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5498a4668067022bed11edabbbc2cb278d973550 | subsection | 31 | 100 | Competitive Analysis | For any \lambda \ge \bar{\lambda }, we have:o_1(\lambda )+o^{{\color {black}\mathcal {S}}}_2(\lambda )-o^{{\color {black}\mathcal {S}}}_2(\bar{\lambda }) & \le \lambda p n_1+(1-p) \eta _1 (\lambda ) + \Delta +\lambda pn_2 +\Delta - (\bar{\lambda }pn_2 - \Delta ) &((\ref {inequality:good-approximation-o_1}),(\ref {inequ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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ca94ada02eefa22a8fedefa32fbcde214420dfbf | subsection | 32 | 100 | Competitive Analysis | We show it will also hold for \lambda :
If the arriving customer is not a classtype-2 customer belonging to the predictablestochastic group, then o_2^{{\color {black}\mathcal {S}}}({\lambda }) = o_2^{{\color {black}\mathcal {S}}}(\lambda -1/n); but q_{2,e}(\lambda ) \ge q_{2,e}(\lambda -1/n), and thus (REF ) holds.
Ot... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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"Vahideh Manshadi"
] | [
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a7521baf8d2e83c3a897b1e24ea002e02b433018 | subsection | 33 | 100 | Competitive Analysis | The proofs are deferred to Appendix .Under event \mathcal {E}, if n_1 < \frac{k}{p^2} \log n, then one of the following conditions holds:q_1(1) + q_{2,e}(1) +q_{2,f}(1) = b,
q_1(1)=n_1 and q_{2,e}(1) +q_{2,f}(1) = n_2, or
q_1(1)=n_1, q_{2,f}(1) =\lfloor \theta b\rfloor and q_{2,e}(1) \ge pn_2 - \frac{k}{p^2} \log n -... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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0534525baa02ab223af7d99f0555cf38d2a71885 | subsection | 34 | 100 | Competitive Analysis | However, the same does not hold
when c^* = 1. For this special case,
we have the following corollary of Theorem REF :
When c^*=1, for c = 1- \@root 3 \of {\frac{1}{ap^{3/2}}\sqrt{\frac{n^2 \log n}{b^3}}} the competitive ratio of ALG_{2,c} is 1-O\left( \@root 3 \of {\frac{1}{ap^{3/2}}\sqrt{\frac{n^2 \log n}{b^3}}}\rig... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
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60fab46858ffbd4d75cae264e533cb6130a54345 | subsection | 35 | 100 | Competitive Analysis | Therefore, we can apply Lemma REF to get:\frac{\mathbb {E}\left[ALG_{2,c}(\vec{V})\right]}{OPT({\color {black}\vec{v}_I})} \ge \frac{\mathbb {E}\left[ALG_{2,c}(\vec{V})|\mathcal {E}\right]\mathbb {P}\left( \mathcal {E} \right)}{OPT({\color {black}\vec{v}_I})} \ge \frac{\mathbb {E}\left[ALG_{2,c}(\vec{V})|\mathcal {E}\r... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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d02499884501dfc1c8103a784c32e9289cfce548 | subsection | 36 | 100 | Competitive Analysis | Under event \mathcal {E}, if n_1\ge \frac{k}{p^2} \log n, then one of the following conditions holds:n_1+n_2 - \frac{2\Delta }{\delta p} \le b , or
n_1+n_2 - \frac{2\Delta }{\delta p} > b and q_2(1) \le \frac{1-c}{1-a} b + c \left(b - n_1 \right)^+ + c \frac{2\Delta }{\delta p} +1.Using Lemma REF and the discussion be... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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1fdc882dbf3c589c5ff54ad50e52560b09b3351e | subsection | 37 | 100 | Competitive Analysis | This means that at time {\color {black}l}, we have:>X>Xu_{1,2}({\color {black}l}) \ge b ,q_2({\color {black}l}) \ge \frac{1-c}{1-a} b + c \left(b - u_1({\color {black}l}) \right)^+.This also provides the following lower bound on the number of accepted classtype-2 customers:q_2(1) = q_2({\color {black}l}) + \left[n_2 - ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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d44a1ab37b7e3489528196f4812b6cc721ce6e3e | subsection | 38 | 100 | Competitive Analysis | Similarly after time {\color {black}l}, we cannot have more than (1 - {\color {black}l}) n customers, and therefore n_1 + n_2 - \left[\eta _1({\color {black}l}) + \eta _2({\color {black}l})\right] \le (1-{\color {black}l}) n. By definition of {\color {black}l}, we have {\color {black}l} \le 1. We also add the condition... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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6d45b116eb0ab41a6289518dd39bd28b7caf4d7e | subsection | 39 | 100 | Competitive Analysis | Note that Inequality (REF ) is Constraint (REF ), where in (REF ), again with a slight abuse of notation, we simplify by substituting \tilde{u}_{1,2} for \tilde{u}_{1,2}({\color {black}l}) and \tilde{o}_j for \tilde{o}_j({\color {black}l}).Further, the most interesting constraint, Constraint (REF ),
conditioncomes from... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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ddc7c07c8d7d636cf94cf34f086e27ec177a1d74 | subsection | 40 | 100 | Competitive Analysis | These two lemmas complete the analysis of competitive ratio in Case (ii).Under event \mathcal {E}, if n_1\ge \frac{k}{p^2} \log n and q_1(1)+q_2(1) < b, then the tuple ({\color {black}l}^{\prime }, n_1^{\prime }, n_2^{\prime }, \eta _1^{\prime }, \eta _2^{\prime }, c^{\prime })\triangleq ({\color {black}l}, n_1, n_2 + ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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d5a80b0694b726af7432be452cf503b9869643d1 | subsection | 41 | 100 | Competitive Analysis | The proofs are deferred to Appendix .Under event \mathcal {E}, if n_1 < \frac{k}{p^2} \log n, then one of the following three conditions holds:q_1(1)+q_2(1)=b;
q_1(1)=n_1 and q_2(1)=n_2; or
q_1(1)=n_1 and q_2(1) \ge cb.Using Lemma REF , we prove the in the following lemma, we establish a lower bound on the competitiv... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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2edd8fb1729a1958aceb8aed346ed3925f84e7a7 | subsection | 42 | 100 | The Adaptive Algorithm | In the design of Algorithm REF , we used the observation that in the partially learnablepredictable model, the demand has a predictablestochastic component that is uniformly spread over the entire horizon. This observation motivated us to define the evolving threshold rule. We remark that in Algorithm REF neither the e... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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67428d91aedccb710fd4a4442dd8264dacd58f44 | subsection | 43 | 100 | Discussion of the Model | In this section, wederive some fundamental properties of further study the performance of online algorithms in our demand model.
First, in Section REF , we present an upper bound on the competitive ratio achievable by any online algorithm under our demand model when the initial inventory b is small—more precisely, b = ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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"Vahideh Manshadi"
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8956878b652259e0929c0852bed840f6aed64da4 | subsection | 44 | 100 | Upper Bounds | In this section, we present an upper bound on the competitive ratio of any online algorithm when b = o(\sqrt{n}).
We start with a warm-up example that illustrates a fundamental limit of any online algorithm in the partially learnablepredictable model.
Figure REF shows two instances with n = 8 arriving customers.
The bo... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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e98d1ba366eaa388e58a6ec9e8c621ece786c22b | subsection | 45 | 100 | Comparison with Existing Algorithms | In this section, we show that,if the customers follow under our demand arrival model, then for a certainthere exists a class of instances for which our algorithms have better performanceachieve higher revenue than algorithms designed for either the worst-case or the random-order model , , which respectively correspond ... | {
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"raw": "Ball, M. O. and M. Queyranne (2009). Toward robust revenue management: Competitive analysis of online booking. Operations Research 57(4), 950–96... | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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fe72b12b37188478ce648450ff5ee7d3ceb662f8 | subsection | 46 | 100 | Comparison with Existing Algorithms | Note that p + \frac{b}{n}(1-p) < p+\frac{1-p}{2-a} for any b < \frac{n}{2-a}.Our Algorithm REF achieves a ratio of at least the performance guaranteeits competitive ratio as given in Theorem REF
and the ratio is tight for this instance (up to an additive error term of O\left(\frac{1}{a(1-p)p}\sqrt{\frac{\log n}{b}}\ri... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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a7a1802e6a6c31c5e339e317b88537a33f4a5738 | subsection | 47 | 100 | The Secretary Problem under Partially Predictable Demand | In this section, we study the online secretary problem under our new arrival model.
In our setting, the secretary problem corresponds to having one unit of inventory, i.e., b = 1, and n customers, where v_{I,j} \in \mathbb {R}^{+} for 1\le j \le n, i.e., we relax the assumption that there are only two types. The object... | {
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"Dawsen Hwang",
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e7ad2c385123d489d8d4eb95dc89c5bbc6b58a04 | subsection | 48 | 100 | The Secretary Problem under Partially Predictable Demand | By optimizing over \gamma , we obtain the following corollary:
Let \gamma ^* \in (0,1) be the unique solution to\log (\gamma ^* p + 1-p) + \frac{\gamma ^* p}{\gamma ^* p + 1-p}=0;then, \text{OSA}_{\gamma ^*} achieves the highest success probability among \text{OSA}_\gamma for all \gamma \in (0,1).
[Table: The optimal ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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e40dabfbe70337149576f0c95a56b03b8d8703d9 | subsection | 49 | 100 | Proof of Lemma | The proof of Lemma REF is based on the following lemma:
Define constants \alpha _{\ref {lemma:low-tail-o1}} \triangleq 5 + \sqrt{6}, \bar{\epsilon }_{\ref {lemma:low-tail-o1}} \triangleq 1/24, and k_{\ref {lemma:low-tail-o1}} \triangleq 4.
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"Dawsen Hwang",
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446282403ad01345fabebd888452dc6ac7e8c402 | subsection | 50 | 100 | Proof of Lemma | For \epsilon \in (0, \bar{\epsilon }_{\ref {coro:binomial},k}), under the same setting as in Theorem , we have:\mathbb {P}\left( |S_n-np| \ge \alpha _{\ref {coro:binomial},k} \sqrt{n \log \left( \frac{1}{\epsilon } \right) } \right) \le k \epsilon .Second, we use a concentration result for random variables drawn from t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.spl.2005.05.019",
"end": 363,
"openalex_id": "https://openalex.org/W1999348522",
"raw": "Hush, D. and C. Scovel (2005). Concentration of the hypergeometric distribution. Statistics & Probability Letters 75(2), 127–132.",
"so... | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
fab5dc8e863a46ace0c0ace75bf7903a1350ddee | subsection | 51 | 100 | Proof of Lemma | Let us consider \lambda = 5/8.
In the following, we count the number of customers in the predictablestochastic group and the unpredictableadversarial group in O_1(\lambda ) separately.We begin by counting the number of classtype-1 customers in the predictablestochastic group that arrive no later than time 5/8 in \vec{V... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
4b7ccabc06542c97b7113dd734ba954ad1176d25 | subsection | 52 | 100 | Proof of Lemma | Call this number \zeta _1.
Finally, we obtain O_1(\lambda ) with the equation O_1(\lambda ) = Z_1+\zeta _1.
In summary, the random variables have the following distributions:R \sim \text{Bin}(n, p),
R_1 \sim \text{Bin}(n_1, p),
Z \sim \text{Bin}(\lambda n, p),
Z_1 \sim \text{Hyper}(Z, R, R_1)Note that Z, R, and R_1 ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.001... | |
960742cb69a5c88639dc6657db5675d6df3e0212 | subsection | 53 | 100 | Proof of Lemma | Thus we have:\mathbb {E}\left[Z_1\right] = \mathbb {E}\left[\mathbb {E}\left[Z_1|R,R_1,Z\right]\right] = \mathbb {E}\left[\frac{ZR_1}{R}\right].The last expectation is a non-linear function of the three random variables R, R_1, and Z. Instead of computing the expectation directly, we
use the concentration bounds of (RE... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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1d5a61c91bdcb0718347e9c09eedabefeb836c9a | subsection | 54 | 100 | Proof of Lemma | Using the law of total probability, we have: for any \tilde{ \alpha } > 0,& \mathbb {P}\left( \left| Z_1 - \mathbb {E}\left[Z_1 | R,R_1,Z\right] \right| \ge \tilde{\alpha } \sqrt{n_1\log \left( \frac{1}{\epsilon } \right)} \right)
\\ & = \mathbb {P}\left( \mathcal {E}^c \right)\mathbb {P}\left( \left| Z_1 - \mathbb {E}... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
ff17d41efe44691d1485aa91b73f204f8b2c92de | subsection | 55 | 100 | Proof of Lemma | For \epsilon \in (0, \bar{\epsilon }_{\ref {claim:concentraionZ1}}],
if n_1 > \frac{k_{\ref {claim:concentraionZ1}}}{p^2}\log \left(\frac{1}{\epsilon }\right), and (R,R_1,Z) \in \mathcal {E}, we have:\mathbb {P}\left( \left| Z_1 - \mathbb {E}\left[Z_1 | R,R_1,Z\right] \right| \ge \alpha _{\ref {claim:concentraionZ1}} ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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-... | |
cac5e6e2c7b07177146d093f52434e08bb9bbbfb | subsection | 56 | 100 | Proof of Lemma | Now we can apply the union bound on (REF ), (REF ), and (REF ) and obtain: when 0<\epsilon ^{\prime } \le \bar{\epsilon }_{\ref {lemma:low-tail-o1}} and n_1 > \frac{k_{\ref {lemma:low-tail-o1}}}{p^2} \log \left( \frac{1}{\epsilon ^{\prime }} \right), with probability at least 1-\epsilon ^{\prime },& \left| Z_1 - \mathb... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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b5d60d7967ff2db44da9af5894b5d85a9bd2abc9 | subsection | 57 | 100 | Proof of Lemma | When applied to O_2^{{\color {black}\mathcal {S}}}(\lambda ), the only modification is that we do not need to consider Lemma REF and (REF ).Second, we apply the union bound on the probability that at least one of the 4n events in (REF )-() is violated (note that \lambda takes n different non-zero values: when \lambda =... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
71b54deb4aa2615161a549d881ed38cd33a2116f | subsection | 58 | 100 | Further Remak on Deterministic Approximations | In Lemma REF , we use the deterministic value \tilde{o}_j(\lambda ) rather than \mathbb {E}\left[O_j(\lambda )\right] to estimate O_j(\lambda ) because \tilde{o}_j(\lambda ) is a very simple function of n_j and \eta _j(\lambda ).
Here we provide an example to show that \tilde{o}_j(\lambda ) and \mathbb {E}\left[O_j(\la... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
067d68134d2a989efb8235c52766f812263eac4c | subsection | 59 | 100 | Proof of Auxiliary Corollaries and Lemmas | Proof of Corollary :
In order to apply Theorem , we define t such that
2e^{-2nt^2} \le k\epsilon , which corresponds tot \ge \sqrt{\frac{1}{2n} \log \frac{2}{ k\epsilon }}.By setting t to be \sqrt{\frac{1}{2n} \log \frac{2}{ k\epsilon }}, what is remaining to prove is that we can find \alpha _{\ref {coro:binomial},k}, ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
0577631b820f29077f8293024571d3c6a0b55f59 | subsection | 60 | 100 | Proof of Auxiliary Corollaries and Lemmas | Putting these two together,2 e^{-2\alpha _{n_1,n,m}(\gamma ^2-1)} \le 2 e^{-\frac{1}{m} \frac{\gamma ^2}{2}}.Therefore, if \alpha _{\ref {coro:hyper},k} \sqrt{m \log \left( \frac{1}{\epsilon } \right) } \ge 2 and m\ge 1,\mathbb {P}\left( |K - \mathbb {E}\left[K\right]| \ge \alpha _{\ref {coro:hyper},k} \sqrt{m \log \le... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.01884467899799347,
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754d1cc082a4a6b33fb60b1534ddd3baa2bdeb1b | subsection | 61 | 100 | Proof of Auxiliary Corollaries and Lemmas | The first two conditions hold by defining \underline{m}_{\ref {coro:hyper},k} \triangleq \max \left\lbrace \left( \log \frac{1}{\bar{\epsilon }_{\ref {coro:hyper},k}}\right)^{-1} , 1\right\rbrace .Proof of Lemma :
Let k = 1/12, Corollary implies that there exist \bar{\epsilon }_{\ref {coro:binomial},k} and \alpha _{\r... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.0... | |
74c0d5687e7ac4b453d010da9f1582af94cea524 | subsection | 62 | 100 | Proof of Auxiliary Corollaries and Lemmas | In particular, we subtract \alpha _{\ref {claim:BinomialBounds}} \sqrt{n \log \left( \frac{1}{\epsilon } \right)} from both the denominator and the numerator,
Therefore,\frac{Z}{R} > \frac{Z - \alpha _{\ref {claim:BinomialBounds}} \sqrt{n \log \left( \frac{1}{\epsilon } \right)} }{R - \alpha _{\ref {claim:BinomialBound... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.025321844965219498,
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0... | |
7959bba90c6b83f8c5800b86025b4cc2e9adb866 | subsection | 63 | 100 | Proof of Auxiliary Corollaries and Lemmas | \\Therefore,n > \frac{4 \alpha _{\ref {claim:BinomialBounds}}^2}{p^2} \log \left( \frac{1}{\epsilon } \right) \Rightarrow R - \alpha _{\ref {claim:BinomialBounds}} \sqrt{n \log \left( \frac{1}{\epsilon } \right)} > 0 .The first inequality in (REF ) holds because n \ge n_1, and, by assumption in the lemma, n_1 > \frac{k... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
edcb6a3ccb94a2862e93bb0502c0eafcf37bc93b | subsection | 64 | 100 | Proof of Auxiliary Corollaries and Lemmas | By definition, \alpha _{\ref {claim:expectationofZ1}} \ge 3 \alpha _{\ref {claim:BinomialBounds}} \ge (2+\lambda ) \alpha _{\ref {claim:BinomialBounds}}, and therefore, the right hand side of the above inequality is at least\lambda n_1 p - \alpha _{\ref {claim:expectationofZ1}} \sqrt{ n_1 \log \left( \frac{1}{\epsilon ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
596023778fc298f3cb35e5b1ef677b6a13aae753 | subsection | 65 | 100 | Proof of Auxiliary Corollaries and Lemmas | When it comes to the upper bound, we can use the same argument as the lower bound and obtain,& \frac{Z R_1 }{R} < \left( \lambda + \frac{2\alpha _{\ref {claim:BinomialBounds}}}{p} \sqrt{\frac{ \log \left( \frac{1}{\epsilon } \right)}{n}} \right)
\left( n_1p + \alpha _{\ref {claim:BinomialBounds}} \sqrt{ n_1 \log \left(... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.037844736129045486,
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... | |
77bde739ff62c896800d2ec3fd67f1938d946735 | subsection | 66 | 100 | Proof of Auxiliary Corollaries and Lemmas | Combining the definition \alpha _{\ref {claim:expectationofZ1}}= 3 \alpha _{\ref {claim:BinomialBounds}} + 2\alpha _{\ref {claim:BinomialBounds}}^2 \sqrt{1/k_{\ref {claim:expectationofZ1}}} and (REF ), it is sufficient to have\frac{1}{p}\sqrt{\frac{\log \left(\frac{1}{\epsilon }\right)}{n}}upper bounded by the constant... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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-0.03281273692846298,... | |
9fc6aa7a27b517a6300b16ed5125d53e3398e516 | subsection | 67 | 100 | Proof of Auxiliary Corollaries and Lemmas | Thus we have:3 \alpha _{\ref {claim:BinomialBounds}} + 2\alpha _{\ref {claim:BinomialBounds}}^2 \frac{1}{p}\sqrt{\frac{\log \left(\frac{1}{\epsilon }\right)}{n}} \le 3\alpha _{\ref {claim:BinomialBounds}}+ 2\alpha _{\ref {claim:BinomialBounds}}^2 \sqrt{1/k_{\ref {claim:expectationofZ1}}} \le \alpha _{\ref {claim:expect... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
04396faa73f45c26dcc317843e44c52003898517 | subsection | 68 | 100 | Proof of Auxiliary Corollaries and Lemmas | Hence, the lemma follows straightforwardly from Corollary . | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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2787f9c51ef2e0bfb11143ddc364f5784384b26b | subsection | 69 | 100 | Missing proofs of Subsection | Proof of Lemma REF :
The revenue of Algorithm REF in this case is ALG_1(\vec{v}) = b - (1-a) \left[ q_{2,e}(1) + q_{2,f}(1) \right], which is decreasing in q_{2,e}(1) + q_{2,f}(1).
Note that due to the fixed threshold rule, we already have an upper bound on q_{2,f}(1), i.e., q_{2,f}(1) \le \theta b.
As a result, using ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.024307532235980034,
0.012993008829653263,
0.023483548313379288,
-0.024460121989250183,
0.008789164014160633,
-0.032410044223070145,
0.... | |
ed65f92f8fbdea94d26da56b039867c6e840a079 | subsection | 70 | 100 | Missing proofs of Subsection | Note that by construction, the alternative adversarial instance has the same optimum offline solution, i.e., OPT(\vec{v}) = OPT(\vec{v}_A).
ALG_1(\vec{v}) \ge & n_1+ a(p(n_1+n_2)-n_1 -5\Delta + \theta b) \\
\ge & n_1+ a(p(n_1+n_2)-n_1 -5\Delta + \theta (n_1+n_2)) &(b \ge n_1+n_2)
\\
= & n_1(1-a+pa+\theta a) + n_2(p+\t... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.01137500535696745,
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0.027159... | |
e71a25b0b90058a3e5a366ff524b666553c02bcb | subsection | 71 | 100 | Missing proofs of Subsection | Further, if we find a time \hat{\lambda } for which we have:o_1(\lambda )+o^{{\color {black}\mathcal {S}}}_2(\lambda )-o^{{\color {black}\mathcal {S}}}_2(\hat{\lambda }) \le \lfloor \lambda pb \rfloor \quad \quad \text{ for all }\lambda \ge \hat{\lambda },then, using a similar induction to the one in Lemma REF , we can... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.046639375388622284,
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0... | |
7526ff0e381ec93dae076cc3d6ffdda3358d056b | subsection | 72 | 100 | Missing proofs of Subsection | We consider three cases in Lemma REF separately.If case (a) in Lemma REF happens, then ALG_1(\vec{v})+n_1 \ge OPT(\vec{v}) and OPT(\vec{v}) \ge ab.
As a result,\frac{ALG_1(\vec{v})}{OPT(\vec{v})} \ge 1-\frac{n_1}{OPT(\vec{v})} \ge 1- \frac{\frac{k}{p^2} \log n}{ab} \ge p+\frac{1-p}{2-a}- \frac{\frac{k}{p^2} \log n}{ab}... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.029793094843626022,
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... | |
0f54bc0caa0646d4bf0fa0d376012d8b6a79c1e0 | subsection | 73 | 100 | Missing proofs of Subsection | As a result,\frac{\frac{k}{p^2} \log n }{a b} \le &
\frac{\frac{k}{p} \sqrt{b\log n} }{b} &(\log n\le ap\sqrt{b\log n} ) \\
\le & \frac{k}{a(1-p)p}\sqrt{\frac{\log n}{b}} &(0<p<1 \text{ and } a<1) \\
= & O\left(\frac{1}{a(1-p)p}\sqrt{\frac{\log n}{b}}\right).Similarly, to prove (d) \frac{\frac{k}{p^2} \log n + 4 \Delta... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.013889231719076633,
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... | |
ebf842c08e32cc4c13f1880694317b7400f9ba88 | subsection | 74 | 100 | Missing proofs of Section | Before proceeding with the proofs, we state and prove an auxiliary lemma that establishes an upper bound on n_1 and n_1 + n_2 using the deterministic approximation functions \tilde{o}_j(\cdot ).For \lambda \in \lbrace 1/n, 2/n, \ldots , 1\rbrace , we have:& n_1 \le \min \left\lbrace \frac{\tilde{o}_1(\lambda )}{\lambda... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.043635133653879166,
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0.025... | |
99a74d0efe7bbd0ff13510ca1514f72745da663a | subsection | 75 | 100 | Missing proofs of Section | Note that we can apply Inequality (REF ) to this modified instance, becauseto those b+\frac{2 \Delta }{\delta p} \ge \frac{k}{p^2} \log n under the condition imposed on b.Note that \delta =\frac{\phi b}{n}=\frac{(1-c)b}{(1-a)n} \ge \frac{(1-c)b}{n}, Condition imposed on b, and{\color {black}\bar{\epsilon }}\le \frac{3}... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.02779402956366539,
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0.0... | |
b191293e3f137cbc87ba8d4a4f14883ecf5ea7fb | subsection | 76 | 100 | Missing proofs of Section | By using Constraints (REF ) and (REF ), we can obtain upper bounds \tilde{o}_1 \le (1-p+{\color {black}l} p)n_1 and \tilde{o}_2 \le (1-p+{\color {black}l} p) n_2 .
With these upper bounds and the fact \tilde{u}_1 \le \frac{\tilde{o}_1}{{\color {black}l} p } , Constraint (REF ) gives:c \ge \frac{a(n_2- (1-p+p{\color {bl... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.056939221918582916,
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0.026715053245425224,
-0.02097841165959835,
0.... | |
03f957321e3789e50081daca9a9bc288cf9a6d27 | subsection | 77 | 100 | Missing proofs of Section | Therefore, for the sake of obtaining a lower bound, we can assume, without loss of generality,n_2 \le b-n_1.With (REF ), the right hand side of (REF ) can be written as&{\color {black}f_1({\color {black}l}) \triangleq }\frac{a n_2 p (1-{\color {black}l}) +\frac{ab}{1-a}+n_1}{an_2+n_1+\frac{a^2b}{1-a}+a b} & \text{ if }... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.06934447586536407,
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-0.017793938517570496,
0.011... | |
f50a94c2511ebe6f2d549f0763edc6e87b1901d5 | subsection | 78 | 100 | Missing proofs of Section | Therefore, according to (REF ), we only need to consider the case b=n_1+n_2 (in the degenerated case n_1=n_2=0, the above quantityf_1\left(\frac{(1-p)n_1}{p(b-n_1)}\right) is 1, which is greater than p+\frac{1-p}{2-a}, so we can assume, without loss of generality, n_1+n_2>0), in which case, the above quantity equals to... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.0346083790063858,
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-0.014893199317157269,
0.... | |
fc08b5091ed6ec3e544dde7c98e2f51ee6891b54 | subsection | 79 | 100 | Missing proofs of Section | As a result, (REF ) is convex and is maximized at extreme values of {\color {black}l}, which in our case is at either {\color {black}l} = \frac{(1-p)n_1}{p(b-n_1)} or {\color {black}l}=1.
Therefore, we only need to prove statementInequality (REF ) at these extreme two values of {\color {black}l}.
The former case, {\col... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.026985179632902145,
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-0.006666209083050489,
0.019... | |
18560887e44a131d94dc9258ef335f27373b65d0 | subsection | 80 | 100 | Missing proofs of Section | According to Constraint (REF ) and using a\min \lbrace n_1+n_2, b\rbrace +(1-a)n_1 = a\min \lbrace n_2, b-n_1\rbrace +n_1 , it suffices to prove\frac{a(n_2-\tilde{o}_2+\frac{b}{1-a})+n_1}{a\min \lbrace n_2, b-n_1\rbrace +n_1+\frac{a^2b}{1-a}+a \tilde{u}_1} \ge 1.or equivalently,{a(n_2-\tilde{o}_2+\frac{b}{1-a})+n_1} \g... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.06476804614067078,
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0.003096092725172639,
-0.013551843352615833,
0.02... | |
72cb99a41e516b3a99077eea68963f551dabbd4a | subsection | 81 | 100 | Missing proofs of Section | By Lemma REF , u_{1,2}(\bar{\lambda }) \ge \min \left\lbrace b, n_1 + n_2 -\frac{2\Delta }{\delta p}\right\rbrace = b.
Therefore, according to the definition of \bar{\lambda }, Condition (REF ) must be satisfied.
Thus,q_2(1) = & q_2 ( \bar{\lambda }) \\ \le & \frac{1-c}{1-a} b + c \left(b - u_1( \bar{\lambda }) \right)... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.05604567751288414,
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... | |
ba0c08d88bea05b003f7fedebabfc189c225883f | subsection | 82 | 100 | Missing proofs of Section | For case (a), n_1+n_2 \le b + \frac{2\Delta }{\delta p}, we note thatOPT(\vec{v}) \le & n_1+ n_2 a \\
\le & \left( b+ \frac{2\Delta }{\delta p} -n_2 \right) + n_2 a &(n_1+n_2 \le b+ \frac{2\Delta }{\delta p}) \\
\le & ALG_{2,c} ({\vec{v}})+ \frac{2\Delta }{\delta p} .&(ALG_{2,c} ({\vec{v}})\ge (b-n_2)+an_2)Therefore,\f... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.04042976722121239,
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0.0... | |
0fbe0c58a44df7baa30732866a8424a48ae24f44 | subsection | 83 | 100 | Missing proofs of Section | It is easy to check that ({\color {black}l}^{\prime }, n_1^{\prime }, n_2^{\prime }, \eta _1^{\prime }, \eta _2^{\prime }, c^{\prime }) satisfies Constraints (REF )-(REF ).
The interesting part is to show that it satisfies Constraint (REF ).
When n_1+n_2\ge b, we can prove it directly from Lemma (since \tilde{u}_{1,2}... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.038914307951927185,
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0... | |
2e57e63738d38a14faae64e29742688cbe4f91e0 | subsection | 84 | 100 | Missing proofs of Section | For the second case,Case (2) \xi = \frac{\Delta n }{\phi b p}, we have\tilde{o}_1^{\prime } + \tilde{o}_2^{\prime } = & {\color {black}l}^{\prime } p n_1^{\prime } + (1-p ) \eta _1^{\prime } +{\color {black}l}^{\prime } p n_2^{\prime } + (1-p ) \eta _2^{\prime } \\
\ge & {\color {black}l} p n_1 + (1-p ) \eta _1 +{\colo... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.026397958397865295,
0.0038967507425695658,
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0.00027251505525782704,
-0.021896572783589363,
0... | |
9162bc88285aa5862942ff338e9234ef7abee485 | subsection | 85 | 100 | Missing proofs of Section | This means, for ALG_{2,c} (with any c \le c^*),c = c^{\prime } \le \frac{a(n_2^{\prime } - \tilde{o}_2^{\prime }+\frac{b}{1-a})+n_1^{\prime }}{a\min \lbrace n_1^{\prime }+n_2^{\prime } , b\rbrace +(1-a)n_1^{\prime }+\frac{a^2b}{1-a}+a \min \lbrace \tilde{u}_1^{\prime }, b\rbrace }.After rearranging terms
(REF ) is equ... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.02898300439119339,
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... | |
f4e73f66fb02736c367a92b07e49001a36e44a63 | subsection | 86 | 100 | Missing proofs of Section | Combining this and using an argument similar to the proof of Lemma REF ,\tilde{u}_1^{\prime } \triangleq & \min \left\lbrace \frac{\tilde{o}_1^{\prime }}{{\color {black}l}^{\prime } p}, \frac{ \tilde{o}_1^{\prime } + (1-{\color {black}l}^{\prime }) (1-p) n}{1-p+{\color {black}l}^{\prime } p} \right\rbrace \\
= &
\min \... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.08292070031166077,
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0.0... | |
d4edbe1f25577e1b3217a321482cb78b48ceeebd | subsection | 87 | 100 | Missing proofs of Section | For proving the upper bound on n_2, i.e., n_2 \le b+ \frac{2\Delta }{\delta p}, we first note that, clearly, if n_2 > b+ \frac{2\Delta }{\delta p}, decreasing n_2 to b+ \frac{2\Delta }{\delta p} (while fixing n_1) does not modify the optimal revenue OPT(\vec{v}).
Using Lemma REF , we know that, when n_2 \ge b+ \frac{2\... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.06763836741447449,
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0.0028... | |
8efb88f167cb81bf93a6f053913fb2bcd53f4228 | subsection | 88 | 100 | Missing proofs of Section | Putting all these together, we have:\tilde{o}_2^{\prime } \ge \tilde{o}_2({\color {black}l}) \ge o_2({\color {black}l}) - \alpha \sqrt{n_2 \log n} \ge o_2({\color {black}l}) - \alpha \sqrt{4b \log n} = o_2({\color {black}l}) -2\Delta .This proves (REF ). Having proved (REF ) and (REF ), at last, we complete the proof a... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
83596cce5637ee1f2d24687c692e2c570f99e3b3 | subsection | 89 | 100 | Missing proofs of Section | Therefore, what is remaining is to show that if q_1(1)+q_2(1)<b and q_2(1)<n_2, then q_2(1) \ge cb , i.e., we are in case (c).Let \bar{\lambda } be the last time whenthat a customer is rejected.
Then, similar to earlier discussion, Inequality (REF ) is satisfied.
Therefore,u_1(\bar{\lambda }) = &\min \left\lbrace \frac... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
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35b7d924a013f7e0ab098e99ed46dadda9acbd12 | subsection | 90 | 100 | Missing proofs of Section | For the firstcase (a), q_1(1)+q_2(1)=b, since n_1 < \frac{k}{p^2} \log n, it is easy to see that\frac{ALG_{2,c}({\vec{v}})}{OPT(\vec{v})} \ge \frac{ab}{ab+\frac{k}{p^2} \log n} \ge \frac{ab - \frac{k}{p^2} \log n}{ab} = 1- \frac{k \log n}{ab p^2},which is at least c if b \ge \frac{k \log n}{a(1-c) p^2}.
Inequality (REF... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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af9014ce0e82168c1ae03f3beb3e3b9e8d02319d | subsection | 91 | 100 | Missing proofs of Section | \end{array}\right.} ~~~~~
w_{I,j} = {\left\lbrace \begin{array}{ll}
a, \qquad & 1 \le j \le b, \\
1, \qquad & b < j \le 2b, \\
0, \qquad & j > 2b.
\end{array}\right.}Let us denote \mathcal {E}\mathcal {U} denote to bethe event in which
in the arrival sequence, none of the first b arrivals belongs to positions [b+1, 2b]... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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c57606875b1ab181a00ae5ad82828b6d15cc6e79 | subsection | 92 | 100 | Missing proofs of Section | Second, denoting R the random variable corresponding to the size of the predictablestochastic group, we have \mathbb {P}\left( \sigma _{{{\color {black}\mathcal {S}}}}^{-1}(i)=i | i \in {{\color {black}\mathcal {S}}}, R \right)=\frac{1}{R} \ge \frac{1}{n}, and thus \mathbb {P}\left( \sigma _{{{\color {black}\mathcal {S... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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66f0a898c7695ded680653505b7a5fe066fd7f3e | subsection | 93 | 100 | Missing proofs of Section | We start by {\vec{w}_{I}}:\mathbb {E}\left[ALG({\vec{W}})\right]
&\le \mathbb {E}\left[ALG(\vec{W}) \,|\, {\mathcal {E}}{\color {black}\mathcal {U}} \right] \mathbb {P}\left( {\mathcal {E}}{\color {black}\mathcal {U}} \right) + OPT({\vec{w}_{I}}) \left(1- \mathbb {P}\left( {\mathcal {E}}{\color {black}\mathcal {U}} \ri... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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] | [
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020104eac90096ae818f6775ebd1ff8699ea39a4 | subsection | 94 | 100 | Missing proofs of Section | As a result,\mathbb {E}\left[ALG({\vec{V}}) \right]
&\le \mathbb {E}\left[ ALG({\vec{V}}) \,|\,{\mathcal {E}}{\color {black}\mathcal {U}}\right] \mathbb {P}\left( {\mathcal {E}}{\color {black}\mathcal {U}} \right) + OPT(\vec{v}_{I}) \left( 1 - \mathbb {P}\left( {\mathcal {E}}{\color {black}\mathcal {U}} \right) \right)... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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fcd7bfdb6f711418d3af18a6d5788da6f55fefcf | subsection | 95 | 100 | Missing proofs of Section | The k^{\text{th}}-highest-revenue customer arrives in the observation period and the k-1 customers with the highest valuesrevenue do not.
The highest-revenue customer arrives first among the k-1 customers with the highest valuesrevenue.Clearly, for any k\ge 2, {\color {black}\mathcal {F}_k} is a success event and thos... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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51eca0bddd4b6d2257f1a455540279be0784258c | subsection | 96 | 100 | Missing proofs of Section | Since the above inequality holds for all m, we have\mathbb {P}\left( \text{success} \right) \ge \lim _{n\rightarrow \infty }\sum _{k=2}^n \mathbb {P}\left( {\color {black}\mathcal {F}_k} \right) \ge \lim _{m \rightarrow \infty }\sum _{k=2}^m p^k \gamma (1-\gamma )^{k-1}\frac{1}{k-1}=\gamma p \log \frac{1}{\gamma p + 1-... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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b2f27a717866e1b163095c078c5b90549385afa3 | subsection | 97 | 100 | Missing proofs of Section | Further,it is easy to check that for any 2\le l \le (1-\gamma )n, conditioned on {\color {black}\mathcal {H}_l}, to have a success, either one of the events {\color {black}\mathcal {F}}_2, {\color {black}\mathcal {F}}_3 , \dots {\color {black}\mathcal {F}}_l occurs or the highest-revenue customer must be one of thearri... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
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ea8f50fb38738a197dcf5bccdadf2fa980603859 | subsection | 98 | 100 | Missing proofs of Section | To formalize this idea, we introduce two lemmas.
If the second-highest-revenue customer is among the first \gamma _2 n customers in {\color {black}\vec{v}_I}, then OSA_{\gamma _2} has a success probability of at least s_2+p(1-p)(1-\gamma _2), when n\rightarrow \infty .
Proof:
Note that
the events \lbrace {\color {bla... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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19adb38ef88c07ad8049afcd769880ed7befdf7f | subsection | 99 | 100 | Missing proofs of Section | Note that the probability that the highest-revenue customer arrives between time \gamma _1 and \gamma _2, and it arrives first among the k highest-revenue customers (except for the second-highest-revenue customer) is at least \frac{\gamma _2 - \gamma _1}{k-1}+o(1).conditioning on the highest-revenue customer arriving b... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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a6c6258a10fc87bc7787860322f8de23846bc4de | abstract | 0 | 69 | Abstract | Consider a population of individuals that observe an underlying state of
nature that evolves over time. The population is classified into different
levels depending on the hierarchical influence that dictates how the
individuals at each level form an opinion on the state. The population is
sampled sequentially by a pol... | {
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} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
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1e8fdc6ed54f02f3ce2dbc3dbdae783c0a26a37b | subsection | 1 | 69 | Introduction | Blackwell dominance and LeCam deficiency are widely used in statistical analysis of estimators , , in characterizing correlated and Nash equilibria in games , and in stochastic control , . Blackwell dominance also has deeper information theoretic interpretations . In this paper, we use Blackwell dominance to construct ... | {
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7b66768152a2439fe29b806e88f07ef010ba5094 | subsection | 2 | 69 | Context. Blackwell Dominance | In general, POMDPs are computationally intractableThey are PSPACE hard requiring exponential computational cost (in sample path length) and memory , . to solve . The main contribution of this paper is to exploit the structure of the social influence network to construct computationally efficient myopic policies that pr... | {
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Dominance | [
"Sujay Bhatt",
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a6218175958abaa400aaf0019ec7984018441312 | subsection | 3 | 69 | Context. Blackwell Dominance | Then from Data Processing Inequality , it follows thatB(1) \succeq _B B(2) \Rightarrow I(\mathcal {X};\mathcal {Y}^{(1)}) \ge I(\mathcal {X};\mathcal {Y}^{(2)}).Theorem REF below provides a relation between Blackwell Dominance and Shannon capacity.Theorem 1 (, , )
For any two conditional distributions B(1) \in \mathbb... | {
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11895e58cfe6c9c7a42942d1048aac4e45ff408e | subsection | 4 | 69 | Main Results and Organization | (i) In Sec., the underlying state is modeled as a Markov chain and the adaptive polling problem is formulated as a POMDP.
Open loop polling, where polling at a particular instant is not influenced by the information previously collected, is ineffective when the states evolve over time. In comparison, the proposed adapt... | {
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b63c06d1c118cb6e01aa28a8c6213041d88ca06b | subsection | 5 | 69 | Related Literature | analyzes a Bayesian approach to intent and expectation polling, but without feedback control. Polling has been considered in and a comparison of intent and expectation polling (non-Bayesian) algorithms is discussed analyzes all US presidential electoral college results from 1952-2008 where both intention and expectatio... | {
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78fc1ccd3e7edef4c57ea740539f297e0887dce9 | subsection | 6 | 69 | Adaptive Polling in Hierarchical Social Networks | This section formulates the adaptive polling problem as a partially observed Markov decision process (POMDP). Sec.REF introduces the model for the adaptive polling problem and Sec.REF formulates the adaptive polling problem as a POMDP. Then the main result, namely, sufficient conditions on the model parameters that ena... | {
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} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
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2a7a3e5c5ab9d9fcee3edbe4eeef96de08f087e8 | subsection | 7 | 69 | Polling Model and Notation | Consider the hierarchical social networkIt is to be noted that, the interconnection in the actual social network connecting the people or nodes is irrelevant given the hierarchical influence. shown in Fig.REF .State: Let x_k \in \mathcal {X}=~\lbrace 1, 2, \cdots , X \rbrace denote a Markov chain evolving at discrete t... | {
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c8d964ef3ec68e54621e4a6359f1b156a645e4b2 | subsection | 8 | 69 | Polling Model and Notation | Observation \textbf {y}^{l}_{k}, l \ge 0 influences \textbf {y}^{l+1}_{k} (see Fig.REF ), i.e,\mathbb {P}(\textbf {y}^{l+1}_{k} = j | \textbf {y}^{l}_{k} = i, x_k = {x}) \\mathbb {P}(\textbf {y}^{l+1}_{k} = j | \textbf {y}^{l}_{k} = i).Discussion of (REF ): The approximation (REF ) says that the likelihood probabilitie... | {
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1f76d84901a36d617c3e2ae8baccdbe633cf53f2 | subsection | 9 | 69 | Polling Model and Notation | For tractability, assume that the confusion matrix between successive levels is modeled using the same distribution B in (REF ), i.e,
\forall ~l \in \lbrace 1, \cdots , N\rbrace ,~\mathcal {H}^{l}_{l-1} = B = \mathcal {H}_0.
So
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c07693da2acb26e3a58f9d48a9024c02e8167bbc | subsection | 10 | 69 | Polling Model and Notation | The pollster's observations in case of majority opinion gathering could be modeled as y \in \mathcal {Y} = \lbrace \text{Cand.1 has majority vote}, \text{Cand.2 has majority vote}\rbrace and in case of fraction opinion gathering, the pollster's observations are fractions y \in \mathcal {Y} = \lbrace \text{Fraction cons... | {
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7b02dfcb567df6658796dc5c5224ea47cf60ea21 | subsection | 11 | 69 | Polling Model and Notation | Associated with a stationary (time independent) policy \mu : \Pi (X) \rightarrow \mathcal {U} and initial belief \pi _0 \in \Pi (X), is the infinite horizon discounted cost :
J_{\mu }(\pi _0;\theta ) = \mathbb {E}_{\mu } \lbrace \sum _{k=0}^{\infty } \rho ^k C(\pi _k,u_k = \mu (\pi _k)) \rbrace .
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0a35babb7a82840c923647a7b07c8a72e4881eb5 | subsection | 12 | 69 | Polling Model and Notation | The pollster employs the control u_k = \mu ^*(\pi _{k-1}) to obtain opinions (y_k \in \mathcal {Y}) from the nodes, and then updates the belief \pi _{k-1} \rightarrow \pi _k about the underlying state x_k \in \mathcal {X} using (REF ). Theorem REF below provides sufficient conditions on the observation distribution of ... | {
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0d9cfbc4da0c621b74b3d104e7e0c1775fef99b7 | subsection | 13 | 69 | Polling Model and Notation | With a slight abuse of notation in (REF ), let O_i(u) denote the i^{th} row of the observation likelihood matrix O(u). In words, O_i(u) is the distribution over the observation alphabet \mathcal {Y} conditional on the state x = i.
Rényi Divergence: For an observation likelihood O(u), the Rényi Divergence of order \alph... | {
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103c663d7d8a6dcef14ca627e21733d0ccea8501 | subsection | 14 | 69 | Polling Model and Notation | \subsubsection {Intent Polling Costs}
The instantaneous cost in the adaptive intent polling problem consists of two components-- the measurement cost and the entropy cost (uncertainty in the state estimate):
\begin{}
\item [i.)] \textit {\underline{Measurement Cost}}: Let u \in \lbrace 1,2, \cdots ,~U\rbrace model the ... | {
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5dc6ff52ae5057d9de270c63261f62cfc59707e6 | subsection | 15 | 69 | Polling Model and Notation | Let f_u(z) = \sum _{l=0}^{N} \beta ^{(u)}_l z^{l} denote the polynomial corresponding to the polling policy \beta ^{(u)}. For an opinion distribution B (defined in (REF )), let the matrix polynomials be f_u(B)~\forall u \in \mathcal {U}.
Theorem 4 (Adaptive Intent Polling)
Consider the adaptive intent polling problem ... | {
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} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
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b7b8bcf9bf55977247fa092718611294e4150152 | subsection | 16 | 69 | Polling Model and Notation | Matrix polynomials and Blackwell Dominance
Let \mathcal {P}_N = \lbrace h | h(z) = \sum _{i=0}^N \beta _i z^i,~\sum _{i=0}^N \beta _i = 1,~\beta _i \ge 0 \rbrace denote the collection of all polynomials with co-efficients that are a convex combination.
Proposition 1
Let Q be a stochastic matrix.
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d877eb4a980e945730bbbbeada25f01017526054 | subsection | 17 | 69 | Polling Model and Notation | From Corollary REF , the Hurwitz polynomial channels are ordered such that the channel that is a sub channel of the other results in a larger reduction in uncertainty on the state.
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Dominance | [
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c8569ac15ecdffdf500bee1ff9cdb31108001af4 | subsection | 18 | 69 | Polling Model and Notation | The instantaneous cost C(\pi ,u) in (REF ) incurred by the pollster in case of adaptive expectation polling is thus given as:
C(\pi ,u) = S(u) + \eta _2(\pi ,u)
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Dominance | [
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7343bfda437d1f530983b108e4514726ad1e0516 | subsection | 19 | 69 | Polling Model and Notation | It is easiest (see Sec.) to poll nodes at level N, so a convenient choice is O(u) = B_{N+1}^{l_u/N+1}.
Fractional Exponents of Stochastic Matrices and Blackwell Dominance
For any ultrametric matrix Q, the K^{th} root, Q^{1 / K}, is also stochastic for any positive integer K; see .
Proposition 2
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Dominance | [
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b15582b0289bb49d9b29518edc74000af65b4b84 | subsection | 20 | 69 | Polling Model and Notation | From Corollary REF , the ultrametric channels are ordered such that the information of nodes at Level 0, for example, revealed by the nodes at Level N (\ne 0) result in a larger reduction in uncertainty on the state, than opinions from nodes at Level N (\ne 0).
Adaptive Friendship Polling
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Dominance | [
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a1c52d5bdbeba846ce291e517f013d9a0d61b4dd | subsection | 21 | 69 | Polling Model and Notation | Channels specified by multinomial distributions model the likelihood of opinion counts in favor of different states from different nodes at the same level.
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26769b692c8b75f566ce63e7efd34ce96e1b973c | subsection | 22 | 69 | Polling Model and Notation | (specifically, Algorithm REF ) we will see how to obtain (approximate) Blackwell dominance of observation distributions using Le Cam deficiency if they are not Blackwell comparable a priori. for an opinion distribution B (defined in (REF )), the opinion fractions corresponding to choosing different levels (polling acti... | {
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Dominance | [
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3a210c5cc1fdf5b8a78a6c158dc1ebd00c8f4c30 | subsection | 23 | 69 | Polling Model and Notation | This section discusses approximate Blackwell dominance and the performance loss due to this approximation.
Le Cam Deficiency
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0ce75a57ae4ed92c1bb5df762c4a89aae605290e | subsection | 24 | 69 | Polling Model and Notation | Let J_{\mu ^*(\gamma )}(\pi ;\theta ) and J_{\mu ^*(\gamma )}(\pi ;\gamma ) be defined as in (REF ), and denote the cumulative costs incurred by the two models \theta and \gamma respectively, when using the polling policy \mu ^*(\gamma ). Let J_{\mu ^*(\theta )}(\pi ;\theta ) and J_{\mu ^*(\theta )}(\pi ;\gamma ) be de... | {
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105ff32588f6e77ec6d712c0fb841a37497508cb | subsection | 25 | 69 | Polling Model and Notation | Let the true POMDP model be \theta = (P,O(1),O(2),C) and the approximation be \gamma = (P,O(1),\hat{O}(2),C). Let \mu (\cdot ;\gamma ) denote the policy parameterized by the approximate model \gamma .
Proposition 4 (Adaptive Expectation v/s Intent)
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689f75cdbb509cf67685131f019eb21328fa731d | subsection | 26 | 69 | Polling Model and Notation | Then
J_{\mu _1^*(\theta _1)}(\pi ;\theta _1) \le J_{\mu _2^*(\theta _2)}(\pi ;\theta _2).
Here O^{(1)} \succeq _B O^{(2)} denotes O^{(1)}(u) \succeq _B O^{(2)}(u)~\forall ~u\in \mathcal {U}.
Discussion: The proof of Theorem REF follows from arguments similar to Theorem 14.8.1 in , and is omitted. Since the observati... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02055574",
"end": 301,
"openalex_id": "https://openalex.org/W2065087844",
"raw": "V. Krishnamurthy, Partially Observed Markov Decision Processes. Cambridge University Press, 2016.",
"source_ref_id": "11e6b835558c8897f3331eda... | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
"cs.SI"
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4553b994ef731628cd8b3cf7130aef8752097596 | subsection | 27 | 69 | Polling Model and Notation | (ii) Define the following discounted cost
\begin{aligned}\tilde{J}_{\mu ^*}(\pi _0) = \mathbb {E}\left\lbrace \sum _{k=1}^{\infty } \rho ^{k-1} \tilde{C}\left(\pi _k, {\mu ^*(\pi _k)}\right)\right\rbrace ,~
\text{where}, \\
\tilde{C}\left(\pi , \mu ^*(\pi )\right) = {\left\lbrace \begin{array}{ll} {C}\left(\pi ,1\right... | {
"cite_spans": []
} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
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dd54a0197fe5d48bc7b932b6b29407845c082cf4 | subsection | 28 | 69 | Polling Model and Notation | First, a sample of 30 recent comedy movies were selected. Depending on their box-office revenues, each of these movies were assigned a state from the state-space \mathcal {X}=\lbrace \text{High}, \text{Medium}, \text{Low} \rbrace . For each of these movies, YouTube comments on their trailers that expressed personal opi... | {
"cite_spans": []
} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
"cs.SI"
] | 2,018 | en | Computer Science | [
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d6b208d3f77cff99fabccfde8678f264ba78ddbb | subsection | 29 | 69 | Polling Model and Notation | RandomThe matrices are generated by stochastic simulation as follows: twenty (1 \times 20) probability vectors were simulated from the Dirichlet distribution on a 19 dimensional unit simplex and stacked as rows. stochastic matrices of size 20 \times 20 were generated for the transition probability matrix P and the obse... | {
"cite_spans": []
} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
"cs.SI"
] | 2,018 | en | Computer Science | [
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0... | |
20a112c9334ee9a77daf615daa7926c9bf041dc4 | subsection | 30 | 69 | Polling Model and Notation | Finally, the results and the performance of the myopic polling policy was illustrated on a dataset from YouTube.
Proofs
Proof of Theorem REF:
Denote by y^{(u)} as the observations recorded when using action u. Then O(u+1) = O(u) R implies the following
\mathbb {P}\left(y^{(u+1)}|x\right) = \sum _{y^{(u)}}\mathbb {P}... | {
"cite_spans": []
} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
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0... |
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