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This document discusses side-channel attacks on encrypted cloud traffic and challenges in mitigating these attacks. It presents research on how patterns in encrypted traffic sizes and directions can leak users' private information when entering inputs into web applications. Even with encryption, traffic analysis can reveal search queries, medical records, tax details, and other data. The root causes are fundamental characteristics of web apps like low entropy inputs and stateful communications. Effective solutions require understanding each application, as padding policies must be application-specific. The document also discusses using techniques from privacy-preserving data publishing to achieve "ceiling padding", but there are challenges around cost and sequential inputs that require new techniques.
![53
Padding Method
Ceiling padding [15][16]:
Inspired by PPDP: grouping and breaking
Dominant-vector of a padding group:
Size of each group is not less than k, and
every flow-vector in a group is padded to dominant-vector of that group.
Achieves k-
indistinguishability, but
not sufficient if the
adversary possess
prior knowledge.
Random ceiling padding method:
A mechanism M: when responding to an action a (per each user request),
It randomly selects k-1 other actions, and
Pads the flow-vector of action a to be dominant-vector of transient group
(those k actions).
Randomness:
Randomly selects members of transient group from certain candidates
based on certain distributions.
To reduce the cost, change the probability of an action being selected as
a member of transient group.](https://image.slidesharecdn.com/6620handout5o-140728085622-phpapp02/85/6620handout5o-53-320.jpg)
![57
Instantiations of Scheme
Bounded uniform distribution:
ct: cardinality of candidate actions
cl: number of larger candidates
Scheme can be realized in many different ways.
Choose group members from different subsets of candidates and based on
different distributions, in order to reduce costs.
Normal distribution:
µ: mean
σ: standard deviation
0 n
i
action
cl
ct
0 n
i
action
largest smallest largest smallest
probability
[max(0,min(i-cl, |VA|-ct)), min(max(0,min(i-cl, |VA|-ct))+ct, |VA|)]
(return)](https://image.slidesharecdn.com/6620handout5o-140728085622-phpapp02/85/6620handout5o-57-320.jpg)
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- 1. Faculty of Engineering and Computer Science Concordia Institute for Information Systems Engineering Cloud Traffic Security Wen Ming Liu INSE 6620 July 23, 2014
- 2. Agenda 2 Cloud Applications Side-Channel Attacks Challenges and Solutions Ceiling Padding
- 5. Agenda 5 Cloud Applications Side-Channel Attacks Challenges and Solutions Ceiling Padding
- 6. 6 Web-Based Applications untrusted Internet Client Server Encryption “Cryptography solves all security problems!”Really?
- 7. 7 Side-Channel Attack on Encrypted Traffic Internet Client ServerEncrypted Traffic User Input Observed Directional Packet Sizes a: 801→, ←54, ←509, 60→ 00: 812→, ←54, ←505, 60→, 813→, ←54, ←507, 60→ b-byte s-byte Network packets’ sizes and directions between user and a popular search engine By acting as a normal user and eavesdropping traffic with sniffer pro 4.7.5. Collected in May 2012 Indicator of the input itself Fixed pattern: identified input string
- 8. 8 Updated Patterns Dec 2013 Internet Client ServerEncrypted Traffic User Input Observed Directional Packet Sizes a: 590→, 67→, ←60, ←60, ←728, 60→ 00: 590→, 67→, ←60, ←60, ←698, 60→, 590→, 68→, ←60, ←60, ←717, 60→ b-byte s-byte Patterns may change over time, but attacks will still work in similar ways Patterns may be different for different Web applications, but they are always there Indicator of input itself Fixed pattern: identified input string
- 9. 9 To Make Things Worse The “Autocomplete” feature allows adversaries to combine the packets corresponding to multiple keystrokes Web applications are highly interactive User Input Observed Directional Packet Sizes a: 590→, 67→, ←60, ←60, ←728, 60→ 00: 590→, 67→, ←60, ←60, ←698, 60→, 590→, 68→, ←60, ←60, ←717, 60→ b-byte s-byte
- 10. 10 Longer Inputs, More Unique the Patterns S value for each character entered as: a b c d e f g 509 504 502 516 499 504 502 h i j k l m n 509 492 517 499 501 503 488 o p q r s t 509 525 494 498 488 494 u v w x y z 503 522 516 491 502 501 First keystroke: Second keystroke: First Keystroke Second Keystroke a b c d a 509 487 493 501 497 b 504 516 488 482 481 c 502 501 488 473 477 d 516 543 478 509 499 Unique s value 12 out of 1616 out of 16 The unique patterns leak out users’ private information: the input string In reality, it may take more than two keystrokes to uniquely identify an input string.
- 11. Side-Channel Leaks To protect the information in critical applications against network sniffing, a common practice is to encrypt their network traffic. However, as discovered in the research, serious information leaks are still a reality. Even though the communications generated during these state transitions are protected by HTTPS, their observable attributes can still give away the information about the user’s selection. The eavesdropper cannot see the contents, but can observe: number of packets, timing/size of each packet. 11 Slides 11-27 are partially based on: S. Chen, R. Wang, X. Wang, and K. Zhang. Side-channel leaks in web applications: A reality today, a challenge tomorrow. In IEEE Symposium on Security and Privacy’10, pages 191–206, 2010.
- 12. Main Findings Analysis of the side-channel weakness in web applications. Several high-profile and really popular web applications actually disclose surprisingly detailed user’s sensitive information Personal health data, family income, investment details, search queries The root causes of the side-channel information leaks are the fundamental characteristics in today’s web applications. Stateful communication, low entropy input and significant traffic distinctions In-depth study on the challenges in mitigating the threat. Evaluate the effectiveness and the overhead of common mitigation techniques such as packet padding. Show that effective solutions to the side-channel problem have to be application-specific, relying on an in-depth understanding of the application being protected. This suggests the necessity of a significant improvement of the current practice for developing web applications. 12
- 13. Fundamental Characteristics of Web Applications The root causes are some fundamental characteristics in today’s web applications : Low entropy inputs for better interactions. Small input space Autosuggestion, auto-complete Stateful communications. Transitions to next states depend both on the current state and on its input. Although information for each transition may be insignificant, their combination can be really powerful. Significant traffic distinctions. The chance of two different user actions having the same traffic pattern is really small. Such distinctions often come from the objected updated by client- server data exchanges. 13
- 15. Basic of Wi-Fi Encryption Schemes: WEP: susceptible to key-recovery attacks WPA: TKIP (RC4) WPA2: CCMP (128-bit AES block cipher in counter mode) The ciphertext fully preserves the size of its plaintext! 15
- 16. Scenario: search using encrypted Wi-Fi WPA/WPA2. Example: user types “list” on a WPA2 laptop. Consequence: Anybody on the street knows our search queries. Attacker’s effort: linear, not exponential. 821 910 822 931 823 995 824 1007 16
- 17. OnlineHealthA (“A” denoting a pseudonym) A web application by one of the most reputable companies of online services Illness/medication/surgery information is leaked out, as well as the type of doctor being queried. Vulnerable designs Entering health records By typing – auto suggestion By mouse selecting – a tree-structure organization of elements Finding a doctor Using a dropdown list item as the search input 17
- 18. tabs Entering health records: no matter keyboard typing or mouse selection, attacker has a 2000× ambiguity reduction power. Find-A-Doctor: attacker can uniquely identify the specialty. Attacker’s power
- 19. OnlineTaxA It is the online version of one of the most widely used applications for the U.S. tax preparation. Design: a tax-preparation wizard Tailor the conversation based on user’s previous input. The forms that you work on tell a lot about your family Filing status Number of children Paid big medical bill The adjusted gross income (AGI) 19
- 20. Entry page of Deductions & Credits Summary of Deductions & Credits Full credit Not eligible Partial credit All transitions have unique traffic patterns. Consult the IRS instruction: $1000 for each child Phase-out starting from $110,000. For every $1000 income, lose $50 credit. $0 $110000 $150000 Not eligibleFull credit Partial credit (two children scenario) child credit state machine 20
- 21. Entry page of Deductions & Credits Summary of Deductions & Credits Full credit Not eligible Partial credit Even worse, most decision procedures for credits/deductions have asymmetric paths. Eligible – more questions Not eligible – no more question Enter your paid interest $0 $115000 $145000 Not eligibleFull credit Partial credit Student-loan-interest credit 21
- 22. Disabled Credit $24999 Retirement Savings $53000 IRA Contribution $85000 $105000 College Expense $116000 $115000Student Loan Interest $145000 First-time Homebuyer credit $150000 $170000 Earned Income Credit $41646 Child credit * $110000 Adoption expense $174730 $214780 $130000 or $150000 or $170000 … $0 A subset of identifiable AGI thresholds We are not tax experts. OnlineTaxA can find more than 350 credits/deductions. 22
- 23. A major financial institution in the U.S. Which funds you invest? • No secret. • Each price history curve is a GIF image from MarketWatch. • Everybody in the world can obtain the images from MarketWatch. • Just compare the image sizes! OnlineInvestA Your investment allocation • Given only the size of the pie chart, can we recover it? • Challenge: hundreds of pie-charts collide on a same size. 23
- 24. Inference based on the evolution of the pie-chart size in 4-or-5 days The financial institution updates the pie chart every day after the market is closed. The mutual fund prices are public knowledge. ≅ 800 charts ≅ 80 charts ≅ 8 charts 1 chart Sizeofday1 Sizeofday2; Pricesoftheday Sizeofday3; Pricesoftheday Sizeofday4; Pricesoftheday ≅80000charts 24
- 25. Challenging to Mitigate the Vulnerabilities 25
- 26. Traffic differences are everywhere. Which ones result in serious data leaks? Need to analyze the application semantics, the availability of domain knowledge, etc. Hard. Is there a vulnerability-agnostic defense to fix the vulnerabilities without finding them? Obviously, padding is a must-do strategy. We found that even for the discussed apps, the defense policies have to be case-by-case. Why challenging? 26
- 27. See if problem can be solved without analyzing individual application Application-agnostic manner: Padding Rounding: Random padding: Average overhead: Given , reduction power being calculated after padding Universal Mitigation Policies 27 Any Problem?
- 28. Agenda 28 Cloud Applications Side-Channel Attacks Challenges and Solutions Ceiling Padding
- 30. 30 Don’t Forget the Cost (Prefix) char s Value Rounding (ΔΔΔΔ) 64 160 256 (c) c 473 512 480 512 (c) d 477 512 480 512 (d) b 478 512 480 512 (d) d 499 512 640 512 (a) c 501 512 640 512 (b) a 516 576 640 768 Padding Overhead (%) 6.5% 14.1% 13.0% 1 S. Chen, R. Wang, X. Wang, and K. Zhang. Side-channel leaks in web applications: A reality today, a challenge tomorrow. In IEEE Symposium on Security and Privacy’10, pages 191–206, 2010. No guarantee of better privacy at a higher cost ↑ ⇏ privacy ↑ ↑ ⇏ overhead ↑ To make all inputs indistinguishable by rounding will result in a 21074% overhead for a well-known online tax system 1
- 31. Two Conflicting Goals 31 To prevent side-channel attacks, we face two seemingly conflicting goals, Privacy protection: Reduce the differences in packet sizes Cost: Minimize the overhead (communication and processing…) The similar goals may allow us to borrow existing expertise on privacy-preserving data publishing(PPDP). How? Grouping and Breaking Slides 31-45 are partially based on: W. M. Liu, L. Wang, K. Ren, P. Cheng, M. Debbabi, S. Zhu, “PPTP: Privacy-Preserving Traffic Padding in Web-Based Applications,” IEEE Trans. on Dependable and Secure Computing (TDSC),
- 32. Solution: Ceiling Padding 32 473 477 478 (c) c 477 477 478 (c) d 478 499 478 (d) b 499 499 516 (d) d 501 516 516 (a) c 516 516 516 (b) a S Value Padding (Prefix) charOption 1 Option 2 Quasi-ID Function 1 Function 2 Sensitive AttributeGeneralization PPTP: Padding group PPDP: anonymized group PPTP goals: Privacy Cost PPDP goals: Privacy Data utility So we can apply existing techniques in data publication to achieve ceiling padding However, there are a few difference, and hence challenges... Ceiling padding: pad every packet to the maximum size in the group
- 33. 33 PPTP Components Internet Interaction: action a: Atomic user input that triggers traffic A keystroke, a mouse click … action-sequence ܽԦ : A sequence of actions with complete input info Consecutive keystrokes… action-set Ai: Collection of all ith actions in a set of action-seq Observation: flow-vector v: A sequence of flows (directional packet sizes) Triggered an action vector-sequence ݒԦ : A sequence of flow-vectors Triggered by an equal-length action-sequence vector-set Vi: Collection of all ith vectors in a set of vector-seq Vector-Action Set VAi: Pairs of ith actions and corresponding ith flow-vectors User Input Observed Directional Packet Sizes a: 801→, ←54, ←509, 60→ 00: 812→, ←54, ←505, 60→, 813→, ←54, ←507, 60→
- 34. 34 Privacy and Cost k-indistinguishability: Given a vector-action set VA Padding group: any S⊆VA satisfying all the pairs in S have identical flow-vectors and no S’ ⊃S can satisfy this property We say VA satisfies k-indistinguishability (k is an integer) if the cardinality of every padding group is no less than k Goal of privacy protection: Upon observing any flow-vector in the traffic, the eavesdropper cannot determine which action in the table (vector-action set) has triggered this flow- vector. l-diversity: Address the cases that: No all inputs should be treated equally in padding (for example, some statistical information regarding the likelihood of different inputs may be publicly known).
- 35. 35 Privacy and Cost Vector-distance: Given two equal-length flow-vectors v1 and v2, vector-distance is the total number of bytes different in the flows: ݏ݅݀ݒ ݒଵ, ݒଶ = ∑ (|ݏଵ − ݏଶ|) ௩భ ୀଵ . Padding cost: Given a vector-set V, the padding cost is the sum of the vector- distances between each flow-vector in V and its countpart after padding. Processing cost: Given a vector-set V, the processing cost is the number of flows in V which corresponding packets should be padded.
- 36. Agenda 36 Cloud Applications Side-Channel Attacks Challenges and Solutions Ceiling Padding
- 37. Challenge 1 37 473 477 478 (c) c 477 477 478 (c) d 478 499 478 (d) b 499 499 516 (d) d 501 516 516 (a) c 516 516 516 (b) a S Value Padding (Prefix) charOption 1 Option 2 Differences and challenges: Data utility measures & padding cost Traffic padding: cost of option 1 is worse than that of option 2 Data publication: utility of function 1 is better than that of option 2
- 38. 38 Challenge 2 Recall that adversaries may combine multiple keystokes Example: One obvious, but invalid solution: Pad every keystroke (separately) Another obvious, but invalid solution: Pad on the whole string! First Keystroke Second Keystroke a b c d a 487 493 501 497 b 516 488 482 481 c 501 488 473 477 d 543 478 509 499 First Keystroke Second Keystroke a b c d a 487 493 501 497 b 516 488 482 481 c 501 488 473 477 d 543 478 509 499 First Keystroke Second Keystroke a b c d a … … 501 … b … 488 … … c 501 488 … … d … … … … First Keystroke Second Keystroke a b c d a 509 … … 501 … b 504 … 488 … … c 502 501 488 … … d 516 … … … … First Keystroke Second Keystroke a b c d a 516 … … 501 … b 504 … 488 … … c 504 501 488 … … d 516 … … … … Strings 1st keystroke 2nd keystroke ac 509 501 ca 502 501 ad 509 497 dd 516 499 … … …
- 39. 39 PPTP - Overview of Algorithms Intention: To demonstrate the existence of abundant possibilities in approaching PPTP issue, and not to design an exhaustive list of solutions. Design three algorithms for partitioning inputs into padding groups. Main difference: the algorithms handle in increasingly complicated cases . Computational complexity: svsdSimple algorithm: Ο ݈݊݊݃ svmdGreedy algorithm: Ο(݊ ⨯ ݊2) (worse case), Ο(݊ ⨯ ݈݊)݊݃ (average case) mvmdGreedy algorithm:Ο(݊ ⨯ ݊2) (worse case), Ο(݊ ⨯ ݈݊)݊݃ (average case)
- 40. 40 Experiment Settings Collect data from two real-world web applications: A popular search engine (users’ search keyword needs to be protected) Collect flow-vectors for query suggestion widget for all possible combinations of four letters by crafting requests to simulate the normal AJAX connection request. Authoritative drug information system from national institute (user’s possible health information needs to be protected) Collect vector-action set for all the drug information by mouse-selecting following the application’s three-level tree-hierarchical navigation.
- 41. 41 Overhead - Padding Cost The padding cost against k: To compare to rounding, Δ=512 (engineB) and Δ=5120 (drugB) which achieves only 5-indistinguishility. Our algorithms have less padding cost in both cases. Observe that our algorithms are superior specially when the number of flow-vectors is larger.
- 42. 42 Overhead – Execution Time Generate n-size flow data by synthesizing n/|VA| copies of engineB and drugB. The computation time of mvmdGreedy increases slowly with n. Practically efficient (1.2s for 2.7m flow-vectors), Require slightly more overhead than rounding when it is applied to a single Δ value. The computational time of mvmdGreedy against privacy property k A tighter upper bound: Ο(݊ ⨯ ݊ ⨯ 2݇ ⨯ λ) (worse case), Ο(݊ ⨯ ݊ ⨯ log (2݇ ⨯ λ)) (average case) The computation time increases slowly with k for engineB, and decreases slowly for drugB.
- 43. 43 Overhead – Processing Cost An application can choose to incorporate the padding at different stage of processing a request, however, we must minimize the number of packets to be padded. Pad the flow-vectors on the fly, Modify the original data beforehand. The processing cost against k: Rounding must pad each flow-vector regardless of the k’s and the applications, while our algorithms have much less cost for engineB and slightly less for drugB.
- 44. Extension 44 Adapt l-diversity to address cases that: No all inputs should be treated equally in padding. Model: Catch the information about the inequality. Algorithms: Need additional constraints on partition.
- 45. 45 Experiments Collect data from two real-world web applications: Another popular search engine (users’ search keyword needs to be protected) Authoritative patent information system from national institute (company’s patent interest needs to be protected)
- 46. 46 Challenge 3: Ceiling Padding Defeated Condition s Value Rounding (ΔΔΔΔ) Ceiling Padding112 144 176 Cancer 360 448 432 528 360 Cervicitis 290 336 432 352 360 Cold 290 336 432 352 290 Cough 290 336 432 352 290 Padding Overhead (%) 18.4% 40.5% 28.8% 5.7% 2-indistinguishability Observation: a patient received a 360-byte packet after login Cancer? Cervicitis? ⇒ 50% , 50% Extra knowledge: this patient is a male Cancer? Cervicitis? ⇒ 100% , 0% Facts (Ceiling Padding): Slides 46-58 are partially based on: W. M. Liu, L. Wang, K. Ren, M. Debbabi,, “Background Knowledge- Resistant Traffic Padding for Preserving User Privacy in Web-Based Applications,” Proc. The 5th IEEE International Conference and on Cloud Computing Technology and Science (IEEE CloudCom 2013),
- 47. 47 Solution: Add Randomness Condition s Value Cancer 360 Cervicitis 290 Cold 290 Cough 290 Cancerous Person 360 360 360 Random Ceiling Padding Instead of deterministically forming padding groups, the server will randomly (at uniform, in this example) selects one out of the other three conditions (together with the real condition) to form a padding group for ceiling padding. Always receive a 360-byte packet Condition s Value Cancer 360 Cervicitis 290 Cold 290 Cough 290 Cervicitis Patient 360 290 66.7%: 290-byte packet 33.3%: 360-byte packet 290
- 48. 48 Better Privacy Protection Diseases s Value Cancer 360 Cervicitis 290 Cold 290 Cough 290 Can tolerate adversaries’ extra knowledge Suppose an adversary knows a patient is male and he saw s = 360 Patient Has Server Selects Cancer Cervicitis Cancer Cold Cancer Cough Cold Cancer Cough Cancer The adversary now can only be 60%, instead of 100%, sure that patient has Cancer. Cost is not necessarily worse In this example, these two methods actually lead to exactly the same expected padding and processing costs
- 49. Analysis 49 Scenario: Algorithm: randomness drawn from uniform distribution Data: action-sequence and flow-vector are of length one Analysis of privacy preservation: Lemma 6.1 Analysis of costs: Lemma 6.2, Lemma 6.3
- 50. Model, Scheme and Experiment 50 Model: component, privacy, method, cost Scheme: Main idea: Server randomly selects members to form the group. Different choices of random distribution lead to different algorithms. Two instantiations of scheme: Bounded uniform distribution; Normal distribution Computation complexity: O(k) Experiment: Data from real-world web applications Low overheads and high uncertainty
- 51. 51 Privacy Properties k-Indistinguishability: For any flow-vector, at least k different actions can trigger it. Given vector-action set VA, padding algorithm M, range Range(M,VA) ∀ ݒ ∈ ܴܽ݊݃݁ ,ܯ ܸܣ , ܽ: Pr (ܯିଵ ݒ = ܽ > 0 ∧ ܽ ∈ ܣ ≥ ݇ Model the privacy requirement of a traffic padding from two perspectives Uncertainty: Apply the concept of entropy in information theory to quantify an adversary’s uncertainty about the real action performed by a user. Given vector-action sequence ܸ,ܣ padding algorithm M ߮ ,ݒ ܸ,ܣ ܯ = − Pr (ܯିଵ ݒ = ܽ logଶ Pr (ܯିଵ ݒ = ܽ ∈ ߶ ܸ,ܣ ܯ = ߮(,ݒ ܸ,ܣ )ܯ × Pr (ܯ ܣ = )ݒ ௩∈ோ(ெ,) Φ ܸ,ܣ ܯ = ෑ ߶(ܸ,ܣ )ܯ ∈
- 52. 52 Privacy Properties (cont.) δ-uncertain k-Indistinguishability: An algorithm M give δ-uncertain k-Indistinguishability for a vector-action sequence ܸܣ if M w.r.t. any ܸܣ ∈ ܸܣ satisfies k-indistinguishability, and The uncertainty of M w.r.t. ܸܣ is not less than δ.
- 53. 53 Padding Method Ceiling padding [15][16]: Inspired by PPDP: grouping and breaking Dominant-vector of a padding group: Size of each group is not less than k, and every flow-vector in a group is padded to dominant-vector of that group. Achieves k- indistinguishability, but not sufficient if the adversary possess prior knowledge. Random ceiling padding method: A mechanism M: when responding to an action a (per each user request), It randomly selects k-1 other actions, and Pads the flow-vector of action a to be dominant-vector of transient group (those k actions). Randomness: Randomly selects members of transient group from certain candidates based on certain distributions. To reduce the cost, change the probability of an action being selected as a member of transient group.
- 54. 54 Cost Metrics Expected processing cost: How many flow-vectors need to be padded ݏܿݎ ܸ,ܣ ܯ = ∑ ∑ ୰ ெ()ஷ௩(ೌ,ೡ)∈ೇಲೇಲ∈ೇಲ ∑ ,௩ : ,௩ ∈ ೇಲ∈ೇಲ Expected padding cost: The proportion of packet size increases compared to original flow-vectors Given vector-action sequence ܸ,ܣ padding algorithm M, (ܽ, :)ݒ ݏܿ ܽ, ܸ,ܣ ܯ = Pr ܯ ܽ = ݒᇱ × ݒᇱ − ݒ ௩ᇱ∈ோ(ெ,) ଵ ܸ:ܣ ݏܿ ܸ,ܣ ܯ = (,ܽ(ݏܿ ܸ,ܣ ))ܯ ,௩ ∈ ܸ:ܣ ݏܿ ܸ,ܣ ܯ = (,ܣܸ(ݏܿ ))ܯ ∈
- 55. 55 Overview of Scheme Main idea: To response a user input, server randomly selects members to form the group. Different choices of random distribution lead to different algorithms. Goal: The privacy properties need to be ensured. The costs of achieving such privacy protection should be minimized. Two stage scheme: Stage 1: derive randomness parameters, one-time, optimization problem; Stage 2: form transient group, real-time.
- 56. 56 Scheme (cont.) Computational complexity: O(k) Stage 1: pre-calculated only once Stage 2: select k-1 random actions without duplicate, O(k). Discussion on privacy: The adversary cannot collect vector-action set even acting as normal user, ܸܣ = 100, ݇ = 20, ݉ݎ݂݅݊ݑ ⟹ )݊݅ݐܿܽ ݊ܽ( ݏݑݎ݃ ݐ݊݁݅ݏ݊ܽݎݐ = 99 19 ≈ 2 Approximate the distribution is hard: all users share one random process. Discussion on costs: Deterministically incomparable with those of ceiling padding.
- 57. 57 Instantiations of Scheme Bounded uniform distribution: ct: cardinality of candidate actions cl: number of larger candidates Scheme can be realized in many different ways. Choose group members from different subsets of candidates and based on different distributions, in order to reduce costs. Normal distribution: µ: mean σ: standard deviation 0 n i action cl ct 0 n i action largest smallest largest smallest probability [max(0,min(i-cl, |VA|-ct)), min(max(0,min(i-cl, |VA|-ct))+ct, |VA|)] (return)
- 58. 58 Uncertainty and Costs v.s. k The padding and processing costs of all algorithms increase with k, while TUNI and NORM have less than those of SVMD. Our algorithms have much larger uncertainty for Drug and slightly larger for Engine.