase.
Algorithm 3 DR-Submodular Explore-Then-Commit
1: Input: Horizon  $T$ , inner time horizon  $T_0$
2: Run Algorithm 2 for  $T_0$ , with according to parameters described in Theorem 2.
3: for remaining time do
4: Repeat the last action of Algorithm 2.
5: end for
4.3 Regret Analysis for DR-submodular maximization with Bandit Feedback
In this section, we provide the regret analysis for the proposed algorithm. We note that by Theorem 2, Algorithm 2 requires a sample complexity of  $\tilde{O}(1/\epsilon^5)$  with a stochastic value oracle for
offline problems (any of (A)-(D) in Theorem 1). Thus, the parameters and the results with bandit feedback are the same for all the four setups (A)-(D). Likewise, when a stochastic gradient oracle is available, Algorithm 2 requires a sample complexity of  $\tilde{O}(1/\epsilon^3)$ . Based on these sample complexities, the overall regret of online DR-submodular maximization problem is given as follows.
Theorem 3. For an online constrained DR-submodular maximization problem over a horizon  $T$ , where the expected reward function  $F$ , feasible region type  $\mathcal{K}$ , and approximation ratio  $\alpha$  correspond to any of the four cases (A)-(D) in Theorem 1, Algorithm 3 achieves  $\alpha$ -regret (13) that is upper-bounded as:
Semi-bandit Feedback (Case 2):  $\tilde{O}(T^{3/4})$ , Bandit Feedback (Case 4):  $\tilde{O}(T^{5/6})$ .
Moreover, in either type of feedback, if  $F$  is non-monotone or  $\mathbf{0} \in \mathcal{K}$ , we may replace  $\tilde{O}$  with  $O$ .
See Appendix N for the proof.
This work provides a novel and unified approach for maximizing continuous DR-submodular functions across various assumptions on function, constraint set, and oracle access types. The proposed Frank-Wolfe based algorithm improves upon existing results for nine out of the sixteen cases considered, and presents new results for offline DR-submodular maximization with only a value oracle. Moreover, this work presents the first regret analysis with bandit feedback for stochastic DR-submodular maximization, covering both monotone and non-monotone functions. These contributions significantly advance the field of DR-submodular optimization (with multiple applications) and open up new avenues for future research in this area.
Limitations: While the number of function evaluations in the different setups considered in the paper are state of the art, lower bounds have not been investigated.
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