SlowGuess/NeurIPS_2023_OCR huggingface dataset:
However, the choice of  $\rho_{t}$  must be restricted to only feasible distributions of posterior means, that is, all possible distributions over posterior means that can be induced by any advertising scheme given the prior  $\lambda$ . It is well known that the distribution over posterior means  $\rho$  is feasible if and only if it is the mean-preserving contraction of the prior [17, 6]. This condition can be equivalently written in terms of a Bayes-consistency condition [39] on the conditional means  $\rho(\cdot|\omega), \omega \in \Omega$ . For simplicity of exposition, we consider discrete quality space  $\Omega = \{\bar{\omega}_1, \dots, \bar{\omega}_m\} \subseteq [0,1]$ , where  $0 = \bar{\omega}_1 < \bar{\omega}_2 < \dots < \bar{\omega}_m = 1$  and  $m = |\Omega|$  is the cardinality of the quality space. Then, a distribution  $\rho$  over posterior means is feasible if one can construct a set of conditional distributions  $(\rho_i)_{i \in [m]}$  satisfying the following Bayes-consistency condition, and vice versa [39]:
$$
\frac {\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q) \bar {\omega} _ {i}}{\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q)} = q, \quad \forall q \in \operatorname {s u p p} (\rho). \tag {BC}
$$
Throughout this paper, we use the collection of distributions  $(\rho_{i})_{i\in [m]}$  satisfying (BC) condition as a convenient way to construct feasible distributions over posterior means:  $\rho (q) = \sum_{i}\rho_{i}(q)\lambda_{i}$ .
With the above observations, we can without loss of generality assume that seller's advertising strategy is to directly choose a distribution  $\rho_{t}$  over the posterior means that satisfies (BC), without considering the design of the underlying signaling scheme  $\{\phi (\sigma |\omega),\Sigma \}$ .
We summarize the new equivalent game timeline as follows: at  $t \in [T]$ , (1) the seller commits to a price  $p_t \in [0, U]$  and an advertising strategy  $\rho_t = (\rho_{i,t})_{i \in [m]}$  satisfying (BC); (2) a buyer  $t$  with private type  $\theta_t \sim F$  arrives; (3) a product with quality  $\omega_t \sim \lambda$  is realized; a posterior mean  $q_t \sim \rho_t$  is realized; (4) the buyer observes the posterior mean  $q_t$ ; the buyer purchases the product  $(a_t = 1)$  to generate revenue  $p_t$  if only if  $v(\theta_t, q_t) \geq p_t$ . Note that the seller knows the form of the buyer's valuation function  $v$ . Moreover, the seller observes the realized product quality  $\omega_t$ , the realized posterior mean  $q_t$ , and the buyer's purchase decision  $a_t$ , but does not know type CDF  $F$  (i.e., the demand function  $D$ ) and the realized buyer type  $\theta_t$ .
Revenue and regret. Given the new formulation, we can also rewrite the revenue and regret in terms of the choices of  $\rho_t, t = 1, \dots, T$ . We define the following function  $\kappa(p, q)$ , which we refer to as the critical type for a given price  $p$  and posterior mean  $q$ .
Definition 2.1 (Critical type). For any  $p \in [0, U]$  and  $q \in [0, 1]$ , define function  $\kappa(\cdot, \cdot)$  as  $\kappa(p, q) \triangleq \min\{\theta \in \Theta : v(\theta, q) \geq p\}$ .
Now under Assumption [1b], due to the monotonicity of the valuation function in buyer's type, we have that given any  $p, q, \theta, \mathbf{1}[v(\theta, q) \geq p] = \mathbf{1}[\theta \geq \kappa(p, q)]$ . Therefore, the buyer  $t$  will purchase the product if and only if  $\theta_t \geq \kappa(p_t, q_t)$ .
Therefore, given the price, advertising  $p_t = p$ ,  $\rho_t = \rho$  and prior distribution  $\lambda$ , the expected revenue in any round  $t$  is given by
$$
\operatorname {R e v} (p, \rho) = \mathbb {E} _ {\omega \sim \lambda , \theta \sim F, q \sim \rho} [ p \cdot \mathbf {1} [ \theta \geq \kappa (p, q) ] ] = p \sum_ {i} \lambda_ {i} \int_ {0} ^ {1} \rho_ {i} (q) D (\kappa (p, q)) d q \tag {1}
$$
Let the seller's online policy offer price  $p_t$  and advertising  $\rho_t$  at time  $t$ , where  $p_t, \rho_t$  can depend on the history of observations/events up to time  $t$ . Then expected regret defined in Section [1] can be equivalently written as
$$
\operatorname {R e g r e t} [ T ] = T \operatorname {R e v} \left(p ^ {*}, \rho^ {*}\right) - \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \operatorname {R e v} \left(p _ {t}, \rho_ {t}\right) \right].
$$
Here, the expectation is taken with respect to any randomness in the algorithm's choice of  $p_t, \rho_t$  and  $p^*, \rho^* = (\rho_i^*)_{i \in [m]}$  are defined as the best price and advertising strategy for a given  $F$  (and  $\kappa(\cdot, \cdot)$  which is determined by  $F$ ). Given the expression for  $\mathrm{Rev}(p, \rho)$  derived above, these can be characterized by the following optimization program
$$
p ^ {*}, \rho^ {*} \triangleq \operatorname * {a r g} _ {p, \rho} \operatorname {R e v} (p, \rho)
$$
$$
\text {s . t .} \quad \frac {\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q) \bar {\omega} _ {i}}{\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q)} = q, \quad q \in [ 0, 1 ]; \quad \rho_ {i} \in \Delta^ {[ 0, 1 ]}, \quad i \in [ m ]. \tag {P_{OPT}}
$$
where the first constraint is due to (BC).
Here, we slightly abuse the notation to redefine Rev as a function of price and  $\rho$ , instead of price and  $\phi$  defined earlier.
2.2 Algorithm design: challenges and ideas
Challenge: high-dimensional continuous decision space. In the last section, we obtained a considerable simplification of the problem by reducing the seller's advertising strategy in every round  $t$  to a distribution  $\rho_t \in \Delta^{[0,1]}$  over posterior means satisfying the (BC) condition. However, the decision space (a.k.a space of arms) still remains high dimensional and therefore a naive application of (uniform or adaptive) discretization-based bandit techniques, e.g. from Lipschitz bandit literature [44, 56], would not achieve the desired results.
Algorithm design idea: exploring over one-dimensional type space. Our algorithm uses a 'model-based approach' instead, where we use buyer purchase responses to develop (upper confidence bound) estimates of the demand model  $D(\theta) = 1 - F(\theta)$  on the points of a discretized type space  $S \subseteq \Theta$ . We then use these upper confidence bounds to construct a piecewise-constant demand function that is an upper confidence bound (UCB) for the demand function  $D$ . Then, in each round we solve for the optimal price and advertising strategy by solving an optimization problem similar to  $P_{\mathrm{OPT}}$  but with the UCB demand function.
Challenge: efficient discretization of type space. The challenge then is to design a discretization scheme for the type space such that we have a) efficient learning, i.e., the discretized space can be efficiently explored to accurately estimate the demand function on those points, and b) Lipschitz property, i.e., the optimal revenue with the UCB estimate of demand function is close to the true optimal revenue as long as the estimation error on the discretized space is small.
There are two main difficulties in achieving this: (1) Lack of any smoothness/Lipschitz assumption on CDF  $F$ . (2) Sensitivity of the Bayes-consistency condition (BC). To see these difficulties, recall that given a price  $p_t$  and realized posterior mean  $q_t$ , the  $t^{th}$  buyer's purchase decision is given by  $a_t = \mathbf{1}[\theta_t \geq \kappa(p_t, q_t)]$ ; thus the seller can obtain demand function estimate at point  $\kappa(p_t, q_t)$ . Without any smoothness or Lipschitz property of demand function, estimates of demand function cannot be extrapolated accurately to other points. This means that in our revenue optimization problem (estimated version of  $\mathsf{P}_{\mathsf{OPT}}$ ), we need to find a price and advertising strategy that we can only use estimates of demand function on a discretized type space, say  $S \subseteq \Theta$ . However, if we restrict to a discretized type space  $S$ , then the support of posterior mean distributions (a.k.a advertising strategy) must be restricted to the points  $q$  such that the corresponding critical types  $\kappa(p, q)$  are in the set  $S$ .
Unfortunately, the (BC) condition makes the set of feasible advertising strategies very sensitive to their support. In particular, if we use uniform-grid based discretization (which is commonly used in previous dynamic pricing literature such as [43, 8]), it is easy to construct examples of prior distribution and valuation function such that there are no or very few feasible advertising strategies with the corresponding restricted support.
Example 2.2. Consider additive valuation function, i.e.,  $v(\theta, \omega) = \theta + \omega$ , and thus  $\kappa(p, q) = p - q$ . Consider quality space  $\Omega = \{\bar{\omega}_i\}_{i \in [3]}$ . Given a small  $\varepsilon$ , consider a uniform-grid based discretization for the type space  $S$ , i.e.,  $S = \{0, \varepsilon, 2\varepsilon, \dots, 1\}$ . If we also use a price  $p$  that is from uniform-grid based discretized price space, i.e.,  $p = k\varepsilon$  for some  $k \in N^+$ , then to ensure  $\kappa(p, q) \in S$ , the support of advertising strategy (i.e., the distribution of the posterior means) must also be restricted within the set  $S$ . However, if the prior distribution  $\lambda$  has negligible probabilities on qualities  $\bar{\omega}_1, \bar{\omega}_3$ , and quality  $\bar{\omega}_2 \notin S$ , then we cannot construct any posterior distribution with the mean in the set  $S$ . Therefore, there does not exist any feasible advertising strategy.
Note that this difficulty cannot be fixed by simply modifying the discretized type space to  $S \cup \Omega$ , because even then we would need to construct an advertising such that  $\kappa(p, q) = p - q \in S \cup \Omega$  in the grid for all  $p$ . That would need that the support of the strategy (i.e., posterior means  $q$ ) is restricted to be in  $\{k\epsilon - \bar{\omega}_i\}_{k \in \mathbb{N}^+, i \in [3]}$ ; such posterior means again may not be achieved here.
Algorithm design idea: novel discretization scheme. Our algorithm employs a carefully-designed quality-and-price-dependent discretization scheme. The above example shows that we cannot uniformly discretize price and type using  $\varepsilon$ -grids, as we may not have any feasible advertising strategy under such discretization. And furthermore, it also shows that this difficulty cannot be fixed by simply adding the  $m$  points in quality space to the discretized type space  $\mathcal{S}$ . Instead, in our discretization scheme, we first uniformly discretize the price space to an  $\varepsilon$ -grid  $\mathcal{P}$ . Then to construct a discretized
type space  $S$ , in addition to the points on an  $\varepsilon$ -grid over  $[0,1]$ , we also include points  $\{\kappa(p,\omega)\}$  for every price  $p \in \mathcal{P}$  and quality  $\omega \in \Omega$ . This gives us a discretized type space  $S$  of size  $mU / \varepsilon$ . We claim that our construction ensures that there exist near-optimal price and advertising strategy with support in the discretized type space  $S$ . The proof of this claim requires a careful rounding argument, which forms one of the main technical ingredients for our regret analysis in Section 3.
2.3 Details of the proposed algorithm
Our dynamic pricing and advertising algorithm jointly discretizes the price space and type space using the following quality-and-price-dependent discretization scheme: given parameter  $\varepsilon$ , we define the following set:
$$
\mathcal {P} \triangleq \{\varepsilon , 2 \varepsilon , \dots , U \}
$$
$$
\mathcal {S} \triangleq \{0, \varepsilon , \dots , 1 - \varepsilon , 1 \} \cup \left\{\left(\kappa (p, \omega) \wedge 1\right) \vee 0 \right\} _ {p \in \mathcal {P}, \omega \in \Omega}. \tag {2}
$$
At time  $t$ , we restrict the price and advertising strategies  $(p_t, \rho_t)$  to the set of  $(p, \rho = (\rho_i)_{i \in [m]})$  such that  $p \in \mathcal{P}$ , and given price  $p$ , each conditional distribution  $\rho_i$  has restricted support  $\mathcal{Q}_p$  defined as  $\mathcal{Q}_p \triangleq \{ q : \kappa(p, q) \in S \}$ . Given the price and advertising  $p_t, \rho_t$ , let the realized posterior mean at time  $t$  be  $q_t \sim \rho_t$ , and let the corresponding critical type be  $x_t \triangleq \kappa(p_t, q_t) \in [0, 1]$ . Then, note that the above restrictions on price and advertising strategies guarantee that  $x_t \in S$ .
Next, to compute the offered price and advertising strategy in round  $t$ , we optimize an upper confidence bound on the revenue function that we develop using upper confidence bounds  $D^{\mathsf{UCB}}(x)$ ,  $x \in S$  of the demand function computed as follows. For every type  $x \in S$ , let  $\mathcal{N}_t(x)$  denote the set of time rounds before time  $t$  that the induced critical type is exactly  $x$ , and let  $N_t(x)$  be the number of such time rounds. That is,  $\mathcal{N}_t(x) \triangleq \{ \tau < t : \kappa(p_\tau, q_\tau) = x \}$ ,  $N_t(x) \triangleq |\mathcal{N}_t(x)|$ ,  $x \in S$ .
Recall that buyer's purchase decision follows  $a_{\tau} = \mathbf{1}[\theta_{\tau} \geq \kappa(p_{\tau}, q_{\tau})]$ . We estimate the demand function at  $x$  as:  $\bar{D}_t(x) \triangleq \frac{\sum_{\tau \in \mathcal{N}_t(x)} a_{\tau}}{N_t(x)}$ . We can now define the following UCB index:
$$
D _ {t} ^ {\mathrm {U C B}} (x) = \min  _ {x ^ {\prime} \in \mathcal {S}: x ^ {\prime} \leq x} \bar {D} _ {t} \left(x ^ {\prime}\right) + \sqrt {\frac {1 6 \log T}{N _ {t} \left(x ^ {\prime}\right)}} + \frac {\sqrt {\left(1 + N _ {t} \left(x ^ {\prime}\right)\right) \ln \left(1 + N _ {t} \left(x ^ {\prime}\right)\right)}}{N _ {t} \left(x ^ {\prime}\right)} \wedge 1, \quad x \in \mathcal {S} \tag {3}
$$
Then, for any pair of discretized price  $p \in \mathcal{P}$  and advertising strategy with discretized support for that price  $\rho = (\rho_{i} \in \Delta^{\mathcal{Q}_{p}}, i \in [m])$ , we define the following seller's revenue estimates:
$$
\mathsf {R e v} _ {t} ^ {\mathsf {U C B}} (p, \rho) \triangleq p \sum_ {i \in [ m ]} \lambda_ {i} \int_ {0} ^ {1} \rho_ {i} (q) D _ {t} ^ {\mathsf {U C B}} (\kappa (p, q)) d q.
$$
Above is well-defined since by definition  $\kappa(p, q) \in S$  for each such  $(p, q) \in \mathcal{P} \times \mathcal{Q}_p$ . Finally, we let  $p_t, \rho_t$  be the optimal solution to the following optimization problem:
$$
(p _ {t}, \rho_ {t}) = \arg \max  _ {p, \rho} \mathsf {R e v} _ {t} ^ {\mathsf {U C B}} (p, \rho)
$$
$$
\begin{array}{l} \text {s . t .} \quad p \in \mathcal {P}; \frac {\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q) \bar {\omega} _ {i}}{\sum_ {i \in [ m ]} \lambda_ {i} \rho_ {i} (q)} = q, q \in \mathcal {Q} _ {p}; \rho_ {i} \in \Delta^ {\mathcal {Q} _ {p}}, i \in [ m ]. \end{array} \tag {P_t^UCB}
$$
We summarize our algorithm as Algorithm 1. The main computational bottleneck of Algorithm 1 is to solve the high-dimensional program  $\mathbf{P}_t^{\mathsf{UCB}}$  at each time  $t\geq |S| + 1$ . As we illustrate in Proposition 2.1, there exists an efficient method to optimally solve this program. The proof of this result utilizes the monotonicity of the function  $D_{t}^{\mathrm{UCB}}$ .
Proposition 2.1 (Adopted from [4, 2]). Let  $\varepsilon$  be the discretization parameter for the set  $\mathcal{P}$  defined in [2]. There exists a polynomial time (in  $|S|U / \varepsilon$ ) algorithm that can solve the program  $\mathsf{P}_t^{\mathsf{UCB}}$
Proof. Since function  $D_t^{\mathsf{UCB}}$  is monotone with discontinuities at the points in the set  $S$ , when we fix a price  $p \in \mathcal{P}$ , the function  $D_t^{\mathsf{UCB}}(\kappa(p, \cdot))$  is also monotone with discontinuities at the points in  $\mathcal{Q}_p = \{q : \kappa(p, q) = x\}_{x \in S}$ . Given a prior  $\lambda$ , optimizing a monotone function with discontinuities over all feasible advertising strategies induced from the prior  $\lambda$  subject to the constraint where the support of advertising strategies must be in the set  $\mathcal{Q}_p$  has been studied in [4, 21]. It has been shown that there
1 Input: Discretization parameter  $\varepsilon$
2 For the first  $|\mathcal{S}|$  rounds, for each  $x \in S$ , offer a price  $p$  with no information advertising s.t.
Algorithm 1: Algorithm for Dynamic Pricing and Advertising with Demand Learning.
$\kappa(p, \mathbb{E}_{\omega \sim \lambda}[\omega]) = x$ . // No information advertising provides completely uninformative signal - the distribution  $\phi(\cdot|\omega)$  of signals does not depend on the realized quality  $\omega$
3 for each round  $t = |\mathcal{S}| + 1,|\mathcal{S}| + 2,\dots ,T$  do
4 For all  $x\in S$  , compute  $D_t^{\mathrm{UCB}}(x)$  as defined in [3].
5 Offer the price  $p_t$  and an advertising  $\rho_t$  computed as an optimal solution to program  $\boxed{P_{t}^{UCB}}$ . /*  $p_t, \rho_t$  satisfies  $\kappa(p_t, q) \in S$  for every  $q \in \operatorname{supp}(\rho_t)$ . */
6 Observe realized posterior mean  $q_{t}\sim \rho_{t}$  and buyer's purchase decision  $a_{t}\in \{0,1\}$
7 Update  $\left\{\mathcal{N}_{t + 1}(x),N_{t + 1}(x),\bar{D}_{t + 1}(x)\right\}_{x\in S}$
exists a polynomial (w.r.t. the number of discontinuities) algorithm based on convex programming that can find an optimal advertising strategy (see Proposition 2 in 4). Thus, an exhaustively search over the discretized price space  $\mathcal{P}$  can lead to an optimal solution to the program  $\mathsf{P}_{t}^{\mathrm{UCB}}$ .
We conclude this section with the following remark on the extension to unbounded type support.
Remark 2.3. Our algorithm and analysis can be extended to the case with unbounded type support (e.g.,  $\Theta = [0,\infty)$ ). In particular, since the price is bounded by  $[0,U]$ , and the quality is bounded within  $[0,1]$ , by the monotonicity of the valuation function, we know the critical types  $\kappa(p,q)$  induced by any possible  $p\in [0,U]$  and  $q\in [0,1]$  is bounded within  $[\kappa(0,1),\kappa(U,0)]$ . Thus, an instance with unbounded type support is equivalent to an instance with bounded type support  $[\kappa(0,1),\kappa(U,0)]$ .
3 Regret Analysis: Proof Overview of Theorem 1.1
In this section, we present our main regret bound for Algorithm  $\square$ , as stated in Theorem  $\square$ . Specifically, we show that for any type CDF  $F$ , given a valuation function  $v(\theta, \omega)$  that is linear in product quality  $\omega$  and satisfies Assumption  $\square$ , our algorithm (Algorithm  $\square$  with parameter  $\varepsilon = \Theta((m \log T / T)^{1/3}))$ , has an expected regret of  $O\left(T^{2/3}(m \log T)^{1/3}\right)$ . Importantly, for this result we do not assume any smoothness or Lipschitz properties of distribution  $F$ .
For this result, we consider arbitrary but discrete quality space  $\Omega$  of cardinality  $m$ . Later in Appendix B, we show improved regret bounds for the case of additive valuations and equally-spaced quality space (see Theorem B.1), and also extend to arbitrary large and continuous quality spaces (see Theorem B.2).
Proof outline. Recall that in every round  $t$ , Algorithm [1] sets the price  $p_t$  and advertising strategy  $\rho_t$  as an optimal solution of program  $\mathsf{P}_t^{\mathsf{UCB}}$  that approximates the benchmark  $\mathsf{P}_{\mathsf{OPT}}$  in two ways. Firstly, it restricts the price and support of advertising strategy to be in a discretized space  $\mathcal{P} \times \{\mathcal{Q}_p, p \in \mathcal{P}\}$ . Secondly, it approximates the true demand function with an upper bound  $D_t^{\mathsf{UCB}}$ . Our proof consists of two main steps that bound the errors due to each of the above approximations. Due to the space limit, all missing proofs in this section are deferred to Appendix A.
- Step 1: bounding the discretization error using a rounding argument (see Appendix A.1).
To separate the discretization error from the error due to demand function estimation, we consider an intermediate optimization problem  $\mathbb{P}$  (in Appendix A.1) obtained on replacing the UCB demand function  $D_{t}^{\mathrm{UCB}}$  with the true demand function  $D$  (while keeping the discretized space for  $p, \rho$ ). Let  $\widetilde{p}^{*}, \widetilde{\rho}^{*}$  be an optimal solution of program  $\widetilde{\mathbb{P}}$ . We show that the revenue  $\operatorname{Rev}(\widetilde{p}^{*}, \widetilde{\rho}^{*})$  is sufficiently close (within  $2\varepsilon$ ) to the optimal revenue  $\operatorname{Rev}(p^{*}, \rho^{*})$ . This bound is obtained using a careful rounding argument: we show that the optimal price  $p^{*}$  and the optimal advertising  $\rho^{*}$  can be rounded to a new price  $p^{\dagger}$  and a new advertising  $\rho^{\dagger}$  that satisfy
(i) feasibility (Lemma A.4):  $p^{\dagger} \in \mathcal{P}$ ,  $\operatorname{supp}(\rho^{\dagger}) \subseteq \operatorname{supp}(\mathcal{Q}_{p^{\dagger}})$ ; and
(ii) revenue guarantee (Lemma A.5):  $\operatorname{Rev}(p^{\dagger}, \rho^{\dagger}) \geq \operatorname{Rev}(p^{*}, \rho^{*}) - 2\epsilon.$
- Step 2: bounding estimation error and establishing optimism (see Appendix A.3).
Next, we show that the UCB estimates of the demand function  $D_t^{\mathrm{UCB}}(x), x \in S$  converge to the true demand function  $D$  with high probability, along with concentration bounds on the gap between the true and estimated function (Lemma A.7). This allows us to show that
(i) Revenue optimism (Lemma A.8): we show that the algorithm's revenue estimates are (almost) optimistic, i.e.,  $\mathrm{Rev}_t^{\mathrm{UCB}}(p_t,\rho_t)\geq \mathrm{Rev}(\tilde{p}^*,\tilde{\rho}^*)\geq \mathrm{Rev}(p^*,\rho^*) - 2\varepsilon$
(ii) Revenue approximation (Lemma A.9): we show that the optimistic revenue estimates are close to the true revenue in round  $t$ , with the gap between the two being inversely proportional to the number of observations, in particular,  $\mathrm{Rev}_t^{\mathrm{UCB}}(p_t,\rho_t) - \mathrm{Rev}(p_t,\rho_t) \leq 5p_t\mathbb{E}_{q\sim \rho_t}\Big[\sqrt{\log T / N_t(\kappa(p_t,q))}\Big]$ .
Finally, in Appendix A.7 we put it all together to bound the regret as stated in Theorem I.1. We first use the above observations to show that regret over each round can be roughly bounded as  $2\epsilon T + 5p_t\mathbb{E}_{q\sim \rho_t}\left[\sqrt{\log T / N_t(\kappa(p_t,q))}\right]$ . Then, using the constraint  $\sum_{x\in S}N_T(x)\leq T$ , we show that in the worst case, total regret over time  $T$  is bounded by  $O(T\epsilon +\sqrt{|S|T\log T})$ . The theorem is then obtained by substituting  $|S| = O(m / \epsilon)$  and optimizing  $\varepsilon$ .
4 Conclusions and Future Direction
In this work, we use a foundational information design framework, Bayesian persuasion, to model the effect of an advertising strategy on customers' beliefs about product quality and consequently their purchase decisions. Our formulation combines the Bayesian persuasion model with dynamic pricing and learning to quantify the tradeoffs between the design of the pricing and advertising strategies and their combined impact on the revenue outcomes. Without any apriori knowledge of the demand function, we show that there exists an efficient online policy that has the regret  $O(T^{2/3}(m \log T)^{1/3})$  for a finite state space with cardinality  $m$ . This result implies that when the number of the states  $m$  is a constant, there is almost no additional learning regret for the seller to additionally learn the optimal advertising compared to the dynamic pricing (without advertising) for non-parametric demand learning problem. There are interesting future directions from this work. First, in our current formulation, buyers' valuation is linear w.r.t the product states. It would be interesting to explore whether our results could be generalized to more general valuation function. Secondly, in our setting, a buyer with i.i.d private type arrives at each time round, and leaves the market no matter what the purchase decision is. However, in practice, buyers may strategize their purchase time for a more favorable price [2, 29, 60]. How to model the advertising effect with strategic buyers and achieve efficient demand learning is another interesting future direction.