Section4

section4

This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. Authors: (1) Pascal Baumann, Max Planck Institute for Software Systems (MPI-SWS), Germany; (2) Rupak Majumdar, Max Planck Institute for Software Systems (MPI-SWS), Germany; (3) Ramanathan S. Thinniyam, Max Planck Institute for Software Systems (MPI-SWS), Germany; (4) Georg Zetzsche, Max Planck Institute for Software Systems (MPI-SWS), Germany. Table of Links Introduction Dynamic Networks of Concurret Pushdown Systems (DCPS) Warm-up: Non-Termination Fair Non-Termination From Progressive Runs for VASSB to Reachability Starvation Conclusion, Acknowledgment & References A. Strengthening Fairness to Progressive Runs B. Proofs from Section 4.1 C. Proofs for Section 5 D. Proofs for Section 5.3 E. Proofs for Section 6 B PROOFS FROM SECTION 4.1 Intuitively, the edge in (1) spawns the initial thread, the edges in (2) remove a thread with csnumber 0 when it becomes active for the first time and the edges in (3) convert it to a balloon, the edges in (4) apply the spawns during the 푗th segment of a thread by transferring them from the balloon to the regular places, the edges in (5) and (6) finalize this application process with the edges in (6) handling the special case of the 퐾th segment of a thread, the edges in (7) burst balloons containing no more spawns, and the edges in (8) initialize the application of spawns for a thread with cs-number ≥ 1 that is already being represented by a balloon. Lemma B.1. The DCPS A has an infinite, progressive, K-context switch bounded run iff the VASSB V has an infinite progressive run. Proof. For the only if direction let p be an infinite, progressive, K-context switch bounded run of A. We want to decompose p into parts corresponding to a single thread each. To do this we look at the continuous segments of p where a thread is active, starting from the first configuration with an active thread and ending each segment at a configuration with no active thread, upon which a new segment begins at the very next configuration. We can now group segments together into executions of single threads: • Start with the earliest not yet grouped segment where the active thread has context switch number 0. Note the local configuration that gets added to the bag at the end of this segment. • Take the very next ungrouped segment in p where the active thread at the beginning matches the previously noted local configuration. Again, note the local configuration that gets added to the bag at the end of this segment. • Repeat the previous step until an empty stack occurs. The constructed run of V is also sound: For each segment of p it has a sequence of transitions that make the same state change as that segment, and also apply the same spawns. The latter is due to initially choosing the contents during the creation of a balloon correctly. A token representing the initial thread is also created in the beginning of the run. Since the transitions from one segment to the next require the same resumption rules Δr as in p (see edges (2) and (8)) these are also possible in V. For the if direction we do a very similar construction, but backwards. Starting with an infinite progressive run of V from c0, we decompose it into segments that end in a state in G. Afterwards we group those segments together by matching the local balloon configurations in the same way we did for local thread configurations before. Then by construction of Lt we know that there is a thread execution of type t that makes the same spawns in each segment as the group of one balloon does. Those executions will all terminate after K context switches, guaranteeing progressiveness, and we can interleave them in the same way as the balloon segments to form an infinite progressive run of A.

Context-Bounded Verification of Liveness: B Proofs from Section 4.1

Zoonomia, Vol. I Or, the Laws of Organic Life  by Erasmus Darwin is part of HackerNoon’s Book Blog Post series. You can jump to any chapter in this book here: [LINK TO TABLE OF LINK]. Section I: Laws of Animal Causation SECTION IV. LAWS OF ANIMAL CAUSATION. . The fibres, which constitute the muscles and organs of sense, possess a power of contraction. The circumstances attending the exertion of this power of CONTRACTION constitute the laws of animal motion, as the circumstances attending the exertion of the power of ATTRACTION constitute the laws of motion of inanimate matter. I . The spirit of animation is the immediate cause of the contraction of animal fibres, it resides in the brain and nerves, and is liable to general or partial diminution or accumulation. II . The stimulus of bodies external to the moving organ is the remote cause of the original contractions of animal fibres. III . A certain quantity of stimulus produces irritation, which is an exertion of the spirit of animation exciting the fibres into contraction. IV . A certain quantity of contraction of animal fibres, if it be perceived at all, produces pleasure; a greater or less quantity of contraction, if it be perceived at all, produces pain; these constitute sensation. V . A certain quantity of sensation produces desire or aversion; these constitute volition. VI . All animal motions which have occurred at the same time, or in immediate succession, become so connected, that when one of them is reproduced, the other has a tendency to accompany or succeed it. When fibrous contractions succeed or accompany other fibrous contractions, the connection is termed association; when fibrous contractions succeed sensorial motions, the connexion is termed causation; when fibrous and sensorial motions reciprocally introduce each other, it is termed catenation of animal motions. All these connections are said to be produced by habit, that is, by frequent repetition. These laws of animal causation will be evinced by numerous facts, which occur in our daily exertions; and will afterwards be employed to explain the more recondite phænomena of the production, growth, diseases, and decay of the animal system. VII About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Darwin, Erasmus, 2005. Zoonomia, Vol. Or, the Laws of Organic Life. Urbana, Illinois: Project Gutenberg. Retrieved May 2022 from https://www.gutenberg.org/files/15707/15707-h/15707-h.htm#sect_IV This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org, located at https://www.gutenberg.org/policy/license.html.

Section 4 - Laws of Animal Causation

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments 4.2. Advantages of (K-)SIF over FIF This part highlights the advantages of our class of algorithms over Functional Isolation Forest. We provide numerical experiments supporting our approach’s appealing properties, such as the robustness to noisy data and detecting order variation changes. Robustness to Noise. We now explore the sensitivity of our class of algorithms to noisy data. We also provide a comparison with Functional Isolation Forest. To that end, we simulate a dataset of 500 ‘relatively smooth’ standard Brownian motion paths with µ = 0 and σ = 0.05. We also simulate 50 standard Brownian motion paths with µ = 0 and σ ∈ [0.05, 0.2]. Depending on the level of noise present in the last 50 curves (i.e., standard deviation here), these paths may represent normal, noisy and abnormal data. For example, 0.05 (corresponding to 0 noise level) is normal data, while 0.2 (corresponding to 0.15 noise level) is abnormal. We perform K-SIF and FIF with both ‘Brownian’ and ‘Cosine’ dictionaries and report the AUC under the ROC curves by considering the 50 curves as the second class in Figure 2. An AUC of 1 means that the 50 curves of the second class are identified as anomalies, while an AUC of 0.5 indicates that the algorithm cannot distinguish them from normal data. According to the findings provided in Section 4.1, the parameters for K-SIF have been chosen such that the number of split window sizes is equal to 10 while the depth of the signature k = 2. We can observe that FIF, regardless of the dictionary used, is highly sensitive to noisy data and considers it abnormal, even with a minimal noise level. In contrast, K-SIF is way more robust to such data and requires higher noise levels to start viewing this data as anomalies. (K-)SIF detects Swap Order Variation Changes. The signature method considers the order of occurring events in functional data. To investigate this phenomenon with our proposed class of algorithms, we define a synthetic dataset of 100 smooth functions given by with t ∈ [0, 1] and q equispaced in [1, 1.4]. Then, we simulate the occurrences of events by adding Gaussian noise on different portions of the functions. We randomly select 90% of them and add Gaussian values on a sub-interval, i.e., where ε(t) ∼ N (0, 0.8). We consider the 10% remaining as abnormal by adding the same ‘events’ on another subinterval compared to the first one, i.e., where ε(t) ∼ N (0, 0.8). See Figure 8 in the Appendix for a dataset visualization. To summarize, we have constructed two identical events occurring at different parts of the functions, leading to isolating anomalies. As presented in the introduction, this class of anomalous data are amongst the most challenging to identify. We compute SIF, K-SIF and FIF with Brownian and Cosine dictionaries on these simulated datasets. As indicated in Section 4.1, for (K-)SIF, we choose ω = 10, the number of split windows, and k = 2, the depth of the signature. In Figure 3, we report boxplots of the anomaly score returned by the algorithms for the normal data in purple and abnormal data in yellow. While this could appear a very simple task, in practice, it is highly challenging for an AD algorithm to differentiate such classes of curves. The introduction of the signature method should tackle precisely this type of scenario, since taking into account the order of the events. Using both dictionaries, FIF fails to detect this anomaly as it is not designed to handle these type of phenomena. In contrast, K-SIF and SIF produce significantly distinguished scores between normal and abnormal data, efficiently classifying the second data class as anomaly. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments 4.2. Advantages of (K-)SIF over FIF This part highlights the advantages of our class of algorithms over Functional Isolation Forest. We provide numerical experiments supporting our approach’s appealing properties, such as the robustness to noisy data and detecting order variation changes. Robustness to Noise. We now explore the sensitivity of our class of algorithms to noisy data. We also provide a comparison with Functional Isolation Forest. To that end, we simulate a dataset of 500 ‘relatively smooth’ standard Brownian motion paths with µ = 0 and σ = 0.05. We also simulate 50 standard Brownian motion paths with µ = 0 and σ ∈ [0.05, 0.2]. Depending on the level of noise present in the last 50 curves (i.e., standard deviation here), these paths may represent normal, noisy and abnormal data. For example, 0.05 (corresponding to 0 noise level) is normal data, while 0.2 (corresponding to 0.15 noise level) is abnormal. Robustness to Noise. We perform K-SIF and FIF with both ‘Brownian’ and ‘Cosine’ dictionaries and report the AUC under the ROC curves by considering the 50 curves as the second class in Figure 2. An AUC of 1 means that the 50 curves of the second class are identified as anomalies, while an AUC of 0.5 indicates that the algorithm cannot distinguish them from normal data. According to the findings provided in Section 4.1, the parameters for K-SIF have been chosen such that the number of split window sizes is equal to 10 while the depth of the signature k = 2. We can observe that FIF, regardless of the dictionary used, is highly sensitive to noisy data and considers it abnormal, even with a minimal noise level. In contrast, K-SIF is way more robust to such data and requires higher noise levels to start viewing this data as anomalies. (K-)SIF detects Swap Order Variation Changes. The signature method considers the order of occurring events in functional data. To investigate this phenomenon with our proposed class of algorithms, we define a synthetic dataset of 100 smooth functions given by (K-)SIF detects Swap Order Variation Changes. with t ∈ [0, 1] and q equispaced in [1, 1.4]. Then, we simulate the occurrences of events by adding Gaussian noise on different portions of the functions. We randomly select 90% of them and add Gaussian values on a sub-interval, i.e., where ε(t) ∼ N (0, 0.8). We consider the 10% remaining as abnormal by adding the same ‘events’ on another subinterval compared to the first one, i.e., where ε(t) ∼ N (0, 0.8). See Figure 8 in the Appendix for a dataset visualization. To summarize, we have constructed two identical events occurring at different parts of the functions, leading to isolating anomalies. As presented in the introduction, this class of anomalous data are amongst the most challenging to identify. We compute SIF, K-SIF and FIF with Brownian and Cosine dictionaries on these simulated datasets. As indicated in Section 4.1, for (K-)SIF, we choose ω = 10, the number of split windows, and k = 2, the depth of the signature. In Figure 3, we report boxplots of the anomaly score returned by the algorithms for the normal data in purple and abnormal data in yellow. While this could appear a very simple task, in practice, it is highly challenging for an AD algorithm to differentiate such classes of curves. The introduction of the signature method should tackle precisely this type of scenario, since taking into account the order of the events. Using both dictionaries, FIF fails to detect this anomaly as it is not designed to handle these type of phenomena. In contrast, K-SIF and SIF produce significantly distinguished scores between normal and abnormal data, efficiently classifying the second data class as anomaly. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

How (K-)SIF Outperforms FIF in Noise Robustness and Event Order Detection

Authors:
(1) Yuwei Guo, The Chinese University of Hong Kong;
(2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author;
(3) Anyi Rao, Stanford University;
(4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory;
(5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory;
(6) Yu Qiao, Shanghai Artificial Intelligence Laboratory;
(7) Maneesh Agrawala, Stanford University;
(8) Dahua Lin, Shanghai Artificial Intelligence Laboratory;
(9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction 2 Work Related 3 Preliminary AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.3 Ablative Study 5.4 Controllable Generation 6 Conclusion 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References 4.4 ANIMATEDIFF IN PRACTICE We elaborate on the training and inference here and put the detailed configurations in supplementary materials. It’s worth noting that when training the domain adapter, the motion module, and the MotionLoRA, parameters outside the trainable part remain frozen. Inference. At inference time (Fig. 2), the personalized T2I model will first be inflated in the same way discussed in Section 4.2, then injected with the motion module for general animation generation, and the optional MotionLoRA for generating animation with personalized motion. As for the domain adapter, instead of simply dropping it during the inference time, in practice, we can also inject it into the personalized T2I model and adjust its contribution by changing the scaler α in Eq. (4). An ablation study on the value of α is conducted in experiments. Finally, the animation frames can be obtained by performing the reverse diffusion process and decoding the latent codes. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Authors: Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Work Related 2 Work Related 3 Preliminary 3 Preliminary AnimateDiff AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.2 Qualitative Comparison 5.3 Ablative Study 5.3 Ablative Study 5.4 Controllable Generation 5.4 Controllable Generation 6 Conclusion 6 Conclusion 7 Ethics Statement 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References 8 Reproducibility Statement, Acknowledgement and References 4.4 ANIMATEDIFF IN PRACTICE We elaborate on the training and inference here and put the detailed configurations in supplementary materials. It’s worth noting that when training the domain adapter, the motion module, and the MotionLoRA, parameters outside the trainable part remain frozen. Inference. At inference time (Fig. 2), the personalized T2I model will first be inflated in the same way discussed in Section 4.2, then injected with the motion module for general animation generation, and the optional MotionLoRA for generating animation with personalized motion. As for the domain adapter, instead of simply dropping it during the inference time, in practice, we can also inject it into the personalized T2I model and adjust its contribution by changing the scaler α in Eq. (4). An ablation study on the value of α is conducted in experiments. Finally, the animation frames can be obtained by performing the reverse diffusion process and decoding the latent codes. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

AnimateDiff in the Wild

Testing ADA on Synthetic and Real-World Data

Authors:
(1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch);
(2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch);
(3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.2 Anchor Regression 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation B Experiments 4.4 Out-of-distribution Robustness In this section, we evaluate the performance of ADA and compare it to prior approaches on tasks involving out-of-distribution robustness. We use the same datasets as [49] and closely follow their experimental setup. Data: We use four of the five out-of-distribution datasets used in [49]. First, we use RCFashionMNIST (RCF-MNIST) [49], which is a synthetic modification of Fashion-MNIST that models subpopulation shifts. Second, we investigate domain shifts using Communities and Crime (Crime) [12], SkillCraft1 Master Table (SkillCraft) [12] and Drug-target Interactions (DTI) [17] all of which are tabular datasets. For Crime, we use state identification, in SkillCraft we use "League Index", which corresponds to different levels of competitors, and in DTI we use year, as domain information. We split the datasets into train-, validation- and test data based on the domain information resulting in domain-distinct datasets. We provide a detailed description of datasets in Appendix B.4. Due to computational complexity, we could not establish a fair comparison on the satellite image regression dataset [23] (the fifth dataset in [49]), so we report some exploratory results in Appendix B.4. Models and comparisons: As detailed in Section 4.3. Additionally, we use a ResNet-18 [15] for RCF-MNIST and DeepDTA [38] for DTI, as proposed in [49]. Results: We report the RMSE and the "worst" domain RMSE, which corresponds to the worst within-domain RMSE for out-of-domain test sets in Table 2. Similar to [49], we report the R value for the DTI dataset (higher values suggest a better fit of the regression model). For full transparency, in the last row, we copy the results from [49]. We can assess that ADA is competitive with C-Mixup and the other data augmentation strategies. Under the same conditions (split of data and Neural network structure), ADA is superior to C-Mixup. But, the degree of improvement is marginal on some datasets and as we show in the last row, we could not fully replicate their results. ADA is significantly better than C-Mixup and other strategies on the SkillCraft data, in which ADA reduces the error by around 15% compared to the ERM solution. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Authors: Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.1 Data Augmentation 2.2 Anchor Regression 2.2 Anchor Regression 3 Anchor Data Augmentation 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation A Additional information for Anchor Data Augmentation B Experiments B Experiments 4.4 Out-of-distribution Robustness In this section, we evaluate the performance of ADA and compare it to prior approaches on tasks involving out-of-distribution robustness. We use the same datasets as [49] and closely follow their experimental setup. Data : We use four of the five out-of-distribution datasets used in [49]. First, we use RCFashionMNIST (RCF-MNIST) [49], which is a synthetic modification of Fashion-MNIST that models subpopulation shifts. Second, we investigate domain shifts using Communities and Crime (Crime) [12], SkillCraft1 Master Table (SkillCraft) [12] and Drug-target Interactions (DTI) [17] all of which are tabular datasets. For Crime, we use state identification, in SkillCraft we use "League Index", which corresponds to different levels of competitors, and in DTI we use year, as domain information. We split the datasets into train-, validation- and test data based on the domain information resulting in domain-distinct datasets. We provide a detailed description of datasets in Appendix B.4. Due to computational complexity, we could not establish a fair comparison on the satellite image regression dataset [23] (the fifth dataset in [49]), so we report some exploratory results in Appendix B.4. Data Models and comparisons: As detailed in Section 4.3. Additionally, we use a ResNet-18 [15] for RCF-MNIST and DeepDTA [38] for DTI, as proposed in [49]. Models and comparisons: Results : We report the RMSE and the "worst" domain RMSE, which corresponds to the worst within-domain RMSE for out-of-domain test sets in Table 2. Similar to [49], we report the R value for the DTI dataset (higher values suggest a better fit of the regression model). For full transparency, in the last row, we copy the results from [49]. We can assess that ADA is competitive with C-Mixup and the other data augmentation strategies. Under the same conditions (split of data and Neural network structure), ADA is superior to C-Mixup. But, the degree of improvement is marginal on some datasets and as we show in the last row, we could not fully replicate their results. ADA is significantly better than C-Mixup and other strategies on the SkillCraft data, in which ADA reduces the error by around 15% compared to the ERM solution. Results This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

ADA's Impact on Out-of-Distribution Robustness

Authors:
(1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch);
(2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch);
(3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.2 Anchor Regression 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation B Experiments 4.2 Housing nonlinear regression We extend the results from the previous section to the California and Boston housing data and compare ADA to C-Mixup [49]. We repeat the same experiments on three different regression datasets. Results are provided in Appendix B.2 and also show the superiority of ADA over C-Mixup for data augmentation in the implemented experimental setup. Data: We use the California housing dataset [19] and the Boston housing dataset [14]. The training dataset contains up to n = 406 samples, and the remaining samples are for validation. We report the results as a function of the number of training points. Models and comparisons: We fit a ridge regression model (baseline) and train a MLP with one hidden layer with a varying number of hidden units with sigmoid activation. The baseline models only use only the original data. We train the same models using C-Mixup with a Gaussian kernel and bandwidth of 1.75. We compare the previous approaches to models fitted on ADA augmented data. We generate 20 different augmentations per original observation using different values for γ controlled via α = 4 similar to what was described in Section 4.1. The Anchor matrix is constructed using k-means clustering with q = 10. Results: We report the results in Figure 3. First, we observe that the MLPs outperform Ridge regression suggesting a nonlinear data structure. Second, when the number of training samples is low, applying ADA improves the performance of all models compared to C-Mixup and the baseline. The performance gap decreases as the number of samples increases. When comparing C-Mixup and ADA, we see that using sufficiently many samples both methods achieve similar performance. While on the Boston data, the performance gap between the baseline and ADA persists, on California housing, the non-augmented model fit performs better than the augmented one when data availability increases. This suggests that there is a sweet spot where the addition of original data samples is required for better generalization, and augmented samples cannot contribute any further. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Authors: Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.1 Data Augmentation 2.2 Anchor Regression 2.2 Anchor Regression 3 Anchor Data Augmentation 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation A Additional information for Anchor Data Augmentation B Experiments B Experiments 4.2 Housing nonlinear regression We extend the results from the previous section to the California and Boston housing data and compare ADA to C-Mixup [49]. We repeat the same experiments on three different regression datasets. Results are provided in Appendix B.2 and also show the superiority of ADA over C-Mixup for data augmentation in the implemented experimental setup. Data : We use the California housing dataset [19] and the Boston housing dataset [14]. The training dataset contains up to n = 406 samples, and the remaining samples are for validation. We report the results as a function of the number of training points. Data Models and comparisons: We fit a ridge regression model (baseline) and train a MLP with one hidden layer with a varying number of hidden units with sigmoid activation. The baseline models only use only the original data. We train the same models using C-Mixup with a Gaussian kernel and bandwidth of 1.75. We compare the previous approaches to models fitted on ADA augmented data. We generate 20 different augmentations per original observation using different values for γ controlled via α = 4 similar to what was described in Section 4.1. The Anchor matrix is constructed using k-means clustering with q = 10. Models and comparisons: Results : We report the results in Figure 3. First, we observe that the MLPs outperform Ridge regression suggesting a nonlinear data structure. Second, when the number of training samples is low, applying ADA improves the performance of all models compared to C-Mixup and the baseline. The performance gap decreases as the number of samples increases. When comparing C-Mixup and ADA, we see that using sufficiently many samples both methods achieve similar performance. While on the Boston data, the performance gap between the baseline and ADA persists, on California housing, the non-augmented model fit performs better than the augmented one when data availability increases. This suggests that there is a sweet spot where the addition of original data samples is required for better generalization, and augmented samples cannot contribute any further. Results This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

ADA vs C-Mixup: Performance on California and Boston Housing Datasets

Authors:
(1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch);
(2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch);
(3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.2 Anchor Regression 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation B Experiments 4 Experiments We experimentally investigate and compare the performance of ADA. First, we use ADA in an in-distribution setting for a linear regression problem (Section 4.1), in which we show that even in this case, ADA provides improved performance in the low data regime. Second, in Section 4.2, we apply ADA and C-Mixup to the California and Boston Housing datasets as we increase the number of training samples. In the last two subsections, we replicate the in-distribution generalization (Section 4.3) and the out-of-distribution Robustness (Section 4.4) from the C-Mixup paper [49]. In [49] the authors further assess a task generalization experiment. However, the corresponding code was not publicly provided, and a comparison could not be easily made. 4.1 Linear synthetic data Using synthetic linear data, we investigate if ADA can improve model performance in an overparameterized setting compared to C-Mixup, vanilla augmentation, or classical expected risk minimization (ERM). Additionally, we analyze the sensitivity of our approach to the choice of γ and the number of augmentations. Data: The generated data follows a standard linear structure Models and Comparisons: We investigate and compare the impact of ADA using two different models with varying complexity: a linear Ridge regression and a multilayer perceptron (MLP) with one hidden layer and 10 units with ReLU activation. Using an MLP with more hidden layers shows similar results (see Appendix B.1 for details). For the Ridge regression model, we increase the dataset by a factor of 10 by sampling from the respective augmentation methods and subsequently compute the regression estimators. In contrast, for the MLP, we implement the augmentation methods on a minibatch level. Specifically, we incorporate vanilla augmentation by adding Gaussian noise to each batch, apply C-Mixup after sampling from the beta distribution in each batch, and finally, apply ADA after sampling from the defined gamma values in each batch. In summary, even in the simplest of cases, in which we should not expect gains from ADA (or C-Mixup), these data augmentation strategies provide gains in performance when the number of training examples is not sufficient to achieve the error floor. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Authors: Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.1 Data Augmentation 2.2 Anchor Regression 2.2 Anchor Regression 3 Anchor Data Augmentation 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation A Additional information for Anchor Data Augmentation B Experiments B Experiments 4 Experiments We experimentally investigate and compare the performance of ADA. First, we use ADA in an in-distribution setting for a linear regression problem (Section 4.1), in which we show that even in this case, ADA provides improved performance in the low data regime. Second, in Section 4.2, we apply ADA and C-Mixup to the California and Boston Housing datasets as we increase the number of training samples. In the last two subsections, we replicate the in-distribution generalization (Section 4.3) and the out-of-distribution Robustness (Section 4.4) from the C-Mixup paper [49]. In [49] the authors further assess a task generalization experiment. However, the corresponding code was not publicly provided, and a comparison could not be easily made. 4.1 Linear synthetic data Using synthetic linear data, we investigate if ADA can improve model performance in an overparameterized setting compared to C-Mixup, vanilla augmentation, or classical expected risk minimization (ERM). Additionally, we analyze the sensitivity of our approach to the choice of γ and the number of augmentations. Data : The generated data follows a standard linear structure Data Models and Comparisons: We investigate and compare the impact of ADA using two different models with varying complexity: a linear Ridge regression and a multilayer perceptron (MLP) with one hidden layer and 10 units with ReLU activation. Using an MLP with more hidden layers shows similar results (see Appendix B.1 for details). Models and Comparisons: For the Ridge regression model, we increase the dataset by a factor of 10 by sampling from the respective augmentation methods and subsequently compute the regression estimators. In contrast, for the MLP, we implement the augmentation methods on a minibatch level. Specifically, we incorporate vanilla augmentation by adding Gaussian noise to each batch, apply C-Mixup after sampling from the beta distribution in each batch, and finally, apply ADA after sampling from the defined gamma values in each batch. In summary, even in the simplest of cases, in which we should not expect gains from ADA (or C-Mixup), these data augmentation strategies provide gains in performance when the number of training examples is not sufficient to achieve the error floor. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Evaluating ADA: Experimental Results on Linear and Housing Datasets

Authors:
(1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu);
(2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu);
(3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Table of Links Abstract and 1. Introduction 2. Problem Formulation 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3.2 Assumptions and induced strategies 3.3 Constraints given by phase-II experiments 3.4 Persuasion ratio and the optimal signaling structure 3.5 Comparison with classical Bayesian persuasion strategies 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4.2 Determined versus sender-designed experiments 4.3 Multi-phase model and classical Bayesian persuasion and References 3.4 Persuasion ratio and the optimal signaling structure Careful readers may notice that if we multiply the persuasion ratio with the prior, the value will be the (maximum) expected utility the sender can achieve from the given set of strategies. Since the sender’s expected utility is monotone increasing in the prior p regardless of which set of strategies the sender adopts, the persuasion ratio under a given prior can be viewed as the relative utility gain this set of strategies can offer to the sender. Hence, given a specific prior, if a set of strategies has a higher persuasion ratio with respect to another set of strategies, the sender should use a strategy in the former instead of the latter. According to this discussion, we can draw a persuasion ratio curve for each set of strategies as the prior is varied in [0, 1]. Abusing notation, we represent the persuasion ratio curve by P R(S). It may appear that an optimization needs to be carried out for each value of the prior. However, structural insights presented in the following two lemmas considerably simplify the analysis. Properties presented in Lemma 3 narrow down the space where the sender needs to search for the optimal signaling strategy. This allows us to depict persuasion ratio curves P R(S) for some basic strategies. On top of that, Lemma 4 provides a systematic approach to derive persuasion ratio curves for all types of strategies. Lemma 3. Given a type of strategy S and a prior p, there exists a (sender’s) optimal strategy S ∈ S which satisfies one of the following two conditions: At least one IC requirement of the constituent induced strategies is tight;


There is a signal that will be sent with probability 1 under S. Before discussing the simplifications that Lemma 3 yields in terms of the key properties for solving the problem, we give an intuitive outline of the proof of Lemma 3. When the IC requirements of the two induced strategies are not tight and both signals are sent with non-zero probability in a strategy S, the sender can increase her expected utility by slightly raising the probability of the signal with a higher persuasion ratio (and adjust the probability of the other signal to respect the prior) to form a strategy S+. The sender can keep doing this ‘slight’ modification of her strategies until either one of the IC conditions is satisfied or the signal is sent with probability one. Lemma 5. The persuasion ratio curve of the optimal signaling strategy P R∗ (p) is the upper envelope of the different types of strategies’ persuasion curves. Further, the optimal signaling strategy (under a given prior) is the strategy that reaches the frontier of the persuasion ratio curve (at that prior). For a two-phase trial or to solve the last two phases of a trial with more than two phases studied in Section 4, the following corollary can further simplify the sender’s optimization procedure. With Corollary 1, the Equation (3) in Lemma 4 reduces to a linear equation. Hence, the comparison in Lemma 5 and the computation in Lemma 4 can be reduced to a comparison of the (unique) corresponding IC strategies (if one exists) under the interim belief listed in Corollary1 for different types of strategies. This paper is available on arxiv under CC 4.0 license. Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Authors: Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Problem Formulation 2. Problem Formulation 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3.2 Assumptions and induced strategies 3.2 Assumptions and induced strategies 3.3 Constraints given by phase-II experiments 3.3 Constraints given by phase-II experiments 3.4 Persuasion ratio and the optimal signaling structure 3.4 Persuasion ratio and the optimal signaling structure 3.5 Comparison with classical Bayesian persuasion strategies 3.5 Comparison with classical Bayesian persuasion strategies 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4.2 Determined versus sender-designed experiments 4.2 Determined versus sender-designed experiments 4.3 Multi-phase model and classical Bayesian persuasion and References 4.3 Multi-phase model and classical Bayesian persuasion and References 3.4 Persuasion ratio and the optimal signaling structure Careful readers may notice that if we multiply the persuasion ratio with the prior, the value will be the (maximum) expected utility the sender can achieve from the given set of strategies. Since the sender’s expected utility is monotone increasing in the prior p regardless of which set of strategies the sender adopts, the persuasion ratio under a given prior can be viewed as the relative utility gain this set of strategies can offer to the sender. Hence, given a specific prior, if a set of strategies has a higher persuasion ratio with respect to another set of strategies, the sender should use a strategy in the former instead of the latter. According to this discussion, we can draw a persuasion ratio curve for each set of strategies as the prior is varied in [0, 1]. Abusing notation, we represent the persuasion ratio curve by P R(S). It may appear that an optimization needs to be carried out for each value of the prior. However, structural insights presented in the following two lemmas considerably simplify the analysis. Properties presented in Lemma 3 narrow down the space where the sender needs to search for the optimal signaling strategy. This allows us to depict persuasion ratio curves P R(S) for some basic strategies. On top of that, Lemma 4 provides a systematic approach to derive persuasion ratio curves for all types of strategies. Lemma 3. Given a type of strategy S and a prior p, there exists a (sender’s) optimal strategy S ∈ S which satisfies one of the following two conditions: Lemma 3. Given a type of strategy S and a prior p, there exists a (sender’s) optimal strategy S ∈ S which satisfies one of the following two conditions: At least one IC requirement of the constituent induced strategies is tight; There is a signal that will be sent with probability 1 under S. At least one IC requirement of the constituent induced strategies is tight; At least one IC requirement of the constituent induced strategies is tight; At least one IC requirement of the constituent induced strategies is tight; There is a signal that will be sent with probability 1 under S. There is a signal that will be sent with probability 1 under S. There is a signal that will be sent with probability 1 under S. Before discussing the simplifications that Lemma 3 yields in terms of the key properties for solving the problem, we give an intuitive outline of the proof of Lemma 3. When the IC requirements of the two induced strategies are not tight and both signals are sent with non-zero probability in a strategy S, the sender can increase her expected utility by slightly raising the probability of the signal with a higher persuasion ratio (and adjust the probability of the other signal to respect the prior) to form a strategy S+. The sender can keep doing this ‘slight’ modification of her strategies until either one of the IC conditions is satisfied or the signal is sent with probability one. Lemma 5. The persuasion ratio curve of the optimal signaling strategy P R∗ (p) is the upper envelope of the different types of strategies’ persuasion curves. Further, the optimal signaling strategy (under a given prior) is the strategy that reaches the frontier of the persuasion ratio curve (at that prior). Lemma 5. The persuasion ratio curve of the optimal signaling strategy P R∗ (p) is the upper envelope of the different types of strategies’ persuasion curves. Further, the optimal signaling strategy (under a given prior) is the strategy that reaches the frontier of the persuasion ratio curve (at that prior). For a two-phase trial or to solve the last two phases of a trial with more than two phases studied in Section 4, the following corollary can further simplify the sender’s optimization procedure. With Corollary 1, the Equation (3) in Lemma 4 reduces to a linear equation. Hence, the comparison in Lemma 5 and the computation in Lemma 4 can be reduced to a comparison of the (unique) corresponding IC strategies (if one exists) under the interim belief listed in Corollary1 for different types of strategies. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

How to Maximize Persuasion Ratios in Two-Phase Trials

Authors:
(1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu);
(2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu);
(3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Table of Links Abstract and 1. Introduction 2. Problem Formulation 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3.2 Assumptions and induced strategies 3.3 Constraints given by phase-II experiments 3.4 Persuasion ratio and the optimal signaling structure 3.5 Comparison with classical Bayesian persuasion strategies 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4.2 Determined versus sender-designed experiments 4.3 Multi-phase model and classical Bayesian persuasion and References 2 Problem Formulation Before the receiver takes his action, a trial consisting of multiple phases will be run, and the outcome in each phase will be revealed to him. In each phase, one experiment will be conducted, which is chosen according to the outcomes in earlier phases. Hence, the experiment outcomes in earlier phases not only affect the interim belief but also influence the possible (sequence of) experiments that will be conducted afterward. In the most sender-friendly setup where the sender can choose any experiment in each phase without any constraints, the problem is equivalent to the classical Bayesian persuasion problem with an enlarged signal space. However, when some experiments are pre-determined conditional on a set of outcomes, the sender must take these constraints into account to design her optimal signaling structure. To present our results on the influence of multiple phases on the sender’s signaling strategy, we start with a model of two-phase trials with binary-outcome experiments in the rest of this section. We then analyze the optimal signaling strategy of this model in Section 3. After that, we will introduce the general model of multiple-phase trials with binary-outcome experiments and propose a systematic approach to analyze the optimal signaling structure in Section 4. This paper is available on arxiv under CC 4.0 license. Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Authors: Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Problem Formulation 2. Problem Formulation 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3.2 Assumptions and induced strategies 3.2 Assumptions and induced strategies 3.3 Constraints given by phase-II experiments 3.3 Constraints given by phase-II experiments 3.4 Persuasion ratio and the optimal signaling structure 3.4 Persuasion ratio and the optimal signaling structure 3.5 Comparison with classical Bayesian persuasion strategies 3.5 Comparison with classical Bayesian persuasion strategies 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials 4.2 Determined versus sender-designed experiments 4.2 Determined versus sender-designed experiments 4.3 Multi-phase model and classical Bayesian persuasion and References 4.3 Multi-phase model and classical Bayesian persuasion and References 2 Problem Formulation Before the receiver takes his action, a trial consisting of multiple phases will be run, and the outcome in each phase will be revealed to him. In each phase, one experiment will be conducted, which is chosen according to the outcomes in earlier phases. Hence, the experiment outcomes in earlier phases not only affect the interim belief but also influence the possible (sequence of) experiments that will be conducted afterward. In the most sender-friendly setup where the sender can choose any experiment in each phase without any constraints, the problem is equivalent to the classical Bayesian persuasion problem with an enlarged signal space. However, when some experiments are pre-determined conditional on a set of outcomes, the sender must take these constraints into account to design her optimal signaling structure. To present our results on the influence of multiple phases on the sender’s signaling strategy, we start with a model of two-phase trials with binary-outcome experiments in the rest of this section. We then analyze the optimal signaling strategy of this model in Section 3. After that, we will introduce the general model of multiple-phase trials with binary-outcome experiments and propose a systematic approach to analyze the optimal signaling structure in Section 4. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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