Question: Represent the scientific article to find a short TLDR

Question: Humans have a natural curiosity to imagine what it feels like to exist as someone or something else. This curiosity becomes even stronger for the pets we care for. Humans cannot truly know what it is like to be our pets, but we can deepen our understanding of what it is like to perceive and explore the world like them. We investigate how wearables can offer people animal perspective-taking opportunities to experience the world through animal senses that differ from those biologically natural to us. To assess the potential of wearables in animal perspective-taking, we developed a sensory-augmenting wearable that gives wearers cat-like whiskers. We then created a maze exploration experience where blindfolded participants utilized the whiskers to navigate the maze. We draw on animal behavioral research to evaluate how the whisker activity supported authentically cat-like experiences, and discuss the implications of this work for future learning experiences. Posthumanist philosophies characterize the human body as "the original prosthesis we all learn to manipulate", and suggest the idea that augmenting or substituting aspects of this prosthesis is the normal progression for humanity. Technology allows humans to enhance senses that may be impaired, and to extend our bodies with added senses beyond what we would otherwise be biologically limited by-giving humans the ability to improve their quality of life. " In short, we are cyborgs". Scholars have investigated how immersive virtual environments can enhance social perspective-taking, and computer-augmented, embodied perspective-taking has been shown to encourage a productive "learning stance" and to enhance both conceptual learning and engagement. Some environmental education scholars and indigenous educational scholars have suggested that building relational ties to non-human actors in nature may contribute to environmental and biology education. In a few cases, educators have asked learners to take on the embodied experiences of insects such as bees and animals such as polar bears. Danish found that children enacting a computer-augmented pollination activity embodying the roles of bees helped them learn nuances of individual and aggregate bee behavior; Lyons and colleagues found that wearable polar bear paws that simulated the feeling of traversing melting polar ice enabled people to show an empathetic understanding of the impacts of climate change. For many people, the most common experience they will have with entities who have different sensory capabilities is through everyday interaction with pets or neighborhood animals. For example, in noticing that our house cat is navigating a dark space where we would likely bump into something, we may recognize the limits of our own senses and consider how our pets' experiences are both similar to and different from our own. Our work explores how embodied technology can mediate human experiences in ways that offer people opportunities to explore and relate to the animal-like behaviors of their pets. We present the design of a cat-inspired whiskers wearable, the Whisker Beard, and an embodied navigation activity that provided a firsthand perspective-taking experience for participants curious about what it might be like to have whiskers. In addition, we discuss our philosophy of what it means for an animal-imitating experience to be authentic and we present the evaluation framework we used to understand how our whiskers activity encouraged participants to behave like cats. Our study addresses two research questions: In what ways can we create technologies and environments that remediate human experiences to be like those of non-humans? RQ2: What are humans' impressions of these technologically remediated experiences? In this paper we describe the design of a sensory augmentation whiskers wearable; the creation of a maze exploration activity for testing the experience of wearing whiskers; our analysis methods for evaluating the authenticity of an animallike experience; and outline opportunities to extend this work, as well as discuss the implications of it. Our technology, experience design, and analysis are motivated by a desire to re-shape science and science education in light of feminist critiques and visions of those fields. Whereas science and science education today tend to emphasize distance, objectivity, and dispassion, we strive to create spaces for discovery and learning that also include closeness, subjectivity, and emotion and that thereby enable learners to author identities in science based on more engaged and connected ways of knowing. The overall approach of the project is to investigate how affection for and curiosity about pets can catalyze scientific investigations and engineering for young people and their families that are based on empathy and perspective-taking, both in and out of schools. We see value in using wearable and mixed-reality technologies to provide humans with the ability to experience the lives of other beings. In particular, this would allow humans to experience what their pets' senses might be like, and thereby facilitate learning experiences that encourage empathy and perspective-taking. Once we can support interspecies perspective taking, we then wish to encourage participants to conduct scientific inquiry within that intersubjective sensational realm -a realm which the German biologist Jakob von Uexküll called "Umwelt". This agenda has the potential to unify efforts to advance scientific and social-emotional education. Humans have strong emotional attachments to their pets and these human-animal bonds coincide with higher amounts of empathy. This can motivate people's curiosities about their pets' lives and experiences. One technological and philosophical challenge involved in pursuing our agenda is the intrinsic disconnect between humans' experiences and those of other species. People only know what it is like to be human, and are unable to know on a phenomenological level what it is truly like to be their pets. Though people cohabitate with their pets, on a biological level they see, hear, smell, taste, and experience the world differently from them-ranging from slight variations of senses to things incomprehensible and alien like sonar and magnetic field detection. Despite this challenge, most pet owners believe that their pets feel something, even if they cannot fully understand what they feel. People naturally evaluate animal behavior and experience through a human lens. In a thought exercise of imagining what it is like to be a bat, Nagel highlights the mind-body problem he encounters: "... I want to know what it is like for a bat to be a bat. Yet if I try to imagine this, I am restricted to the resources of my own mind". While we will likely never solve the mind-body problem of humans truly understanding their pets' experiences, we can address the "body" aspect of the problem and use it to drive thought experiments about the lives of pets. These thought exercises can give people a better sense of animals' lives by acknowledging biological differences that many people have about how their pets see, hear, and feel the world. In this study, we present the of an early step toward feminist science and science education: the design and evaluation of a wearable device meant to offer humans the experience of exploring an environment using whiskers. This first step is intended to both elucidate how new technologies can offer transspecies sensory experiences, and to show how we might assess the validity of those experiences (i.e., the extent to which they immerse humans in the reality of another species) by comparing humans' behaviors in the new technologically-mediated umwelt with those whom the umwelt natively describes. Wearables and mixed reality technologies have been applied in a wide variety of domains ranging from educational contexts, medical settings, natural environments, performances, and museums. The ing hybrids of wearables and mixed reality technologies are sensory augmentation devices. Sensory augmentation devices give humans the ability to experience phenomena that they are physically unable to process, as well as some phenomena that are simply unnatural to the human experience. The field of sensory substitution and augmentation enables humans to use existing senses to substitute for the ones that they cannot experience, and to augment senses they already have to make them more powerful. For example, sensory substitution has been applied as a method for offering hearing and/or visually impaired people additional senses to substitute for the one in which they have an impairment. These technologies range from devices that can be implanted within the body, sending direct signals to internal mechanisms in the brain, to external substitute devices that provide tactile feedback on the surface of the skin. In one example of tactile feedback, researchers developed a non-invasive "vibratory vest" for deaf and hearing impaired individuals that processes auditory information and converts it into vibrotactile feedback on the wearer's torso. Our work builds on approaches that use these technologies to provide physical, sensory feedback experiences for the wearer. Wearables have the ability to provide hands-on, interactive, and embodied learning experiences. There is research to suggest that wearable technologies in the classroom can increase engagement and improve student attitudes towards STEM activities. One of the pedagogical affordances of wearable technologies is the ability to gather contextual information from a firsthand account. Some uses of wearables in education focus on understanding and quantifying the self, such as: calculating our heart rates, monitoring how many steps we have taken, and providing feedback about our emotional states. Other approaches embed the quantifying potential of wearables in scientific inquiry and discourse. In contrast, our research focuses on how wearable technologies can help people understand experiences beyond the self, more specifically, the potential for wearable technologies to foster empathy by helping humans understand the experiences of others. Incorporating empathy and perspective-taking into scientific question-asking is deeply rooted in feminist educational theory and practice. These practices offer a more inclusive view of what it means to actively participate in science, and promote empathy as a valued part of scientific discovery. An example of a wearable device aimed to motivate empathy is a series of mushroom foraging tools designed to build more intimate relationships between humans and the environments they are probing by connecting them physically to the environment. Nobel laureate Barbara McClintock cited her ability to get "a feel for the organism" as influential in supporting her discoveries in the area of maize cytogenetics. McClintock has said, "I know my corn plants intimately, and I find it a great pleasure to know them," a sentiment we believe could inform future wearable-mediated science education. When interacting with non-human animals, it is easy to notice how their sensory capabilities differ from humans'. For example, people may notice their cat hear or see something without hearing or seeing it themselves. In addition to animal companions' sharper senses, people may also notice qualitative differences in how they experience the environment, such as when their pets deftly navigating through small spaces. People may also notice and wonder about the presence of physical characteristics that humans lack, such as whiskers or purring behaviors. Our work in this paper is motivated by human curiosity about these differences between species and explores how taking on the physical or perceptual characteristics of an animal helps people to understand it. In this study, we specifically focus on how participants' behavior compares to navigation and foraging techniques that animals exhibit in the wild and in controlled laboratory maze experiments. An animal's ability to gather environmental information and make decisions in real-time is key for its survival. Animals must constantly make decisions and consider tradeoffs during activities like foraging for food, avoiding predators, and searching for a mate. These trade-offs describe what is called the exploration-exploitation problem, a key idea in organizational learning and animal behavioral science. A common definition describes exploration as the process of randomly searching an unrefined and unexplored area, and exploitation as the process of searching a more refined area for some reward or resource. The less information an animal has about an environment, the more likely it is to be exploring rather than exploiting. Researchers argue that optimal search strategies for animals include alternating patterns of quick movement and searching. In the process of exploring and exploiting an environment, animals exhibit "egocentric" and "geocentric" navigational strategies. In an egocentric strategy, an organism uses previously acquired information, either from memory or another internal mechanism, to move along a path. In a geocentric strategy, an organism uses cues and real-time feedback from the environment to continuously reorient itself along a path. Animals rely heavily on geocentric strategies to explore and search unfamiliar areas; it is too difficult to primarily rely on internal mechanisms when moving through new territory. The term thigmotaxis describes the behavior of either moving toward or away from touch stimulus; an example of this behavior is "wall-hugging" in animals, which is the tendency to avoid open areas and stick to the perimeter of an environment during navigation. Wall-hugging behavior in animals is commonly tested in laboratory maze environments, where high levels of wall-hugging can suggest anxiety or fear in an animal. Therefore, wall-hugging is commonly seen as a strategy that animals invoke when exploring a new and unfamiliar environment. For example, mice spend, on average, 74% of their time hugging a wall in their first five minutes exploring an unfamiliar environment, and about 65% of their time doing so by thirty minutes. We created a sensory augmentation device, the Whisker Beard, that provides a physical simulacrum of what it is like to have whiskers. We wanted our whiskers to be functionally and aesthetically reminiscent of cat whiskers, including the appearance and placement of the whiskers and the physical properties of the materials. In addition, we wanted the functionality to be authentic in the sense that a person wearing it would use the whiskers to enact behaviors similar to animals'. Whiskers, formally known as vibrissae, are rigid but flexible hairs that provide tactile sensory feedback in many mammals and rodents. Ahl describes a number of characteristics and roles for whiskers; whiskers differ from hair in that their follicles are much thicker than normal hair follicles, and that whiskers are sensitive and send signals to the brain. Whiskers are commonly localized to facial regions of animals, but can also be found on other parts of the bodies of animals. They are essential to survival, monitoring, communication, and aggression. They help serve as a sensory substitute for animals with poor close-sightedness; cats have difficulty seeing objects very close to their faces, so whiskers help them sense objects close to them, as well as protecting their face from harmful objects. A project close to our vision is a helmet that uses infrared sensors to detect close-by objects and provides vibrotactile feedback to the wearer; like our project,'s design was inspired by mammals' abilities to use whiskers to detect their surroundings. Our project and theirs are similar in that they are whisker-inspired designs for sensory augmenting wearable devices, however our goals differ in that we are not trying to offer an efficient solution for humans to navigate in low-light conditions, but rather to provide perspective-taking experiences for wearers to experience what it is like to have and use whiskers. The version of the Whisker Beard used in this study has a total of four whiskers, two for each side of the face. In order to mimic the genal (cheek) whiskers of the cat, our whiskers attach to the sides of two acrylic cheek plates that rest 2-3 cm above the skin of the wearer, therefore the whiskers protrude from the sides of the wearer's cheeks close to the mouth (see Figure 1). Each whisker's total length (38.5 cm) is approximately the average human's shoulder width (39 cm). We chose this length in order to make the whiskers extend beyond the shoulders of an average sized person, thereby imitating the appearance and functional reach of cats' whiskers. Figure 1 shows a researcher wearing the Whisker Beard. The whiskers are composed of three flexible components: a flex sensor that detects the deflection that the whiskers are receiving, polystyrene strips that extend the usable length of the flex sensor significantly and returns the flex sensor to a neutral unbent position, and Sugru, which is a moldable silicone glue that holds the polystyrene strip and flex sensor together. In addition, a small connector at the end of the whisker attaches to the cheek (see Figure 2). Our device detects when the wearer brushes their whiskers against a surface. When the whiskers are bent the change in resistance is measured and converted to a voltage. The calculated voltage measurement is then used to set the pulsewidth-modulation signal to the vibration motor control. The device provides vibrotactile feedback to the wearer by varying the intensity of vibration proportional to the bend angle through an array of vibration motors on the scalp, inspired by the work from. Each motor is coupled to one whisker. We chose to place the vibration motors on the scalp so the vibrotactile sensation would be felt on the wearer's head, and so the motors could be placed far enough apart to allow for better point discrimination (making it easier for the wearer to identify which specific motors are vibrating). The whiskers sense bidirectionally in order to better mimic a cat's actual sensory perceptions when approaching and backing out of different confined locations. The current hardware design allows for bidirectional sensing, but the software does not explicitly indicate direction of the whiskers' bends through haptic feedback, however, the participant can feel the direction of tension on the wearable. In order to study how the Whisker Beard can support animallike environmental exploration, we built a human-sized cardboard maze (4m x 6m) for our participants to explore while wearing the whiskers wearable. Mazes are a common tool in behavioral research involving rats and smaller mammals. Our hypothesis was that participants would be able to use the whiskers as a workable form of sensing during maze navigation. We wanted participants to rely on the whiskers as heavily as possible and other senses as little as possible. Therefore, we chose to blindfold the participants since vision is the dominant sense for non-visually impaired humans for gathering environmental information. We designed the maze to promote enough confusion for participants so they would need to use feedback from the whiskers to guide them. We did so in order to observe a more honest range of animal behavior. Before creating the human-size maze, we created a series of small-scale prototypes. We used these to consider different possible maze routes as well as what kinds of obstacles would promote explorations of the whiskers' affordances. We prototyped obstacles that would encourage participants to enact cat-like behaviors such as rubbing one's face on a surface, and having to back out of narrow spaces. We came to the that experimenting with unfamiliar topography was the best way to create an experience that removes participants from a human sensory experience within an exploration task. Therefore, the final maze design included multiple dead ends, corridors of different widths, vertically hinged flaps, and cutouts in the walls. In addition, due to the bidirectionality of the whiskers, we designed our obstacles such that wearers would mostly need to rely on the horizontal changes in the whiskers' shapes. Figure 3 shows a photo of the finished cardboard maze, and Figure 4 shows a digital rendering of an aerial view of the maze. We placed 10 toy mice throughout the maze to induce exploration and exploitation behavior. We gave participants this task in order to give them a survival-oriented goal to collect as many resources as possible from an unfamiliar environment in a limited time frame to strengthen the worldview of being an animal. We investigated how participants used the Whisker Beard to navigate the maze, and in our analysis we focus on what ways participants did or did not exhibit animal-like behaviors during the experimental activity, and how they reacted to the experience of cat-like sensory augmentation. To do so, we evaluate how participants behaved like cats, as well as what participants' impressions were about the activity. We draw on work from animal and biological sciences, particularly key areas of research related to navigation and animal behavior. In addition, we highlight moments from participant discussion that illustrate ideas and questions they had after the experience. We recruited six undergraduate college students who lived on campus in the dormitory that we constructed the maze in. Due to the circumstances of living in a dorm room, none of the participants currently owned a pet, however three of the participants had previously owned a cat, or still had a cat at home, and the participants who had not owned a cat had owned a dog and also knew family and friends who have had cats. All six participants completed the maze activity on the same day in subsequent 20 minute sessions. As participants arrived for their sessions, we greeted them individually in a separate room. There, we assisted each participant in attaching the Whisker Beard to their face. We spent a few minutes explaining how the device works and gave participants time to learn the correlation between whisker bend and vibration location and intensity. We did this by individually flexing each whisker while the participant was wearing it (prior to blindfolding them) and asking them to describe if they could feel the changes in the vibrotactile feedback. Once participants were familiar with the wearable, we explained the maze procedure to participants, and brought them -blindfolded -into the maze room. We replaced all 10 toy mice in the same locations before the start of each participant's session (see Figure 4). We gave each participant a maximum of 10 minutes to crawl through the maze and explore, and to collect as many toy mice as they could. We asked participants to think-aloud during the maze so we could have a better sense of their thought process throughout the activity to both validate our coding of the data, as well as collect information about participants' impressions of the activity. After each participant completed the maze activity, we asked them to refrain from discussing the experience with other participants until everyone was finished. We had them each fill out a worksheet to provide responses to questions like, how did it feel to navigate using whiskers, and what moments stood out to you when using your whiskers to navigate the maze? We selected open-ended questions such as these to gather information about participants' experiences on an individual level and to get their initial impressions right after the activity. After all participants were finished with the activity, we reconvened as a full group for a facilitated discussion. We began the discussion with three prompts on the board: "How did you feel when you first put the whiskers on?" "What particular moments in the maze stood out to you?" and "Other thoughts?" We asked participants to write their thoughts about these prompts on sticky-notes and place them on the board. Once participants completed this, we used the responses to facilitate discussion among the whole group. In order to investigate how the wearable and maze activity mediated participants' experiences in an animal-like way (RQ1), we analyzed participants' interactions, behaviors, and commentary throughout the activity through coding of the data and analysis of their movement. We assessed interrater reliability by having each coder annotate a map of the maze using the process shown in Figure 7. We compared each annotation on both maps. If the coders identified an action as a different event type as shown in Figure 6, or if one coder identified an event at a particular time and the other did not identify an event, these were marked as disagreements. We then added up all the agreements and disagreements to calculate with Figure 5. A visualization of participants' usage of their whiskers and hands over time. Note that Jodie's time is much shorter than other participants due to the fact she moved through the maze very quickly and returned to the starting point after about four minutes. Cohen's Kappa. For example, to assess agreement on where "whisker interactions" occurred, we looked at where the raters identified whisker interactions on their maps, and counted all matching identifications as agreements, and all discrepancies as disagreements. We did this for all symbols in the key and totaled the . After reaching an inter-rater reliability of k=0.62, which indicates substantial agreement, the two researchers coded the rest of the data in parallel. To analyze the paths of participants and their interactions, we captured audio and video recordings of participants' movements and interactions in the maze from five cameras: we placed four cameras in different positions along the maze's perimeter, and a researcher held one camera and followed the participant as they moved through the maze. We analyzed the videos to produce content logs and time stamps of different events. In addition, we used the videos to manually generate aerial maps that depict participants' movements and interactions through the maze. Figures 6 and 7 illustrate the orientation, position, and scale of the participant, as well as depict participants' interactions with their whiskers and hands at a given time. In addition, we denoted four subsections of the maze that we thought represented the different types of obstacles and areas (see Figure 4). The red zone is where the maze starts and contains two straight hallways. The yellow zone contains a large area of open space, as well as a wide opening that narrows to a dead end. The blue zone contains a straight hallway that opens into three smaller corridors. The green zone is a long hallway with cardboard flaps hanging on either side that participants had to push through. In Shapiro and Hall's museum mapping work, they created aerial maps to illustrate museum visitor movement and engagement to understand how the space created by the gallery facilitates learning. We apply their mapping methods in our study in order to identify how participants interacted with the space with the Whisker Beard and how their interactions and movement compared to animal-like behavior (see Figure 7 for examples of the paths we generated for analysis and refer to Figure 6 for our key). Figure 6 for our key. The map on the right contains color so that we could draw overlapping paths and keep track of the order in which it occurred (such as when participants doubled back to specific areas). In our analysis, we define moments of primarily whisker interaction as exploration, and moments of hand-swiping interactions as exploitation. We encouraged participants to solely rely on their whiskers to navigate the maze, and to only use their hands when they needed to collect mice. Because of this, we find the distinction between whisker use versus hand use as a reasonable indicator of exploration-exploitation behaviors. Finally, we coded moments of geocentric and egocentric strategies using the perspectives defined earlier, and coded wall-hugging to be moments where a participant makes con- tinued contact with their body and or whiskers against the wall. We separate our into four categories of participants': patterns of exploration and exploitation and how they related to whisker use; geocentric and egocentric strategies; wall-hugging behaviors; and reflections on the experience. The first three are aimed at providing information about the authenticity of the animal-like behaviors that participants exhibited (RQ1), while the fourth addresses what impressions people had about the wearable and the experience (RQ2). Participants alternated between periods of long exploration and relatively shorter exploitation. In Figure 5 we illustrate the exploration-exploitation search behaviors of each participant (listed with pseudonyms). Blue segments denote periods of navigation where participants were primarily relying on their whiskers to move around. Red segments denote moments where participants moved their arms across the floor in order to search for and obtain mice. Yellow segments are places where we coded participants as using both exploration and exploitation, which was rare because crawling and swiping is challenging. In Table 1 we show the breakdown of the amount of time participants spent exploring and exploiting the maze. Participants spent an average of 70.3% of time exploring and 35.8% of time exploiting. Participants switched back and forth between exploration and exploration, with an average exploration interval length of 15.2 seconds and an average exploitation interval length of 7.4 seconds. On average, participants caught seven out of the ten mice. During periods of whisker exploration, participants enacted three common whisker techniques that primarily occurred in the long, straight parts of the maze (red zone in 4). In Figure 8 we illustrate what these three techniques look like. In one technique (A) the wearer constantly drags the whiskers against the wall as they move. In the second (B) the wearer alternates between motion and pausing to brush the whiskers against the wall, and in the third (C) the wearer moves between walls alternating between the sides of the whiskers they use. All six participants enacted at least one form of A and B during the maze, and two participants utilized the C technique as well. The two participants that enacted all three techniques both have pet cats at home. Participants used a variety of geocentric and egocentric search strategies as they navigated the maze. Table 2 shows the of this categorization and demonstrates the different geocentric and egocentric strategies that participants enacted, with examples and quotes to illustrate how they used particular environmental feedback. We found that all six participants relied on enacting geocentric strategies throughout the entire activity, and used egocentric strategies more sparingly. During the think-aloud all six participants made comments about using touch and sound feedback from the whiskers, as well as tactile feedback from their hands and body. Only two participants made comments about attempting to use egocentric strategies, with one participant commenting that he was able to create an internal map during the maze (which he later admitted was incorrect after seeing the maze). All six participants exhibited wall-hugging behavior (thigmotaxis) as they moved through the maze blindfolded. Sound of the whiskers touching the walls "The navigation is also noise of the whiskers touching things." Egocentric Memory of the space from an earlier point in time "I know I'm in the center of the room, because I've been in this room before with this outlet on the floor" Memory of revisiting the same part of the maze "I think I'm going in circles" Internal recall of physical orientation "[I was] thinking about my previous and next moves, as well as an internal map." Table 2. A breakdown of geocentric and egocentric navigational strategies that participants used. Five of the six participants spent over half of their time in the maze hugging the wall, with one participant hugging the wall for 97.3% of her maze time. On average, participants spent 68.6% of their time in the maze hugging the wall (min=34.6, max=97.3, standard deviation=20.87). Participants' tendencies to hug the wall varied, but interestingly, the participant who hugged the wall the least had a higher tendency than others to bump into obstacles head first (14 times). Participants frequently wall-hugged while exploring using whisker technique A from Figure 8. Participants reflected on their experiences in the whisker activity through individual worksheet responses and a full group discussion. During their reflections, participants commented on the physical experience of the whisker activity, and drew connections to phenomena they have experienced with their own pet cats or others' cats. In response to the question, "How did it feel to wear the whiskers?" participants described both the physical sensation of how it felt to have whiskers, and the functional use of the whiskers and their ability to adapt to it. Responses that described the feeling of adapting to the whiskers include: • " As I got used to them, the whiskers started to become a part of me." • "I liked having another sense. They got much easier to use as I played around with angles and pressure on the sensed surfaces." • "It was an easy way to'see' side to side. Times in empty space though stood out, with nothing to feel in front of me made me more cautious" • "The first time I went through a small space and both whiskers activated [while] having to back out...stood out to me" • "It was a bit odd at first but I quickly got used to them. It was kinda nice having an additional aid apart from the feeling in my hands and feet. " Responses that described the more physical feeling of having vibrating whiskers include: • "I had to push through the flaps and it was almost overwhelming with vibrations." • "It was an interesting feeling having the vibration kind of tickle. When I was really close to something it was also kind of shocking and made me want to back away." Most of the above responses to the question of how it "felt" focused on the overall experience of using the whiskers to navigate, as opposed to the specific physical feeling of the vibrations on the skin. In addition, we were curious whether participants would think about their own pet cats, or other cats that they know. We asked, "Did you think about your own pet cat while you were in the maze? If so, what did you think about?" and participants responded in ways that included perspective-taking commentary that were empathetic. Their responses included: • "I thought about my friend's cat and how when she was five, she cut off the cat's whiskers thinking they were long hairs. For a good week the cat had to stumble around, falling over, and running into things often." • "How cats sometimes bump their heads into things then back-up confused. I could sympathize." • "Yes I did, I thought about how the vibration was a little like how they would use their whiskers. I also thought about how hard it would be to navigate without her whiskers" Most of the empathetic responses show participants acknowledging that it would be difficult to navigate as a cat stripped of its whiskers, similarly to how it was difficult for them to navigate without their sight. Participants described moments of feeling disoriented when they entered the large open area of the maze (yellow zone in Figure 4), and during moments of technical difficulties when a whisker fell out. According to participants, the open areas were disorienting because they lost their sense of physicality and location, suggesting some understanding of, and potential for empathy with, animals' thigmotactic strategies. The of the Whisker Beard and maze activity show examples of participants exhibiting behaviors and strategies similar to those that animals perform when searching for resources in unfamiliar environments. We separate our into discussions about their physical behaviors and strategies, as well as their impressions of the experience. As depicted in Figure 5 and Table 1, as participants explored the maze, they alternated between periods of explorative and exploitative behavior as they switched between using their whiskers and using their hands. Participants spent, on average, a longer amount of time exploring and moving through than maze than they spent hand swiping to look for mice. These are in line with animal foraging behaviors. Benichou et al. says that animals searching for resources switch between periods of motion and periods of searching. In addition, their work shows that intervals of exploration tend to be longer than intervals of exploitation. This aligns with the amount of time our participants dedicated to these behaviors. While we cannot claim that participants would not have enacted similar exploration-exploitation behaviors without whiskers, we can say that the behaviors that they enacted with whiskers were in line with foraging behaviors. Interestingly, several of the participants made use of the whiskers in ways that strikingly resembled cat behavior, as depicted in Figure 8. As participants moved down long passages, some used their whiskers to gauge the width of the passage by moving back and forth brushing each side of their whiskers on the opposing walls. This demonstrates that participants used the whiskers to enhance their spatial awareness, one of the supposed evolutionary factors behind the presence of whiskers. We noticed that when participants used techniques B and C, they mimicked the behavior of cats who rub their olfactory face glands on objects to mark their scent, as well to get a sense for the physical properties of a specific object. While this behavior in cats is not necessarily used for navigation purposes, it is used for gauging the size and shape of an object. Participants did this in order to look for hidden passageways and moveable obstacles. Our observations of participants' geocentric and egocentric behaviors provided us with a fuller picture of how participants used the whiskers in tandem with other strategies during the activity. Participants relied on the vibrotactile feedback from the Whisker Beard in determining their path of movement through the maze. In addition to the vibrotactile feedback, we found that participants also relied on the sounds the whiskers made as they brushed against the maze's cardboard surfaces. We validated this observation through think-aloud commentary that participants provided throughout the maze, and through post-maze group discussion. The fact that participants relied on additional tactics beyond the vibrations is not an inauthentic outcome, but rather a reasonable one. Participants' use of different egocentric and geocentric tactics is naturally aligned with how animals navigate the world-getting the most information from their environment by whatever means are accessible to them. The blindfolded maze procedure afforded participants the ability to experience the Whisker Beard in an unfamiliar environment. As expected, due to the unfamiliarity of the environment, participants relied on more geocentric strategies. These are in line with animal navigation research which suggests that egocentric strategies are too difficult to use when exploring new terrain, and therefore animals rely more heavily on geocentric strategies to gather real-time physical feedback. In time, participants who revisited areas of the maze began to recognize their surroundings, which led them to use internal recall from their memory to identify their approximate position; however, because they were blindfolded they still had to rely on geocentric strategies as well. Unsurprisingly, participants told us that being blindfolded and losing their sense of sight was disorienting for them; sight is one of humans', and cats', dominant senses for obtaining information about an environment. Participants described the open-space areas of the maze as "disorienting" and tended to try to find a wall as quickly as they could to reorient themselves. The level of consistent wall-hugging that participants exhibited is in line with experiments where increased levels of anxiety correlated to higher levels of thigmotaxis. Usually, animals' tendency to hug the wall would decrease as an experiment went on, except in circumstances where animals do not have enough time to fully process their environment. In our experiment, blindfolding the participants made it challenging for them to produce an accurate internal map of the space, leading them to continuously avoid open areas and rely on vibrotactile and audio feedback from the walls during navigation. The participants' reflections during the maze and post-maze show promising beginnings to meaningful discussions of animal empathy, as many drew connections to prior experiences of pets who were blind, deaf, or had their whiskers cut off and discussed how disorienting and difficult it would be for them to navigate with a sense removed. Participants described the whiskers as feeling like an extra sense, one that they were able to adapt to even in a short timeframe. Although losing their sight was disorienting, they were able to utilize the whiskers as a substitute for being able to "see." The combination of the Whisker Beard and maze activity suggests that through disorienting them and having them behave like a cat, they were able to consider what it would be like to be a cat relying on its whiskers every day, and how challenging it would be for a cat who has no whiskers at all. The most severe technology design issue we encountered during the study was whisker placement; each participant noted that the lack of having a front-facing whisker on the forehead made it difficult to avoid obstacles directly in front of the wearer. Cats have a set of superciliary or suborbital whiskers on their brow for this very purpose, and the lack of these front-facing whiskers on our wearable was frustrating for participants. We did not include them in the original design because we wanted to focus on the whiskers on the sides of the faces, not realizing how important it would be to include them in other areas on the head. Although participants were able to make use of the whiskers during this study, we believe that a longer study would give participants a more adequate amount of time to adapt to the whiskers and therefore exhibit more natural behaviors, such as participants relying more on the whiskers than their hands during resource collection. It might also allow a different set of behaviors to emerge, as wearers become more comfortable in the environment and reduce thigmotaxic behavior. We continue to investigate how the whisker wearable, and technologies like it, can remediate human experience in order to support deeper intersubjectivity with animals, and how such remediations can offer an experiential framework for science and science education. To address the aforementioned technological limitations and move towards a more authentic design, we are working towards a more modular and customizable design. The next generation of the wearable will include additional sensing capabilities that support cat senses beyond whiskers, like hearing. A new custom board, Dr. Bones (Figure 10), will serve as a connection hub and accommodate the micro:bit as the primary controller for all input and output modules. Incorporating the micro:bit into our design will enable participants to program their individual modules using the Makecode programming environment. In exposing the individual sensing and output elements of this project we aim to encourage young people to create their own sensory augmentation systems. We conjecture that this customizability will offer a variety of ways for people to engage with scientific ideas relating to animals. In addition to offering customizability for sensors, we will be offering people the ability to customize the placement of sensors and attachments on the wearable. For example, participants who wished they had a front-facing whisker during the activity will be able to create one. This will be more authentic to the way cats' whiskers work because cats and other mammals are able to direct control over the direction of their whiskers, as well as other parts of their bodies like their ears. Further, we are addressing the lack of vibrotactile directional feedback by adding three motors per whisker to our design; therefore, through motor sequencing and drive intensity, we can render the direction and angle of the bend human sensation. One intriguing aspect of our work in adding whiskers to humans was the ability to create new types of empathetic experiences. While our initial work has focused on developing new technology and the analytical infrastructure needed to test it, we see several opportunities for more deeply exploring the experiential aspects of this technology. First, we may consider how to explore the transhumanist aspects of gaining a new sensory capability. Can people find uses for wearable whiskers in their daily activities or as part of their everyday lived experience? By creating a more portable, lightweight version of the hardware, we may explore opportunities to send this device into the world to see what people make of it. Second, we may consider how wearable whiskers can increase understanding of, and empathy for, the experiences of nonhuman animals. For example, by leading a person through an experience similar to the everyday behavior of a feral cat (sneaking through backyards, chasing birds, searching for edible items), can wearers of the whiskers better comprehend and empathize with the experiences of feral cats? Finally, we may explore how experiences with wearable whiskers could increase an individual's understanding of, and relationship with, their own pets or other familiar animals. For example, children often must be taught what kinds of touch are liked and disliked by their pets; until they learn this, they may be frustrated by their pets' apparent distrust or fear when they are nearby. We may explore how to design experiences that can help a human understand a particular aspect of their pets' lived experience as a way of supporting a more respectful relationship between species. Wearable technologies and embodied learning experiences free humans from the confines of their biological limitations. This enables researchers to provide low-cost opportunities that offer firsthand perspective-taking experiences for people, allowing people to experiment with new sensory interactions, including ones that non-human animals have access to. We presented the design of the Whisker Beard, which does just that-provides humans with the opportunity to experience what it would be like to have a new sense, in this case, what it would be like to have whiskers. We introduced concepts from animal behavioral science research and described how we applied it to evaluating the experiences of participants' while immersed in an animal perspective-taking activity. Our observations of participants' enactment of animal-like behaviors, as well as their impressions about the experience suggest that they were immersed in the sensory experience of being a cat with whiskers. We are actively iterating on the designs of our hardware to offer more customizability. This will enable participants to design their own sensory augmenting technologies where they can explore their own curiosities about their pets' other senses. In near-future experiments we will iterate on the design of the wearable activity to offer a more immersive experience where participants can continue to enact animal-like behaviors. Our next steps will then be to investigate how participants developing increased awareness of animals' sensory experiences can support their enactment of empathetically-oriented design activities focused on improving animals' quality of life.

This paper explores using wearable sensory augmenting technology to facilitate first-hand perspective-taking of what it is like to have cat-like whiskers.

We explored the visualization designs that can support chronic patients to present and review their health data with healthcare providers during clinical visits.


Question: Represent the article for finding a single-sentence TLDR

Question: Generalization error (also known as the out-of-sample error) measures how well the hypothesis learned from training data generalizes to previously unseen data. Proving tight generalization error bounds is a central question in statistical learning theory. In this paper, we obtain generalization error bounds for learning general non-convex objectives, which has attracted significant attention in recent years. We develop a new framework, termed Bayes-Stability, for proving algorithm-dependent generalization error bounds. The new framework combines ideas from both the PAC-Bayesian theory and the notion of algorithmic stability. Applying the Bayes-Stability method, we obtain new data-dependent generalization bounds for stochastic gradient Langevin dynamics (SGLD) and several other noisy gradient methods (e.g., with momentum, mini-batch and acceleration, Entropy-SGD). Our recovers (and is typically tighter than) a recent in and improves upon the in. Our experiments demonstrate that our data-dependent bounds can distinguish randomly labelled data from normal data, which provides an explanation to the intriguing phenomena observed in Zhang et al. (2017a). We also study the setting where the total loss is the sum of a bounded loss and an additiona l`2 regularization term. We obtain new generalization bounds for the continuous Langevin dynamic in this setting by developing a new Log-Sobolev inequality for the parameter distribution at any time. Our new bounds are more desirable when the noise level of the processis not very small, and do not become vacuous even when T tends to infinity. Non-convex stochastic optimization is the major workhorse of modern machine learning. For instance, the standard supervised learning on a model class parametrized by R d can be formulated as the following optimization problem: where w denotes the model parameter, D is an unknown data distribution over the instance space Z, and F: R d × Z → R is a given objective function which may be non-convex. A learning algorithm takes as input a sequence S = (z 1, z 2, . . ., z n) of n data points sampled i.i.d. from D, and outputs a (possibly randomized) parameter configurationŵ ∈ R d. A fundamental problem in learning theory is to understand the generalization performance of learning algorithms-is the algorithm guaranteed to output a model that generalizes well to the data distribution D? Specifically, we aim to prove upper bounds on the generalization error err gen (S) = L(ŵ, D) − L(ŵ, S), where L(ŵ, D) = Ez∼D[L(ŵ, z)] and L(ŵ, S) = 1 n n i=1 L(ŵ, z i) are the population and empirical losses, respectively. We note that the loss function L (e.g., the 0/1 loss) could be different from the objective function F (e.g., the cross-entropy loss) used in the training process (which serves as a surrogate for the loss L). Classical learning theory relates the generalization error to various complexity measures (e.g., the VC-dimension and Rademacher complexity) of the model class. Directly applying these classical complexity measures, however, often fails to explain the recent success of over-parametrized neural networks, where the model complexity significantly exceeds the amount of available training data (see e.g., Zhang et al. (2017a) ). By incorporating certain data-dependent quantities such as margin and compressibility into the classical framework, some recent work (e.g., ; ;) obtains more meaningful generalization bounds in the deep learning context. An alternative approach to generalization is to prove algorithm-dependent bounds. One celebrated example along this line is the algorithmic stability framework initiated by. Roughly speaking, the generalization error can be bounded by the stability of the algorithm (see Section 2 for the details). Using this framework, study the stability (hence the generalization) of stochastic gradient descent (SGD) for both convex and non-convex functions. Their work motivates recent study of the generalization performance of several other gradient-based optimization methods:;;;;;;. In this paper, we study the algorithmic stability and generalization performance of various iterative gradient-based method, with certain continuous noise injected in each iteration, in a non-convex setting. As a concrete example, we consider the stochastic gradient Langevin dynamics (SGLD) (see ; ;). Viewed as a variant of SGD, SGLD adds an isotropic Gaussian noise at every update step: where g t (W t−1) denotes either the full gradient or the gradient over a mini-batch sampled from training dataset. We also study a continuous version of, which is the dynamic defined by the following stochastic differential equation (SDE): where B t is the standard Brownian motion. Most related to our work is the study of algorithm-dependent generalization bounds of stochastic gradient methods. first study the generalization performance of SGD via algorithmic stability. They prove a generalization bound that scales linearly with T, the number of iterations, when the loss function is convex, but their for general non-convex optimization are more restricted. presents a generalization bound that also combines PAC-Bayesian analysis with stability. However, their prior and posterior are probability distributions on the hyperparameter space, while ours are distributions on the hypothesis space. Our work is a follow-up of the recent work by , in which they provide generalization bounds for SGLD from both stability and PAC-Bayesian perspectives. Another closely related work by derives similar bounds for noisy stochastic gradient methods, based on the information theoretic framework of. However, their bounds scale as O(T /n) (n is the size of the training dataset) and are sub-optimal even for SGLD. We acknowledge that besides the algorithm-dependent approach that we follow, recent advances in learning theory aim to explain the generalization performance of neural networks from many other perspectives. Some of the most prominent ideas include bounding the network capacity by the norms of weight matrices. Most of these are stated in the context of neural networks (some are tailored to networks with specific architecture), whereas our work addresses generalization in non-convex stochastic optimization in general. We also note that some recent work provides explanations for the phenomenon reported in Zhang et al. (2017a) from a variety of different perspectives (e.g., ; Arora et al. (2018; 2019) ). first consider stochastic gradient Langevin dynamics (SGLD) as a sampling algorithm in the Bayesian inference context. give a non-asymptotic analysis and establish the finite-time convergence guarantee of SGLD to an approximate global minimum. Zhang et al. (2017b) analyze the hitting time of SGLD and prove that SGLD converges to an approximate local minimum. These are further improved and generalized to a family of Langevin dynamics based algorithms by the subsequent work of. In this paper, we provide generalization guarantees for the noisy variants of several popular stochastic gradient methods. The Bayes-Stability method and data-dependent generalization bounds. We develop a new method for proving generalization bounds, termed as Bayes-Stability, by incorporating ideas from the PAC-Bayesian theory into the stability framework. In particular, assuming the loss takes value in [0, C], our method shows that the generalization error is bounded by both 2C Ez[2KL(P, Q z)] and 2C Ez[2KL(Q z, P)], where P is a prior distribution independent of the training set S, and Q z is the expected posterior distribution conditioned on z n = z (i.e., the last training data is z). The formal definition and the can be found in Definition 5 and Theorem 7. Inspired by , instead of using a fixed prior distribution, we bound the KLdivergence from the posterior to a distribution-dependent prior. This enables us to derive the following generalization error bound that depends on the expected norm of the gradient along the optimization path: Here S is the dataset and g e (t) is the expected empirical squared gradient norm at step t; see Theorem 11 for the details., where L is the global Lipschitz constant of the loss, our new bound depends on the data distribution and is typically tighter (as the gradient norm is at most L). In modern deep neural networks, the worstcase Lipschitz constant L can be quite large, and typically much larger than the expected empirical gradient norm along the optimization trajectory. Specifically, in the later stage of the training, the expected empirical gradient is small (see Figure 1(d) for the details). Hence, our generalization bound does not grow much even if we train longer at this stage. Our new bound also offers an explanation to the difference between training on correct and random labels observed by Zhang et al. (2017a). In particular, we show empirically that the sum of expected squared gradient norm (along the optimization path) is significantly higher when the training labels are replaced with random labels (Section 3, Remark 13, Figure 1, Appendix C.2). We would also like to mention the PAC-Bayesian bound (for SGLD with 2 -regularization) proposed by. (This bound is different from what we mentioned before; see Theorem 2 in their paper.) Their bound scales as O(1/ √ n) and the numerator of their bound has a similar sum of gradient norms (with a decaying weight if the regularization coefficient λ > 0). Their bound is based on the PAC-Bayesian approach and holds with high probability, while our bound only holds in expectation. Extensions. We remark that our technique allows for an arguably simpler proof of (, Theorem 1); the original proof is based on SDE and Fokker-Planck equation. More importantly, our technique can be easily extended to handle mini-batches and a variety of general settings as follows. 1. Extension to other gradient-based methods. Our naturally extends to other noisy stochastic gradient methods including momentum due to (Theorem 26), Nesterov's accelerated gradient method in (Theorem 26), and Entropy-SGD proposed by (Theorem 27). 2. Extension to general noises. The proof of the generalization bound in relies heavily on that the noise is Gaussian 1, which makes it difficult to generalize to other noise distributions such as the Laplace distribution. In contrast, our analysis easily carries over to the class of log-Lipschitz noises (i.e., noises drawn from distributions with Lipschitz log densities). 3. Pathwise stability. In practice, it is also natural to output a certain function of the entire optimization path, e.g., the one with the smallest empirical risk or a weighted average. We show that the same generalization bound holds for all such variants (Remark 12). We note that the analysis in an independent work of also satisfies this property, (see Corollary 1 in their work), which scales at a slower O(1/ √ n) rate (instead of O(1/n)) when dealing with C-bounded loss. Generalization bounds with 2 regularization via Log-Sobolev inequalities. We also study the setting where the total objective function F is the sum of a C-bounded differentiable objective F 0 and an additional 2 regularization term λ 2 w 2 2. In this case, F can be treated as a perturbation of a quadratic function, and the continuous Langevin dynamics (CLD) is well understood for quadratic functions. We obtain two generalization bounds for CLD, both via the technique of Log-Sobolev inequalities, a powerful tool for proving the convergence rate of CLD. One of our bounds is as follows (Theorem 15): The above bound has the following advantages: 1. Applying e −x ≥ 1 − x, one can see that our bound is at most O(√ T /n), which matches the previous bound in (, Proposition 8) 3. 2. As time T grows, the bound is upper bounded by and approaches to 2e 4βC CLn −1 β/λ (unlike the previous O( √ T /n) bound that goes to infinity as T → +∞). If the noise level is not so small (i.e., β is not very large), the generalization bound is quite desirable. Our analysis is based on a Log-Sobolev inequality (LSI) for the parameter distribution at time t, whereas most known LSIs only hold for the stationary distribution of the Markov process. We prove the new LSI by exploiting the variational formulation of the entropy formula. Notations. We use D to denote the data distribution. The training dataset S = (z 1, . . ., z n) is a sequence of n independent samples drawn from D. S, S ∈ Z n are called neighboring datasets if and only if they differ at exactly one data point (we could assume without loss of generality that z n = z n). Let F (w, z) and L(w, z) be the objective and the loss functions, respectively, where w ∈ R d denotes a model parameter and z ∈ Z is a data point. Define and L(w, D) are defined similarly. A learning algorithm A takes as input a dataset S, and outputs a parameter w ∈ R d randomly. Let G be the set of all possible mini-batches. G n = {B ∈ G : n ∈ B} denotes the collection of mini-batches that contain the n-th data point, while G n = G \ G n. Let diam(A) = sup x,y∈A x − y 2 denote the diameter of a set A. 1 In particular, their proof leverages the Fokker-Planck equation, which describes the time evolution of the density function associated with the Langevin dynamics and can only handle Gaussian noise. 2 They assume the loss is sub-Gaussian. By Hoeffding's lemma, C-bounded random variables are subGaussian with parameter C. 3 The proof of their O(√ T /n) bound can be easily extended to our setting with 2 regularization. Definition 2 (Expected generalization error). The expected generalization error of a learning algorithm A is defined as Algorithmic Stability. Intuitively, a learning algorithm that is stable (i.e., a small perturbation of the training data does not affect its output too much) can generalize well. In the seminal work of (see also), the authors formally defined algorithmic stability and established a close connection between the stability of a learning algorithm and its generalization performance. Definition 3 (Uniform stability). A randomized algorithm A is n -uniformly stable w.r.t. loss L, if for all neighboring sets S, S ∈ Z n, it holds that where w S and w S denote the outputs of A on S and S respectively. Lemma 4 (Generalization in expectation). Suppose a randomized algorithm A is n -uniformly stable. Then, |err gen | ≤ n. In this section, we incorporate ideas from the PAC-Bayesian theory (see e.g.,) into the algorithmic stability framework. Combined with the technical tools introduced in previous sections, the new framework enables us to prove tighter data-dependent generalization bounds. First, we define the posterior of a dataset and the posterior of a single data point. Definition 5 (Single-point posterior). Let Q S be the posterior distribution of the parameter for a given training dataset S = (z 1, . . ., z n). In other words, it is the probability distribution of the output of the learning algorithm on dataset S (e.g., for T iterations of SGLD in, Q S is the pdf of W T ). The single-point posterior Q (i,z) is defined as For convenience, we make the following natural assumption on the learning algorithm: Assumption 6 (Order-independent). For any fixed dataset S = (z 1, . . ., z n) and any permutation p, Q S is the same as Q S p, where S p = (z p1, . . ., z pn). Assumption 6 implies Q (1,z) = · · · = Q (n,z), so we use Q z as a shorthand for Q (i,z) in the following. Note that this assumption can be easily satisfied by letting the learning algorithm randomly permute the training data at the beginning. It is also easy to verify that both SGD and SGLD satisfy the order-independent assumption. Now, we state our new Bayes-stability framework, which holds for any prior distribution P over the parameter space that is independent of the training dataset S. Theorem 7 (Bayes-Stability). Suppose the loss function L(w, z) is C-bounded and the learning algorithm is order-independent (Assumption 6). Then for any prior distribution P not depending on S, the generalization error is bounded by both 2C E z 2KL(P, Q z) and 2C E z 2KL(Q z, P). Remark 8. Our Bayes-Stability framework originates from the algorithmic stability framework, and hence is similar to the notions of uniform stability and leave-one-out error (see). However, there are important differences. Uniform stability is a distribution-independent property, while Bayes-Stability can incorporate the information of the data distribution (through the prior P). Leave-one-out error measures the loss of a learned model on an unseen data point, yet Bayes-Stability focuses on the extent to which a single data point affects the outcome of the learning algorithm (compared to the prior). To establish an intuition, we first apply this framework to obtain an expectation generalization bound for (full) gradient Langevin dynamics (GLD), which is a special case of SGLD in (i.e., GLD uses the full gradient ∇ w F (W t−1, S) as g t (W t−1)). Theorem 9. Suppose that the loss function L is C-bounded. Then we have the following expected generalization bound for T iterations of GLD: where g e (t) ] is the empirical squared gradient norm, and W t is the parameter at step t of GLD. Proof The proof builds upon the following technical lemma, which we prove in Appendix A.2. Lemma 10. Let (W 0, . . ., W T) and (W 0, . . ., W T) be two independent sequences of random variables such that for each t ∈ {0, . . ., T}, W t and W t have the same support. Suppose W 0 and W 0 follow the same distribution. Then, where W ≤t denotes (W 0, . . ., W t) and W <t denotes W ≤t−1.,0) ], where 0 denotes the zero data point (i.e., f (w, 0) = 0 for any w). Theorem 7 shows that By the convexity of KL-divergence, for a fixed z ∈ Z, we have Let (W t) t≥0 and(W t) t≥0 be the training process of GLD for S = (S, z) and S = (S, 0), respectively. Note that for a fixed w <t, both W t |W <t = w <t and W t |W <t = w <t are Gaussian Applying Lemma 10 and Recall that W t−1 is the parameter at step t − 1 using S = (S, z) as dataset. In this case, we can rewrite z as z n since it is the n-th data point of S. Note that SGLD satisfies the order-independent assumption, we can rewrite z as z i for all i ∈ [n]. Together with,, and using x i, we can prove this theorem. More generally, we give the following bound for SGLD. The proof is similar to that of Theorem 9; the difference is that we need to bound the KL-divergence between two Gaussian mixtures instead of two Gaussians. This proof is more technical and deferred to Appendix A.3. Theorem 11. Suppose that the loss function L is C-bounded and the objective function f is Llipschitz. Assume that the following conditions hold: Then, the following expected generalization error bound holds for T iterations of SGLD: where g e (t) ] is the empirical squared gradient norm, and W t is the parameter at step t of SGLD. Furthermore, based on essentially the same proof, we can obtain the following bound that depends on the population gradient norm: The full proofs of the above are postponed to Appendix A, and we provide some remarks about the new bounds. Remark 12. In fact, our proof establishes that the above upper bound holds for the two sequences W ≤T and W ≤T:. Hence, our bound holds for any sufficiently regular function over the parameter sequences:. In particular, our generalization error bound automatically extends to several variants of SGLD, such as outputting the average of the trajectory, the average of the suffix of certain length, or the exponential moving average. Remark 13. Inspired by Zhang et al. (2017a), we run both GLD (Figure 1) and SGLD (Appendix C.2) to fit both normal data and randomly labelled data (see Appendix C for more experiment details). As shown in Figure 1 and Figure 2 in Appendix C.2, larger random label portion p leads to both much higher generalization error and much larger generalization error bound. Moreover, the shapes of the curves our bounds look quite similar to that of the generalization error curves. In this section, we study the generalization error of Continuous Langevin Dynamics (CLD) with 2 regularization. Throughout this section, we assume that the objective function over training set S is defined as F (w, S) = F 0 (w, S) + λ 2 w 2 2, and moreover, the following assumption holds. Assumption 14. The loss function L and the original objective F 0 are C-bounded. Moreover, F 0 is differentiable and L-lipschitz. The Continuous Langevin Dynamics is defined by the following SDE: where (B t) t≥0 is the standard Brownian motion on R d and the initial distribution µ 0 is the centered Gaussian distribution in R d with covariance We show that the generalization error of CLD is upper bounded by O e 4βC n −1 β/λ, which is independent of the training time T (Theorem 15). Furthermore, as T goes to infinity, we have a tighter generalization error bound O βC 2 n −1 (Theorem 39 in Appendix B). We also study the generalization of Gradient Langevin Dynamics (GLD), which is the discretization of CLD: where ξ k is the standard Gaussian random vector in R d. By leveraging a developed in , we show that, as Kη 2 tends to zero, GLD has the same generalization as CLD (see Theorems 15 and 39). We first formally state our first main in this section. dw) has the following expected generalization error bound: In addition, if L is M -smooth and non-negative, by setting λβ > 2, λ > 8M 2 ), GLD (running K iterations with the same µ 0 as CLD) has the expected generalization error bound: where C 1 is a constant that only depends on M, λ, β, b, L and d. The following lemma is crucial for establishing the above generalization bound for CLD. In particular, we need to establish a Log-Sobolev inequality for µ t, the parameter distribution at time t, for every time step t > 0. In contrast, most known LSIs only characterize the stationary distribution of the Markov process. The proof of the lemma can be found in Appendix B. Lemma 16. Under Assumption 14, let µ t be the probability measure of W t in CLD (with dµ 0 = 1 Z e −λβ w 2 2 dw). Let ν be a probability measure that is absolutely continuous with respect to µ t. Suppose dµ t = π t (w) dw and dν = γ(w) dw. Then, it holds that We sketch the proof of Theorem 15, and the complete proof is relegated to Appendix B. Proof Sketch of Theorem 15 Suppose S and S are two neighboring datasets. Let (W t) t≥0 and (W t) t≥0 be the process of CLD running on S and S, respectively. Let γ t and π t be the pdf of W t and W t. Let F S (w) denote F (w, S). We have The high level idea to prove this bound is very similar to that in. We first observe that the (stationary) Gibbs distribution µ has a small generalization error. Then, we bound the distance from µ t to µ. In our setting, we can use the Holley-Stroock perturbation lemma which allows us to bound the Logarithmic Sobolev constant, and we can thus bound the above distance easily. In this paper, we prove new generalization bounds for a variety of noisy gradient-based methods. Our current techniques can only handle continuous noises for which we can bound the KL-divergence. One future direction is to study the discrete noise introduced in SGD (in this case the KL-divergence may not be well defined). For either SGLD or CLD, if the noise level is small (i.e., β is large), it may take a long time for the diffusion process to reach the stable distribution. Hence, another interesting future direction is to consider the local behavior and generalization of the diffusion process in finite time through the techniques developed in the studies of metastability (see e.g., Lemma 17. Under Assumption 6, for any prior distribution P not depending on the dataset S = (z 1, . . ., z n), the generalization error is upper bounded by where L(w) denotes the population loss L(w, D). Let err train = ES Ew∼Q S L(w, S) and err test = ES Ew∼Q S L(w). We can rewrite generalization error as err gen = err test − err train, where and Thus, we have Now we are ready to prove Theorem 7, which we restate in the following. Theorem 7 (Bayes-Stability). Suppose the loss function L(w, z) is C-bounded and the learning algorithm is order-independent (Assumption 6), then for any prior distribution P not depending on S, the generalization error is bounded by both 2C E z 2KL(P, Q z) and 2C E z 2KL(Q z, P). Proof By Lemma 17, The other bound follows from a similar argument. Now we turn to the proof of Theorem 11. The following lemma allows us to reduce the proof of algorithmic stability to the analysis of a single update step. Proof By the chain rule of the KL-divergence, The lemma follows from a summation over t = 1,..., T. The following lemma (see e.g., (, Section 9)) gives a closed-form formula for the KLdivergence between two Gaussian distributions. The following lemma (, Theorem 3) helps us to upper bound the KL-divergence. Definition 19. Let P and Q be two probability distributions on R d. The directional triangular discrimination from P to Q is defined as where Lemma 20. For any two probability distributions P and Q on R d, Recall that G is the set of all possible mini-batches. G n = {B ∈ G : n ∈ B} denotes the collection of mini-batches that contain n, while G n = G \ G n. diam(A) = sup x,y∈A x − y denotes the diameter of set A. The following technical lemma upper bounds the KL-divergence between two Gaussian mixtures induced by sampling a mini-batch from neighbouring datasets. Lemma 21. Suppose that batch size b ≤ n/2. {µ B : B ∈ G} and {µ B : B ∈ G} are two collections of points in R d labeled by mini-batches of size b that satisfy the following conditions for some constant β ∈ [0, σ]: B∈G p µ B,σ and P = 1 |G| B∈G p µ B,σ be two mixture distributions over all mini-batches. Then, Proof of Lemma 21 By Lemma 20, KL(P, P) is bounded by The numerator of the above integrand is upper bounded by while the denominator can be lower bounded as follows: which implies, by the convexity of 1/x, that Inequalities and together imply Now we bound the right-hand side of for fixed A and B. By applying a translation and a rotation, we can assume without loss of generality that µ A = 0, and the last d − 2 coordinates of µ B and µ B are all zero. Note that the integral is unchanged when we project the space to the twodimensional subspace corresponding to the first two coordinates. Thus, it suffices to prove a bound for d = 2. We rewrite as Let I be the integral in the right-hand side of. Note that 2, we make two claims which we will prove later:, φ(y, δ) is non-increasing in δ. The above claims imply that: 1. For any r ∈ 0, 2. For any r ∈ Plugging the above into gives We conclude that n 2 σ 2. Finally, we prove the two claims used above:, we have e (b) 2 e −δ(δ+2y) (δ + y) < 0 for y ≥ 0, we conclude that for any y ≥ 1/ √ 2: Recall that SGLD on dataset S is defined as Here γ t is the step size. B t = {i 1, . . ., i b} is a subset of {1, . . ., n} of size b, and S Bt = (z i1, . . ., z i b) is the mini-batch indexed by B t. Recall that F (w, S) denotes. We restate and prove Theorem 11 in the following. Theorem 11. Suppose that the loss function L is C-bounded and the objective function F is Llipschitz. Assume that the following conditions hold: Then, the following expected generalization error bound holds for T iterations of SGLD: where g e (t) = Ew∼W t−1 [] is the empirical squared gradient norm, and W t is the parameter at step t of SGLD. Proof By Theorem 7, we have for any prior distribution P. In particular, we define the prior as P (w) = E S∼D n−1 [P S (w)], where P S (w) = Q (S,0). By the convexity of KL-divergence, Fix a data point z ∈ Z. Let (W t) t≥0 and (W t) t≥0 be the training process of SGLD for S = (S, z) and S = (S, 0), respectively. Fix a time step t and w <t = (w 0, . . ., w t−1). Let P t and P t denote the distribution of W t and W t conditioned on W <t = w <t and W <t = w <t, respectively. By the definition of SGLD, we have, and p µ denotes the Gaussian distribution for B ∈ G n and µ B = µ B for B ∈ G n. By applying Lemma 21 with β = γt ∇F (wt−1,z) 2 b and σ = σ t, By Lemma 10, which implies that Together with and, we have Since SGLD is order-independent, we can replace ∇F (w, z n) with ∇F (w, z i) for any i ∈ [n] in the right-hand side of the above bound. Our theorem then follows from the concavity of √ x. Furthermore, if we bound KL(P, Q z) instead of KL(Q z, P) in the above proof, we obtain the following bound that depends on the population squared gradient norm: We can extend the generalization bounds in previous sections, which require the noise to be Gaussian, to other general noises, namely the family of log-lipschitz noises. Definition 22 (Log-Lipschitz Noises). A probability distribution on R d with density p is L-loglipschitz if and only if ∇ ln p(w) ≤ L holds for any w ∈ R d. A random variable ζ is called an L-log-lipschitz noise if and only if it is drawn from an L-log-lipschitz distribution. The analog of SGLD, noisy momentum method (Definition 24), and noisy NAG (Definition 25) can be naturally defined by replacing the Gaussian noise ζ t at each iteration with an independent L-log-lipschitz noise in the definition. The following lemma is an analog of Lemma 21 under L-log-lipschitz noises. Recall that G denotes a collection of mini-batches of size b. Lemma 23 readily implies the analogs of Theorems 11, 26 and 27 under more general noise distributions. Lemma 23. Suppose that batch size b ≤ n/2 and N is an L noise -log-lipschitz distribution on R d. {µ B : B ∈ G} and {µ B : B ∈ G} are two collections of points in R d that satisfy the following conditions for some constant β ∈ 0, 1 Lnoise: For µ ∈ R d, let p µ denote the distribution of ζ +µ when ζ is drawn from N. Let P = 1 |G| B∈G p µ B and P = 1 |G| B∈G p µ B be mixture distributions over all mini-batches. Then, for some constant C 0 that only depends on L noise. Following the same argument as in the proof of Lemma 21, we have where Fixed A ∈ G n and B ∈ G n. Let p noise denote the density of the noise distribution N. Since µ A − µ B ≤ 1 and p noise is L noise -log-lipschitz, we have Similarly, since µ B − µ B ≤ β, we have Then, it follows from βL noise ≤ 1 that Therefore, the integral on the righthand side of can be upper bounded as follows: Plugging the above inequality into and gives and We adopt the formulation of Classical Momentum and Nesterov's Accelerated Gradient (NAG) methods in and consider the noisy versions of them. Definition 24 (Noisy Momentum Method). Noisy Momentum Method on objective function F (w, z) and dataset S is defined as Definition 25 (Noisy Nesterovs Accelerated Gradient). Noisy Nesterovs Accelerated Gradient (NAG) on objective function F (w, z) and dataset S is defined as In both definitions, γ t is the step size, mini-batch B t is drawn uniformly from G, ζ t is a Gaussian noise drawn from N (0,, and η ∈ is the momentum coefficient. Theorem 26. Under the same assumptions on the loss function, objective function, batch size and learning rate as in Theorem 11, the generalization bounds in Theorem 11 still hold for noisy momentum method and noisy NAG. For any time step t and w <t = (w 0, w 1, ..., w t−1), let P t and P t denote the distribution of W t and W t conditioned on W <t = w <t and W <t = w <t, respectively. By definition, we have If t = 1, for both noisy momentum method and noisy NAG, we have µ B = w t−1 − γ t ∇ w F (w t−1, S B), µ B = w t−1 − γ t ∇ w F (w t−1, S B). For t > 1, if noisy momentum method is used, we have µ B = w t−1 + η(w t−1 − w t−2) − γ t ∇ w F (w t−1, S B),. Similarly, the following holds under noisy NAG: In either case, it can be verified that the conditions of Lemma 21 hold for β = 2γtL b and σ = σ t. The rest of the proof is the same as the proof of Theorem 11. In the Entropy-SGD algorithm due to , instead of directly optimizing the original objective F (w), we minimize the negative local entropy defined as follows: Intuitively, a wider local minimum has a lower loss (i.e., −E(w, γ)) than sharper local minima. for more details. The Entropy-SGD algorithm invokes standard SGD to minimize the negative local entropy. However, the gradient of negative local entropy is hard to compute. Thus, the algorithm uses exponential averaging to estimate the gradient in the SGLD loop; see Algorithm 1 for more details. We have the following generalization bound for Entropy-SGD. Algorithm 1: Entropy-SGD Input: Training set S = (z 1, .., z n) and loss function g(w, z). Hyper-parameters: Scope γ, SGD learning rate η, SGLD step size η and batch size b. Theorem 27. Suppose that the loss function L is C-bounded and the objective function F is Llipschitz. If batch size b ≤ n/2 and √ η ≤ ε/(20L), the following expected generalization error bound holds for Entropy-SGD: where ] is the empirical squared gradient norm, and W t,k denotes the training process with respect to S. Since g e (t, k) is at most L 2, it further implies the generalization error of Entropy-SGD is bounded Proof of Theorem 27 Define the history before time step (t, k) as follows: Since µ is only determined by W, we only need to focus on W. This proof is similar to the proof of Theorem 11. By setting P = E S [Q (S,0) ]. Suppose S = (S, z) and S = (S, 0) are fixed, let W and W denote their training process, respectively. Considering the following 3 cases: 1. W t,0 ← W t−1,K+1: In this case, for a fixed w ≤(t−1,K+1), we have In this case, fix a w ≤(t,k), applying Lemma 21 gives In this case, for a fixed w ≤(t,K), we have By applying Lemma 10, we have and Where g e (t, k) is the empirical squared gradient norm of the k-th SGLD iteration in the t-th SGD iteration, respectively. The rest of the proof is the same as the proof of Theorem 11. The continuous version of the noisy gradient descent method is the Langevin dynamics, described by the following stochastic differential equation: where B t is the standard Brownian motion. To analyze the above Langevin dynamics, we need some preliminary knowledge about Log-Sobolev inequalities. Let p t (w, y) denote the probability density function (i.e., probability kernel) describing the distribution of W t starting from w. For a given SDE such as, we can define the associated diffusion semigroup P: Definition 28 (Diffusion Semigroup). (see e.g., ) Given a stochastic differential equation (SDE), the associated diffusion semigroup P = (P t) t≥0 is a family of operators that satisfy for every t ≥ 0, P t is a linear operator sending any real-valued bounded measurable function f on R d to The semigroup property P t+s = P t • P s holds for every t, s ≥ 0. Another useful property of P t is that it maps a nonnegative function to a nonnegative function. The carré du champ operator Γ of this diffusion semigroup (w.r.t) is We use the shorthand notation Γ(f) = Γ(f, f) = β −1 ∇f 2 2, and define (with the convention that 0 log 0 = 0) Definition 29 (Logarithmic Sobolev Inequality). (see e.g., ) A probability measure µ is said to satisfy a logarithmic Sobolev inequality LS(α) (with respect to Γ), if for all functions f: D(E) is the set of functions f ∈ L 2 (µ) for which the quantity dµ has a finite (decreasing) limit as t decreases to 0. A well-known Logarithmic Sobolev Inequality is the following for Gaussian measures. Lemma 30 (Logarithmic Sobolev Inequality for Gaussian measure). Let µ be the centered Gaussian measure on R d with covariance matrix σ 2 I d. Then µ satisfies the following LSI: Lemma 30 states that the centered Gaussian measure with covariance matrix σ 2 I d satisfies LS(βσ 2) (with respect to Γ), where Γ = β −1 ∇f, ∇g is the carré du champ operator of the diffusion semigroup defined above. Before proving our , we need some known from Markov diffusion process. It is well known that the invariant measure of the above CLD is the Gibbs measure dµ = 1 Zµ exp(−βF (w)) dw (, (1.3) ). In other words, µ satisfies R d P t f dµ = R d f dµ for every bounded positive measurable function f, where P t is the Markov semigroup in Definition 28. The following lemma by (see also ) allows us to determine the Logarithmic Sobolev constant of the invariant measure µ. Lemma 31 (Bounded perturbation). Assume that the probability measure ν satisfies LS(α) (with respect to Γ). Let µ be a probability measure such that 1/b ≤ dµ/dν ≤ b for some constant b > 1. Then µ satisfies LS(b 2 α) (with respect to Γ). In fact, Lemma 31 is a simple consequence of the following variational formula in the special case that φ(x) = x log x, which we will also need in our proof: Lemma 32 (Variational formula). (see .g., ) Let φ: I → R on some open interval I ⊂ R be convex of class C 2. For every (bounded or suitably integrable) measurable function f: R d → R with values in I, It is worth noting the integrand of the right-hand side is nonnegative due to the convexity of φ. Recall that F S (w) = F (w, S):= F 0 (w, S) + λ w 2 2 /2 is the sum of the empirical original objective F 0 (w, S) and 2 regularization. Let dµ = Lemma 33. Under Assumption 14, let Γ(f, g) = β −1 ∇f, ∇g be the carré du champ operator of the diffusion semigroup associated to CLD, and µ be the invariant measure of the SDE. Then, µ satisfies LS(e 4βC /λ) with respect to Γ. Let µ t be the probability measure of W t. By definition of P t, for any real-valued bounded measurable function f on R d and any s, t ≥ 0, In particular, if the invariant measure µ = µ ∞ exists, we have The following lemma is crucial for establishing the first generalization bound for CLD. In fact, we establish a Log-Sobolev inequality for µ t, the parameter distribution at time t, for any time t > 0. Note that our choice of the initial distribution µ 0 is important for the proof. Lemma 34. Under Assumption 14, let µ t be the probability measure of W t in (CLD) with initial dw. Let Γ be the carré du champ operator of diffusion semigroup associated to (CLD). Then, for any f: Proof Let µ be the invariant measure of CLD. By Lemma 33 and Definition 29, By applying Lemma 32 with φ(x) = x log x, we rewrite the left-hand side as where the last equation holds by the definition of invariant measure P t f dµ = f dµ. Thus, we have Let µ t be the probability measure of W t. Lemma 32 and together imply that Since dµ dµ0 ≤ exp(2βC) and µ is the invariant measure, we conclude that Lemma 16. Under Assumption 14, let µ t be the probability measure of W t in CLD (with dµ 0 = 1 Z e −λβ w 2 2 dw). Let ν be a probability measure that is absolutely continuous with respect to µ t. Suppose dµ t = π t (w) dw and dν = γ(w) dw. Then it holds that: Proof Let f (w) = γ(w)/π t (w), by Lemma 34 and We can see that the left-hand side is equal to KL(γ, π t) 6, and the right-hand side is equal to This concludes the proof.. We can rewrite F S (w) = 1 n n i=1 h(w, z i). Define µ S,k and ν S,t as the probability measure of W k (in GLD) and W t (in CLD), respectively. provided a bound of KL(µ S,k, ν S,ηK) under Assumption 35. This bound enables us to derive a generalization error bound for the discrete GLD from the bound for the continuous CLD. We use the assumption from. Their work considers the following SGLD: Where g S (w k) is a conditionally unbiased estimate of the gradient ∇F S (w k). In our GLD setting, 1. The function h takes non-negative real values, and there exist constants A, B ≥ 0, such that 2. For each z ∈ Z, the function h(·, z) is M -smooth: for some M > 0, 3. For each z ∈ Z, the function h(·, z) is (m, b)-dissipative: for some m > 0 and b ≥ 0, 4. There exists a constant δ ∈, such that, for each S ∈ Z n, 5. The probability law µ 0 of the initial hypothesis W 0 has a bounded and strictly positive density p 0 with respect to the Lebesgue measure on R d, and dw) has the following expected generalization error bound: 6 Indeed, 8M 2 ), the GLD (running K iterations with the same µ 0 as CLD) has the expected generalization error bound: where C 1 is a constant that only depends on M, λ, β, b, L and d. We apply the uniform stability framework. Suppose S and S are two neighboring datasets that differ on exactly one data point. Let (W t) t≥0 and (W t) t≥0 be the process of CLD running on S and S, respectively. Let γ t and π t be the pdf of W t and W t. We have According to Fokker-Planck equation (see) for CLD, we know that It follows that (integration by parts) and (integration by parts) Together with, we have Solving this differential inequality gives By Pinsker's inequality, we can finally see that By Lemma 4, the generalization error of CLD is bounded by the right-hand side of the above inequality. Now, we prove the second part of the theorem. Let (W k) k≥0 and (W k) k≥0 be the (discrete) GLD processes training on S and S, respectively. Then for any z ∈ Z: Since λβ > 2 and λ > and From, we have Combining, and, we have By Definition 3, GLD is n -uniformly stable. Applying Lemma 4 gives the generalization bound of GLD. Lemma 37 (Exponential decay in entropy). (, Theorem 5.2 .1) The logarithmic Sobolev inequality LS(α) for the probability measure µ is equivalent to saying that for every positive function ρ in L 1 (µ) (with finite entropy), for every t ≥ 0. The following Lemma shows that P t (dµ0 dµ) = µ t in our diffusion process. Lemma 38. Let P denote the diffusion semigroup of CLD. Let µ denote the invariant measure of P and let µ t denote the probability measure of W t. Then P t (dµ0 dµ) = µ t. Proof Let dµ = µ(x) dx and dµ t = µ t (x) dx. As shown in (, page 118), our diffusion process (Smoluchowski dynamics) is reversible, which means µ(x)p t (x, y) = µ(y)p t (y, x). Thus for any g(x), we have Since g is arbitrary, P t (dµ0 dµ) and µ t must be the same. dw) has the following expected generalization error bound: In addition, if F 0 is also M -smooth and non-negative, by setting λβ > 2, λ > 1 2 and η ∈ [0, 1 ∧ 2λ−1 8M 2), the GLD process (running K iterations with the same µ 0 as CLD) has the expected generalization error bound: where C 1 is a constant that only depends on M, λ, β, b, L and d. Proof of Theorem 39 Suppose S and S are two datasets that differ on exactly one data point. Let (W t) t≥0 and (W t) t≥0 be their processes, respectively. Let dµ t = π t (w) dw and dµ t = π t (w) dw be the probability measure of W t and W t, respectively. The invariant measure of CLD for S and S are denoted as µ and µ, respectively. Recall that dµ = 1 Z µ e −βF S (w) dw, dµ = 1 Z µ e −βF S (w) dw. The total variation distance of µ and µ is Since Zµ Z µ exp(−β(F S (w) − F S (w))) ∈ e Since µ and µ satisfy LS(e 4βC/λ) (Lemma 33), applying Lemma 37 with ρ = dµ0 dµ and ρ = dµ 0 dµ and Lemma 38 yields: KL(µ t, µ) ≤ exp −2λt e 4βC KL(µ 0, µ), KL(µ t, µ) ≤ exp −2λt e 4βC KL(µ 0, µ). Since KL(µ 0, µ) and KL(µ 0, µ) are upper bounded by 2βC, Pinsker's inequality implies that TV(µ t, µ) and TV(µ t, µ) are upper bounded by exp −2λt e 4βC βC. Combining with and note that TV(µ t, µ t) ≤ TV(µ t, µ) + TV(µ, µ) + TV(µ t, µ), we have −2λt e 4βC βC + 8βC 2 n. By Lemma 4, the generalization error of CLD is bounded by the right-hand side. The proof for GLD proceeds in the same way as the second part of the proof of Theorem 15. We first present the general setup of our experiments: • Small AlexNet: k is the kernel size, d is the depth of a convolution layer, fc(m) is the fullyconnected layer that has m neurons. The ReLU activation are used in the first 6 layers. • MLP: The MLP used in our experiment has 3 hidden layers, each having width 512. We also use ReLU as the activation function in MLP. Objective function: For a data point z = (x, y) in MNIST, the objective function is Random labels: Suppose the dataset contains n datapoint, and the corruption portion is p. We randomly select n · p data points, and replace their labels with random labels, as in Zhang et al. (2017a). The of this experiment (see Figure 1) is discussed in Section 3, Remark 13. Here we present our implementation details. We repeat our experiment 5 times. At every individual run, we first randomly sample 10000 data points from the complete MNIST training data. The initial learning rate γ 0 = 0.003. It decays 0.995 after every 60 steps, and it stops decaying when it is lower than 0.0005. During the training, we keep σ t = 0.2 √ 2γ t. Recall that the empirical squared gradient norm g e (t) in our bound (Theorem 9) Under review as a conference paper at ICLR 2020 Since it is time-consuming to compute the exact g e (t), in our experiment, we use an unbiased estimation instead. At every step, we randomly sample a minibatch B with batch size 200 from the training data, and use 1 200 i∈B ∇f (W t−1, z i) 2 as g e (t) to compute our bound in Figure 1. The estimation of g e (t) at every step t is shown in Figure 1(d). Since g e (t) is not very stable, in our figure, we plot its moving average over a window of size 100 to make the curve smoother (i.e., g avg (t) = 1 100 t+100 τ =t g e (τ)). In this subsection, we present some experiment for running SGLD on both MNIST and CIFAR10 datasets, to demonstrate that our bound (see Theorem 11), in particular the sum of the empirical squared gradient norms along the training path, can distinguish normal dataset from dataset that contains random labels. We note that in our experiments, the learn rate we choose is larger than that is required by the condition of Theorem 11. Due to the (non-optimal) constant in our bound, the bound is currently greater than 1, and hence we ignore the numbers on the y-axis. However, again, the curves of our bounds look quite similar to the generalization curves (see Figure 2). This indicates that the sum of squared empirical gradient norms is highly related to the generalization performance, and we believe by further optimizing the constants in our bound, we can achieve a generalization bound that is close to the real generalization error.

represent the short summary for finding the corresponding scientific article (long)

We give some generalization error bounds of noisy gradient methods such as SGLD, Langevin dynamics, noisy momentum and so forth.

Represent the short summary for finding the corresponding scientific article (long)

This paper 1) characterizes functions representable by ReLU DNNs, 2) formally studies the benefit of depth in such architectures,  3) gives an algorithm to implement empirical risk minimization to global optimality for two layer ReLU nets.