Brett M Frischmann
email: brett.frischmann@law.villanova.edu

Alain Marciano
email: alain.marciano@umontpellier.fr

Giovanni Battista Ramello
email: giovanni.ramello@uniupo.it

Giovanni Battista

Ramello Brett

M Frischmann

Retrospectives: Tragedy of the Commons After 50 Years

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Garrett Hardin's "The Tragedy of the Commons" (1968) has been incredibly influential in biology, ecology, and various social sciences, including economics. It has become a totemic reference to which tributes are regularly paid (for examples, see Feeny,Berkes,McCay and 3 resource with the governance of that resource, and it further conflated open access with commons, despite significant differences in those forms of governance. We will clarify these distinctions and thus explore the limitations of Hardin's view. Unfortunately, Hardin's distorted perspective on the commons often persists to this day in economic discussions. Thus, we discuss some applications and extensions of research on the commons, including infrastructure, knowledge, and other issues. Interdisciplinary scholars have identified an expanding set of shared resources for which commons governance may effectively address social dilemmas.

Hardin's Tragedy of the Commons

Garrett Hardin (1915Hardin ( -2003) ) is well-known in biology for his work on evolution and natural selection. He spent most of his career at the University of California at Santa Barbara, where he arrived in 1946 and stayed until his retirement in 1978. Hardin was also prominent as a public intellectual. He contributed to magazines, gave popular lectures, appeared on TV and radio, and testified before many congressional committees. In both his academic and non-academic work, Hardin emphasized the need to control population growth. The "Tragedy of the Commons" essay is his most famous attempt to make his case, but it was neither the first nor the last attempt (as emphasized by [START_REF] Oakes | Garrett Hardin's Tragic Sense of Life[END_REF]. A prominent early effort along these lines, "The Competitive Exclusion Principle," appeared in Science magazine in 1960. In this article, Hardin stated that "complete competition cannot coexist" (p. 1292; italics in original), explaining: … (i) if two non-interbreeding populations "do the same thing"--that is occupy precisely the same niche in Elton's sense-and (ii) if they are "sympatric" that is, if they occupy the same geographic territory-and (iii) if population A multiplies even the least bit faster than population B, then ultimately A will completely displace B, which will become extinct.

Anticipating Michael Ghiselin's bioeconomics (1974) and Edward O. Wilson's sociobiology (1975), Hardin combined, extrapolated, and generalized a result that he had found in biology and economics. In 1960, Hardin first anchored his competitive exclusion principle, also known as "Gause's principle." Specifically, Russian evolutionist Georgii [START_REF] Gause | Experimental Studies on the Struggle for Existence[END_REF] had shown in a series of experiments involving yeast and paramecia that when different species live in a shared environment and use similar resources, one species will tend to drive out the others. [START_REF] Hardin | The Competitive Exclusion Principle[END_REF]Hardin ( , 1293) ) acknowledged the principle was hard to "prove or disprove" empirically, but further argued that " [t]he "truth' of the principle is and can be established only by theory."

Second, [START_REF] Hardin | The Competitive Exclusion Principle[END_REF]Hardin ( , 1295) ) thought it was "possible" that principle "originated in economic thought." Thus, as another source of inspiration for the claim that competition would destroy itself, he cited "the French mathematician" Joseph Bertrand. Commenting on Cournot's duopoly model, Bertrand (as quoted in Hardin, pp. 1295-6) noted that if … one of the competitors will lower his price in order to attract the buyers to himself and that the other, trying to regain them, will set his price still lower … there is no limit to the lowering of the price. Whatever common price might be initially adopted, if one of the competitors were to lower the price unilaterally he would thereby attract the totality of the business to himself ... Hardin (1296) thus stated baldly: "Any competitor knows that unrestrained competition will ultimately result in but one victor," in part because unrestricted companies will form cartels or use intellectual property to block competitors. In international trade, Hardin argued, his proposed competition exclusion principles meant that the world would need to reconsider tariffs and trade barriers, to prevent one country from being the victor that drives all others from the market.

Of course, just as many biologists of the time were engaged in controversy over the actual reach of the competitive exclusion principle, many economists then and now would take issue with the claim that all competition tends to monopoly. Even at the time, Gordon Tullock (1960, p. 95) stressed that Hardin's article "contains an error in economics. From the principle that complete competitors cannot coexist he deduces the development of monopolies." Tullock argued: "If the principle has any application to economics at all, it would indicate that one type of economic enterprise might, by multiplication of its members, replace another, but this would not lead to monopoly." In [START_REF] Hardin | The Tragedy of the Commons[END_REF] "tragedy of the commons" essay, instead of focusing on biological competitors with a fixed set of resources and similar needs, Hardin added another dimension: the stress that population growth inevitably placed on environmental resources -"the world available to the terrestrial population is finite" (1243). That was also the case with other natural resources, such as oceans -that people tend to think as "inexhaustible" (1245), National parks -"there is only one Yosemite Valley" (1245), air and rivers, and even "airwaves of radio and television" (1249). Scarcity was the origin of the economic, ecological, and social problem.

In what is probably the most-quoted portion of the essay (at least in classes in economics!), [START_REF] Hardin | The Tragedy of the Commons[END_REF]Hardin ( , 1244) ) proposed a sheepherder allegory to understand the basic motivation at work.

Picture a pasture open to all. It is to be expected that each herdsman will try to keep as many cattle as possible on the commons. As a rational being, each herdsman seeks to maximize his gain. Explicitly or implicitly, more or less consciously, he asks, 'What is the utility to me of adding one more animal to my herd?' This utility has one negative and one positive component.

(1) The positive component is a function of the increment of one animal. Since the herdsman receives all the proceeds from the sale of the additional animal, the positive utility is nearly 1.

(2) The negative component is a function of the additional overgrazing created by one more animal. Since, however, the effects of overgrazing are shared by all the herdsmen, the negative utility for any particular decision-making herdsman is only a fraction of -1.

Adding together the component partial utilities, the rational herdsman concludes that the only sensible course for him to pursue is to add another animal to his herd. And another; and another.... But this is the conclusion reached by each and every rational herdsman sharing a commons. Therein is the tragedy. Each man is locked into a system that compels him to increase his herd without limitin a world that is limited. Ruin is the destination toward which all men rush, each pursuing his own best interest in a society that believes in the freedom of the commons. Freedom in a commons brings ruin to all. In [START_REF] Hardin | The Tragedy of the Commons[END_REF] view, individuals acted vis-à-vis the world and its resources as if they were "independent, rational, free-enterprisers" (1245), as if their action had no impact on them, as if Adam Smith's invisible hand had actually worked and that "decisions reached individually will, in fact, be the best decisions for an entire society" (1244). Rationally, such behavior was perfectly understandable: "Each man," explained Hardin (1244-45), "is locked into a system that compels him to increase his herd without limit [or] of 'fouling our own nest.'" The consequence was unavoidable. Because of the ongoing increase in population and humanity's shared use of the global commons, "the per capita share of the world's goods must steadily decrease" (1243) which would generate "misery" (1243) and "ruin" (1244).

At its core, [START_REF] Hardin | The Tragedy of the Commons[END_REF] tragedy, captured in his sheepherder analogy, illustrates a rather standard economics problem of interdependence, which involves issues of collective goods or external effects. After early analyses by [START_REF] Pigou | The Economics of Welfare[END_REF] and [START_REF] Knight | Some Fallacies in the Interpretation of Social Cost[END_REF], these problems had received more attention and had been more frequently discussed in the second half of the 1950s with the works of [START_REF] Samuelson | The Pure Theory of Public Expenditure[END_REF], [START_REF] Bator | The Anatomy of Market Failure[END_REF] and [START_REF] Coase | The Problem of Social Cost[END_REF] (for an overview, see [START_REF] Marciano | Market Failure in Context: Introduction[END_REF]. Hardin offered quick and casual references to Adam Smith and Thomas Malthus, but he did not refer to any of the modern economists who had contributed to study how to deal with situations involving interdependence. Hardin also neglected prior work in economics that dealt with commons (for example, Gordon 1954 on fisheries). Indeed, [START_REF] Hardin | The Tragedy of the Commons[END_REF] sheepherder allegory strikingly echoed the view Samuelson advanced in 1954: namely, that the economic problem of inefficient resource allocation does not come from the interdependence of resources but rather from the tendency of selfinterested individuals to engage in free-riding -the "hope to snatch some selfish benefit" in Samuelson's words (389). For Hardin, the problem resided more in the freedom to use a resource rather than in the characteristics of the resource itself; significantly, he titled one of the sections of his article "Tragedy of Freedom in a Commons" (1244).

To avoid the tragedy, [START_REF] Hardin | The Tragedy of the Commons[END_REF] argued for governance to constrain consumption and ensure sustainability. His advice was simple: stop making resources open to all. He pointed out that this step already had been taken with "food gathering; when farms were enclosed, pastures, hunting and fishing areas were restricted." It should also be done, in Hardin's (1249) view with "commons as a place for waste disposal" and with "pollution by automobiles, factories, insecticide sprayers, fertilizing operations, and atomic energy installations" and as well with the "commons in matters or pleasure," restricting, for instance, "the propagation of sound waves […] mindless music … in the public medium." All this would imply coercion, "the infringements on somebody's personal liberty." But in Hardin's view, it was the illusion and appearance of freedom associated with the philosophy of open access to resources that was actually coercive: "Individuals locked into the logic of the commons are free only to bring on universal ruin." [START_REF] Hardin | The Tragedy of the Commons[END_REF] primary focus and attack in the "tragedy of the commons" essay was on rising human populations: indeed, the subtitle of the article is "The population problem has no technical solution; it requires a fundamental extension in morality." Hardin argued that the "freedom to breed" is "intolerable." He rejected appeals to conscience: "[A]n appeal to independently acting consciences selects for the disappearance of all conscience in the long run, and an increase in anxiety in the short." He insisted upon mutual coercion as the approach: "Coercion is a dirty word to most liberals now, but it need not forever be so." He referred to the UN Universal Declaration on Human Rights, which held that all choices about the size of families should be made by families. Hardin responded: "It is painful to have to deny categorically the validity of this right; denying it, one feels as uncomfortable as a resident of Salem, Massachusetts, who denied the reality of witches in the 17 th century" (1246)." 2 2 As [START_REF] Bajema | Garrett James Hardin: Ecologist, Educator, Ethicist and Environmentalist[END_REF], notes, Hardin was often willing to "break social taboos." Along with calling for government-mandated population control, he signed with 51 others a Wall Street Journal op-ed called "Mainstream Science on Intelligence,", written by Linda [START_REF] Gottfredson | Mainstream Science on Intelligence: An Editorial With 52 Signatories, History, and Bibliography[END_REF][START_REF] Gottfredson | Mainstream Science on Intelligence: An Editorial With 52 Signatories, History, and Bibliography[END_REF]), that discussed average racial-ethnic differences in intelligence measured in terms of IQ. Another striking example is his 1974 essay, "Living in a Lifeboat," in which he again insisted on the need to control population growth and asserted that the [START_REF] Hardin | The Tragedy of the Commons[END_REF] recognized two solutions to the tragedy of the commons: government regulation and privatization. Both solutions rely on collective action through government to introduce constraints on resource consumption. The approaches differ substantially in terms of the manner in which ongoing (month-to-month, day-to-day, minute-to-minute) resource allocation decisions are made. Government can constrain consumption by directly managing or regulating resource use. Alternatively, government can establish a system of private property rights delineating ownership of the resources. The former presumes government will consider the aggregate effects and manage resource use efficiently over time, and the latter presumes well-defined private property rights will facilitate market exchanges and thereby lead to an efficient allocation of access and use rights, and consequently, efficient resource management over time. In the case of population growth, Hardin's essay does not explicitly contemplate a privatization approach (like a transferable right for any adult to be the biological parent to one child). Government regulation of population is necessary, Hardin (1248) claimed: "Freedom to breed will bring ruin to all."

Early Interpretation and Incorporation within Economics

Hardin's (1968) article attracted a reasonable degree of public attention, but most economists did not much refer to Hardin. The few who did argued that Hardin was cited for having coined an interesting expression for what economists already knew. For example, in one of the first references made by an economist to Hardin, Dales (1975, 495) explained that because of the difficulties in assigning property rights, "the expected outcome followedoveruse, congestion, premature depletion, or extinction, depending on the particular characteristics of the caseand the value of the resource dropped toward zero. 'The tragedy of the commons,' to use Hardin's apt phrase, unfolded inexorably." Along the same lines, a few years later, one reads that "[w]ithout private property, society will experience the 'tragedy of the commons'ownership by all actually means ownership by none" (Carroll, problem of "Christian-Marxist" idealism is "the sharing ethics is that it leads to the tragedy of the commons" (1974,581). Ciscil, and Chisholm, 1979, 607). At the beginning of the 1980s, a few economists discussed again the problem of fisheries and with a similar tone. For instance, [START_REF] Johnson | Contracting Problems and Regulation: The Case of Fishery[END_REF] referred to a series of articles on fisheries as common property resources, and they mentioned Hardin's 1968 article only in passing, as if it were a necessary reference. 3But over time, the "tragedy of the commons" article became one among many articles that formed "the large literature" devoted to "common-property management" (Thiesenhusen 1991, 18). In the economics literature, it was no longer distinguished from the articles written by economists themselves, and Hardin was put on the same footing as the economists who had worked on property rights. He was thus cited as a scholar defending a neoclassical "perspective" [START_REF] Carroll | The Market as a Commons: An Unconventional View of Property Rights[END_REF] or "paradigm" [START_REF] Swaney | Externality and Community[END_REF]) and lumped together with [START_REF] Coase | The Problem of Social Cost[END_REF] and [START_REF] Demsetz | Toward a Theory of Property Rights[END_REF], despite their very different perspectives and approaches. The context of [START_REF] Hardin | The Tragedy of the Commons[END_REF] work, with its focus on overpopulation and advocacy of mandatory government control over population growth, largely faded away.

The Ostroms

Elinor and Vincent Ostrom noted as early as 1971 (p. 207) that "Garret Hardin had indicated that these strategies [such as free riding] give rise to 'the tragedy of the commons' where increased individual effort leaves everyone worse off." Again in 1973 (pp. 210-211), Vincent Ostrom referred to Hardin and described the tragedy as follows:

Individualistic decision making applied to common-property resources will inexorably result in tragedy unless the structure of decision-making arrangements can be modified to enable persons to act jointly in relation to those resources as a common property.

Potential recourse to coercive measures will also be necessary to preclude a hold-out strategy and regulate patterns of use among all users. Unrestricted individualistic decision making in relation to common-property resources or public goods will lead to the competitive dynamic of a negative-sum game: the greater the individual effort, the worse off people become. These references were not different from those of most economists at that time. But the Ostroms, and especially Elinor, disagreed with Hardin. In an interview with [START_REF] Levi | An Interview with Elinor Ostrom[END_REF], she recounted that the first time she heard Hardin she "was somewhat taken aback" because her and Vincent's work proved Hardin was wrong.4

Elinor Ostrom and other social scientists challenged the frame set by Hardin by asking two foundational sets of questions: First, how well does the tragedy of the commons allegory describe reality? Is it a useful theory for making predictions about real-world behavior of individuals sharing common pool resources? Does it describe a normal or exceptional situation? Does it provide a useful basis for choosing or designing regulatory solutions? Second, does the binary choice between government command-and-control regulation and private property-enabled markets reflect the full range of options? Are there alternative (bottom-up) institutions and/or means for collective action [START_REF] Frischmann | Two Enduring Lessons from Elinor Ostrom[END_REF])?

To answer the first set of questions, it may be convenient to work within the confines of Hardin's sheepherder allegory, because doing so makes analysis tractable. As Elinor Ostrom explained (2007, 15183):

Situations characterized by [Hardin's] assumptions, in which individuals independently make anonymous decisions and primarily focus on their own immediate payoffs, do tend to overharvest open-access resources. Researchers have repeatedly generated a 'tragedy of the commons' in experimental laboratories when subjects make independent and anonymous decisions in a common-pool resource setting.

4

For more details on Elinor Ostrom's framework, useful starting points are [START_REF] Aligica | Rethinking Institutional Analysis: Interviews with Vincent and Elinor Ostrom[END_REF] and [START_REF] Tarko | An Intellectual Biography[END_REF].

The allegory, however, is reductionist and distorting. It includes a series of assumptions with respect to both resource and resource management that severely limited its generalizability.

Simply put, Hardin's description of the "tragedy of the commons" ruled out-by assumption-the possibility that people might communicate and find ways to cooperate.

Again, as Ostrom explained (2007, 15183):

Making one small change ... in the structure of laboratory experiments, a change that is predicted by game theory to make no difference in the predicted outcome, has repeatedly had major impacts on interactions and outcomes. Simply enabling subjects to engage in face-to-face communication between decision rounds enables them to approach socially optimal harvesting levels rather than severely overharvesting the commons. In the face-to-face discussions, participants tend to discuss what they all should do and build norms to encourage conformance.

Elinor [START_REF] Ostrom | Collective Action and the Evolution of Social Norms[END_REF] also criticized reliance on the rational actor model at the heart of Hardin's allegory when analyzing collective action and social dilemmas. After establishing a series of 'well-substantiated facts' about human behavior based on extensive fieldwork, she concluded: "I believe that one is forced by these well-substantiated facts to adopt a more eclectic (and classical) view of human behavior" (141). She then developed a "secondgeneration model of rationality" in which humans are "complex, fallible learners who seek to do as well as they can given the constraints that they face and who are able to learn heuristics, norms, rules, and how to craft rules to improve achieved outcomes" (E. Ostrom 1998, p. 9).

The second-generation model of rationality predicts that reciprocity, reputation, and trust as "core relationships" can lead to increased net benefits ( 13). This theoretical model identifies "individual attributes" that are particularly important in explaining behavior in social dilemmas. These attributes include "[1] the expectations individuals have about others' behavior (trust), [2] the norms individuals learn from socialization and life's experiences (reciprocity), and [3] the identities individuals create that project their intentions and norms (reputation)" ( 14).

Elinor [START_REF] Ostrom | Governing the Commons: The Evolution of Institutions for Collective Action[END_REF] rebelled against the distorting reductionism that Hardin's essay represented -and Hardin was hardly the only culprit. Donning the analytical straitjacket would "lead the analyst to miss what is most important and focus on what is least relevant" (Frischmann 2013, p. 5). This was especially true when approaching the second set of questions and the feasibility of institutional solutions besides private property-enabled markets and government command-and-control regulation. Commons governance was ignored as a solution because it was presumed to be the problem that inevitably leads to ruin.

To explore alternative institutional arrangements, Elinor and Vincent Ostrom, and their colleagues both at the Workshop in Political Theory and Policy Analysis at Indiana University and also around the world, advocated "to combine formal approaches, fieldwork and experiments in order to 'penetrate' social reality rather than to use formal techniques to 'distance' ourselves from it" (V. Ostrom 2009, p. 5). Thus, in the three decades that followed publication of Hardin's "tragedy," they engaged in rigorous, interdisciplinary social science to diagnose social dilemmas and to understand the commons as a mode of governing access to and use of shared resources. This approach stressed context and was grounded in empirical study. Systematic studies of real communities demonstrated that commons governance works in some contexts and fails in others (E. The external variables affect actors and action situations, which "generate patterns of interactions and outcomes that are evaluated by participants in the action situation (and potentially by scholars) and feed back on both the external variables and the action situations" and the actors (Ostrom, 2010, 647). There are many detailed examples using the IAD framework is beyond the scope of this paper. As one brief example, consider a lobster fishery. The tragedy of the commons allegory makes assumptions about the biophysical characteristics (depletable), community (independent, self-interested rational actors), and rules-in-use (every fisherman for himself); also, it assumes the only actors are the fishermen and the only relevant collective action problem is the prediction of ruinous competition.

Viewed through the IAD lens, the empirical shortcomings of Hardin's allegory become clear: lobsters are not purely depletable; as a biological matter, they can reproduce and replenish stocks. The relevant community involves more than just the fishermen. Communication and cooperation are feasible. The rules-in-use are more nuanced than everyone for himself (for detailed examination, see [START_REF] Acheson | Capturing the Commons: Devising Institutions to Manage the Maine Lobster Industry[END_REF].

Recognizing that people often can cooperate effectively and build institutions to enable sustainable use of shared resources focused scholarly attention on complexity, context, communities, and institutions. This broader field of vision brought informal institutions into view and encouraged their systematic study, and it also improved our understanding of formal institutions by revealing the many ways that government, market, and community institutions depend on each other to be successful. Figuring out how best to successfully cooperate in governing ourselves and our shared environments remains one of the core questions studied in law, economics, political science, sociology, and many other related fields today.

Two Key Conceptual Mistakes that Further Muddle Hardin's Special Case

As the work of Elinor Ostrom and many others makes clear, the assumptions made in [START_REF] Hardin | The Tragedy of the Commons[END_REF] "tragedy of the commons" article, highlighted by the sheepherder allegory, limit his analysis to a special case. In this section, we go one step further, explaining why

Hardin made two basic conceptual mistakes that further distort the usefulness of his case.

First, Hardin confused resources with governance. In his sheepherding allegory, for example, the relevant resource is a pasture, and the relevant governance is open access sharing: as the allegory begins, "Picture a pasture open to all." To describe commons as the resource subject to tragedy is a category error. Commons are not, and should not be conflated with, resources. They are not common pool resources nor public goods; these types of sharable goods may, however, be governed as or within commons. Instead, commons are a form of resource governance where members of a community share resources on terms set by the community. Thus, commons "applies to resources, and involves a group or community of people, but commons does not denote the resources, the community, a place, or a thing.

Commons [are] the institutional arrangement of these elements" (Frischmann, Madison, Strandburg 2014, p. 2). Unfortunately, many people describe commons as the shared resource subject to tragedy. This perpetuates Hardin's conceptual mistake.

Second, Hardin conflated two different governance systems. He used the term "commons," but he limited his analysis to the consequences of only one mode of governance, open access sharing. Yet these are quite different, and the differences matter (Frischmann 2012, 8). Open access implies no ownership or property rights. No individual or institution has the right to exclude others from the resource. Hence, all who want access can get access, typically for free. By comparison, commons involve some form of communal ownership (community property rights, public property rights, joint ownership rights). As a consequence, access to the resource is restricted to the members of the relevant community, under more or less restrictive conditions, and nonmembers can be excluded. In other words, open access differs from commons in several ways: in terms of ownership (none vs. communal/group), its definition of community (public at large vs. a more narrowly defined and circumscribed group with some boundary between members and nonmembers), and its degree of exclusion (none vs. exclusion of nonmembers).

These distinctions are important for understanding different institutions and how social arrangements operate at different scales. By making the assumptions that he did, Hardin (1968) locked himself into the analysis of a special case and significantly underestimated the power of commons as an efficient form of governance.

Extensions: Infrastructure and Knowledge

The tragic dilemma at the core of Hardin's allegory has been identified and discussed for a wide range of different resources. Much of the early work focused on natural resources like fisheries and other typically common pool resources. It has also received renewed attention in immigration debates (for example, Normadin and Valles 2015). But not surprisingly, the dilemma arises with many human-made resources too. Some are common pool resources subject to congestion and potential deterioration like roads and other infrastructure, while others are public goods subject to free rider concerns like ideas and other knowledge resources. In these settings, there can also be an inclination toward a Hardin-style bias to believe that the relevant collective action problems (i) can be diagnosed in terms of ruinous competition and (ii) can only be addressed by direct government mandates and or government-enforced property rights. Such bias leads analysts to undervalue the usefulness and workability of commons governance over these resources by the relevant community.

Here, we offer a brief and selective discussion of these topics, with an emphasis on how interdisciplinary research in these areas is moving past the basic "tragedy of the commons" model.

Infrastructure Commons

Infrastructure resources often are managed in an openly accessible manner that gives rise to possibilities that economists have likened to Hardin's tragedy of the commons. 6 For example, individual users rationally use toll-free highways at a rate and in a manner that maximizes private gains, but disregards the effects on other users or more generally, the sustainability of the resource. If each individual acts in such a fashion, aggregate consumption may lead to congestion costs from crowding, increased waiting time in queues, slower service, pollution, noise, reduced quality of service due to increased interruptions of service, and accelerated depreciation and depletion of the shared infrastructure [START_REF] Frischmann | Infrastructure: The Social Value of Shared Resources[END_REF].

The basic economic model of congestion, like [START_REF] Hardin | The Tragedy of the Commons[END_REF] tragedy of the commons, assumes homogenous uses [START_REF] Vickrey | Congestion Theory and Transport Investment[END_REF][START_REF] Arnott | A Structural Model of Peak-Period Congestion: A Traffic Bottleneck with Elastic Demand[END_REF]. The shared meadow is used for grazing sheep (not for grazing other animals or for other activities); the shared highway is used to complete trips (in more-or-less identical vehicles). When considering homogenous use, economists utilize a congestion cost function that relates the marginal social cost of resource use (like feeding sheep or completing a trip) to utilization rates (traffic) and resource capacity (like acreage or number of lanes). Such congestion is called "anonymous crowding," because the determinants of crowding are utilization and facility size, and attributes of individual users play no part in the equation (Cornes and Sandler 1996, p. 355). However, complications arise as heterogeneous users and uses are incorporated into economic models [START_REF] Cornes | The Theory of Externalities, Public Goods and Club Goods[END_REF]: for example, variance in capacity consumption rates and cross-crowding between uses. To illustrate cross-crowding, Frischmann (2012) extends Hardin's sheepherder allegory to include different livestock, assuming sheep, donkeys, and buffalo graze differently and also positing that "sheep and buffalo … fight each other." For roadways, a similar extension might involve cars, mass-transit buses, and trucks.

Heterogeneity affects the analysis of costs and benefits and is relevant to diagnosing congestion problems and comparing solutions. When crowding is no longer anonymous, discriminating among uses becomes a regulatory option to consider. In addition, a standard response of modern economists to congestion would be to use some form of congestion pricing to encourage users with heterogenous values of traveling by car at certain times to sort themselves-a policy choice which goes unconsidered in [START_REF] Hardin | The Tragedy of the Commons[END_REF].

There is also a case that certain infrastructural resources ought to be managed openly, because doing so may generate public goods and positive externalities or positive scale returns-greater social value with greater use of the resource [START_REF] Frischmann | Infrastructure: The Social Value of Shared Resources[END_REF][START_REF] Frischmann | Retrospectives: The Marginal Cost Controversy[END_REF]. [START_REF] Rose | The Comedy of the Commons: Commerce, Custom, and Inherently Public Property[END_REF] called this the "comedy of the commons." She used road systems (pp. pp. 775-81) to illustrate and explained how commerce enabled by roads is an … interactive practice whose exponential returns to increasing participation run on without limit… Through ever-expanding commerce, the nation becomes ever wealthier, and hence trade and commerce routes must be held open to the public, even if contrary to private interest.

Instead of worrying that too many people will engage in commerce, we worry that too few will undertake the effort.

Commerce generates private value that is captured by participants in economic transactions, as buyers and sellers exchange goods and services, but it also generates social value that is not easily observed and captured by participants: for example, value associated with traveling to visit friends and relatives or traveling for recreation, as well as the value of widespread attendance at civic events, knowledge exchange, socialization, and acculturation. If open travel creates positive externalities, society may find diverse ways to take this into account as it considers how to manage access to roads. [START_REF] Frischmann | Infrastructure: The Social Value of Shared Resources[END_REF] extends the point to a variety of other infrastructural resources, ranging from basic research to the Internet.

The Public Domain and Knowledge Commons

The connection from intellectual property to the tragedy of the commons has been made explicit by a number of writers. Carrol, Ciscel and Chisholm (1979, p. 611) appear the first to refer to copyright law as a means for dealing with the tragedy that seems to also plague intellectual commons. Yet, ideas are public goods, not common pool resources. Thus, using and overusing ideas (or works of art) may, in special cases, reduce their value (Landes and Posner 2002, p. 15), but does not congest or deplete them. Unconstrained consumption seems goodthe more, the merrier [START_REF] Rose | The Comedy of the Commons: Commerce, Custom, and Inherently Public Property[END_REF]; see also [START_REF] Merton | The Matthew Effect in Science, II: Cumulative Advantage and the Symbolism of Intellectual Property[END_REF])and even just (as discussed by [START_REF] Gosseries | Intellectual Property Rights and Theories of Justice[END_REF] rather than bad or wrong. If an intellectual resource, such as an idea, is openly accessible to all, then everyone who can profitably make use of it will do. But there's a catch. Ideas are products of human intellect; they require investment of time, effort, and capital. Unconstrained consumption by free riders presents a risk for potential investors, who may struggle to recover a sufficient return on their investment and may underinvest as a result. Thus, tragic underproduction of intellectual resources appears to be a social dilemma that mirrors Hardin's tragedy of the commons.

Avoiding cultural, technological and scientific stagnation seems to require collective action.

Many approaching this problem assume [START_REF] Hardin | The Tragedy of the Commons[END_REF] two options: direct government intervention (via public funding) or privatization (via intellectual propertyenabled markets). To be clear, government funding and intellectual property are incredibly important drivers of socially valuable knowledge production. However, one can make a case that much or even most of humanity's intellectual resources have been generated and shared within open-access and community-based commons, often without government subsidy and outside of intellectual-property-mediated markets. The free rider allegory myopically presumes tragedy, leading people to believe that free-riding is necessarily harmful and needs to be eliminated [START_REF] Lemley | Property, Intellectual Property, and Free Riding[END_REF]. In fact, the opposite is often true. Free riding is pervasive and is often a beneficial feature, rather than a bug, of our economic, cultural, and social systems [START_REF] Ramello | Property Rights and Externalities. The Uneasy case of Knowledge[END_REF][START_REF] Frischmann | Infrastructure: The Social Value of Shared Resources[END_REF]). Since at least [START_REF] Schumpeter | The Theory of Economics Development[END_REF], the creative process has been likened to a recombinant process in which (open) access to previously created knowledge plays a crucial role, fueling progress, and driving the combinatorial process of knowledge accumulation that fosters economic growth. This perspective emphasizes the positive externalities characterizing knowledge production and the role of commons governance [START_REF] Weitzman | Recombinant Growth[END_REF]Marchese et al. 2019), including the role of the knowledge commons and rules governing the public domain.

Knowledge commons refers to the institutionalized community governance of the sharing and, in many cases, creation and curation, of a wide range of intellectual and cultural resources (Frischmann, Madison, and Strandburg 2014). Examples include scientific research commons, including data, literature, and research materials (Reichman, Dedeurwaerdere and Uhlir 2016), intellectual property pools [START_REF] Madison | Constructing Commons in the Cultural Environment[END_REF], open source computer software projects [START_REF] Schweik | Internet Success: A Study of Open-Source Software Commons[END_REF], Wikipedia [START_REF] Hoffman | Wikitruth Through Wikiorder[END_REF][START_REF] Safner | Institutional entrepreneurship, wikipedia, and the opportunity of the commons[END_REF], "jamband" fan communities (Schultz 2006), and highly specialized technical knowledge like the Cornish steam engine [START_REF] Nuvolari | Collective invention during the British Industrial Revolution: the case of the Cornish pumping engine[END_REF]. Using the Ostrom-inspired Governing Knowledge Commons framework, case studies examine many governance issues, including interactions with intellectual property, government subsidies, and regulation. (Frischmann, Madison, and Strandburg 2014;Strandburg, Frischmann and Madison 2017;[START_REF] Sanfilippo | Governing Privacy in Knowledge Commons[END_REF]. For example, rare disease research consortia must address numerous governance challenges, including allocating research funding, authorship credit, and other rivalrous resources; overcoming potential "anticommons" 7 dilemmas arising from researchers' incentives to hoard access to patients and their data; maintaining privacy, security, and the trust of patients and their families; reducing transaction costs of cooperation between widely dispersed researchers; and managing interactions with outsiders, such as pharmaceutical companies.

Other researchers have focused on the governance of the public domain, which is the incredibly capacious set for ideas, facts, and many other intellectual resources that are openly accessible by default to everyone [START_REF] Litman | The Public Domain[END_REF]; see also [START_REF] Posner | Intellectual Property: The Law and Economics Approach[END_REF]. As an example of work in this area, a rich legal literature explores categories related to public domain like the 7 Michael Heller (1998) coined the term "anticommons." He argued that too much private property could lead to underuse and waste of resources, with effects as tragic as the lack of property rights described by [START_REF] Hardin | The Tragedy of the Commons[END_REF]. Heller and Rebecca Eisenberg explored the potential tragedy of the anticommons in biomedical research [START_REF] Heller | Can Patents Deter Innovation? The Anticommons in Biomedical Research[END_REF]. [START_REF] Buchanan | Symmetric Tragedies: Commons and Anticommons[END_REF] develop a formal economic model of anticommons.

The anticommons concept may offer a justification for fair-use and other copyright exceptions (Parisi, Schultz, and Depoorter 2000;[START_REF] Depoorter | Fair use and copyright protection: a price theory explanation[END_REF]. 20 semi-commons constructed within intellectual property systems and the creative commons constructed with intellectual property licenses (for a detailed survey, see [START_REF] Benkler | Between Spanish Huertas and the Open Road: A Tale of Two Commons?[END_REF].

Social demand for trusted governance of shared knowledge resources, ranging from medical data to algorithmically generated intelligence, is growing [START_REF] Frischmann | Re-Engineering Humanity[END_REF], even as public trust in governments and markets as sources of governance seems tenuous. Now, more than ever, we need to explore if, when, and how commons governance can scale.

Conclusion

Hardin's (1968) ultimate legacy is not really about his analysis of the tragedy of the commons, which was relatively trivial given already available economic tools, nor is it about his analysis of commons, which focused only on a narrow special case. Hardin contributed a catchphrase that caught the prevailing winds of public discourse and drew attention to the governance of shared resources. In the end, Ostrom's interdisciplinary, international and systematic analysis of commons governance is and should be the abiding legacy of the tragedy of the commons.

Figure 1 :

 1 Figure 1: Institutional Analysis and Development (IAD) Framework Source: From Ostrom and Hess (2007, p. 44).

One set of writers even supposed that Hardin had "documented more fully" the problem of overexploitation(Balachandran, Fisher, and Stanley, 1989, p. 261), which is obviously an exaggeration: Hardin had given no more than a few examples. Furthermore, despite Hardin's depiction, commons existed and where successfully managed in various ways in medieval England and many other parts of the world for a very long time[START_REF] Dahlman | The Open Field System and Beyond: A Property Rights Analysis of an Economic Institution[END_REF][START_REF] Cox | No Tragedy on the Commons[END_REF][START_REF] Bannon | Fresh Water in Roman Law: Rights and Policy[END_REF].

Unlike Hardin's pasture, most infrastructure resources are human-made, and that gives rise to a host of supply-side issues, which we leave aside for the sake of brevity.


Adrien Chan-Hon-Tong

Gradient regularisation increases deep learning performance under stress on remote sensing applications

Gradient regularisation increases deep learning

performance under stress on remote sensing applications.

Introduction

Deep learning (see [START_REF] Lecun | Deep learning[END_REF] for a review) is currently the state of the art of many intelligence artificial applications. Yet, deep learning performances are plagued by adversarial examples (see [START_REF] Goodfellow | Explaining and harnessing adversarial examples[END_REF][START_REF] Papernot | The limitations of deep learning in adversarial settings[END_REF][START_REF] Szegedy | Intriguing properties of neural networks[END_REF] as samples of adversarial example literature). At test time, it is possible to design a specific invisible/marginal perturbation such as a targeted network eventually predicts different outputs on original and disturbed input. This threat is worsen as producing adversarial examples does not require to have access to the internal structure of the network [START_REF] Narodytska | Simple black-box adversarial attacks on deep neural networks[END_REF] and can have physical implementation [START_REF] Kurakin | Adversarial examples in the physical world[END_REF].

In addition, an other issue is the lack of global smoothness of deep network, typically linked to Lispchitz constant of the network, and, not just to local smoothness which is strengthen by adversarial defences. Thus, quite orthogonally with the very large effort of the community on adversarial defence (e.g. [START_REF] Zhang | Theoretically principled trade-off between robustness and accuracy[END_REF][START_REF] Wong | Provable defenses against adversarial examples via the convex outer adversarial polytope[END_REF][START_REF] Shaham | Understanding adversarial training: Increasing local stability of supervised models through robust optimization[END_REF]), this paper focus on Lispchitz related penalty like [START_REF] Peter L Bartlett | Spectrallynormalized margin bounds for neural networks[END_REF][START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF]. Now, most Lispchitz related methods only measures their effects with accuracy i.e. global smoothness is measured as a way to reduce overfitting. But, there is no trivial link between smoothness (model dependant) and overfitting (model and data dependant). Instead, this paper offers as framework to measure Lispchitz penalty effect by accuracy under stress (including adversarial attack but not only) i.e. global smoothness is measured as a way to increase local smoothness (even if adversarial defence are obviously a more straightforward way to do so for small perturbation).

In contrast, this paper presents remote sensing experiments (object detection and image segmentation on remote sensing datasets) which shows consistently that the offered Lispchitz related penalty increases performance under stress while being related to the global smoothness of the network (an not just local smoothness).

In this paper, the focus is given to remote sensing applications. On one hand, remote sensing image can not be hacked like social network images (by modifying encoding), or, autonomous driving images (by hacking traffic signs). But, on the other hand, remote sensing system should deal with large illumination changes, death pixels, atmospheric blur, and, camouflage which is totally an adversarial attack. Also, many remote sensing applications like car counting, or, image based smart farming are currently waiting certification framework to be used in real life while autonomous driving may require more time. And, such framework will probably include robustness evaluation.

So, the contributions of this paper are:

• it offers a new Lispchitz related penalty

• which is evaluated as a way to increase performances under stress i.e. to increase local smoothness and not to reduce overfitting

• using a toolbox (made public) for stress evaluation including both adversarial attack (typically Fast gradient sign method (FGSM) [START_REF] Goodfellow | Explaining and harnessing adversarial examples[END_REF]), and, common perturbation (a subset of [START_REF] Hendrycks | Benchmarking neural network robustness to common corruptions and perturbations[END_REF])

• the consistent trend of our remote sensing experiments is that this new penalty term increases performance under stress

The new penalty term is presented in section 3 after related works of section 2. Experiments are presented in section 4 before conclusion.

2 Related works

Stress test and adversarial defence

Definition of dependable evaluation is not that clear. Two theoretical dead end forbids a formal dependable evaluation: without a formal definition of the task, formal proof of correctness is impossible (typically [START_REF] Katz | Reluplex: An efficient smt solver for verifying deep neural networks[END_REF] can prove some property but it can prove nothing about accuracy on unknown samples), and, all classifiers are equally bad averaged on all classification problem [START_REF] David | The lack of a priori distinctions between learning algorithms[END_REF].

So dependable evaluation should have something to do with very large testing datasets. But, very large datasets are expensive to collect. Thus, more realistically, dependable evaluation will be related to middle size testing datasets augmented to be stress tests (and simulation, plus, formal property which are out of the scope of this paper).

Typically, [START_REF] Hendrycks | Benchmarking neural network robustness to common corruptions and perturbations[END_REF] offers a toolbox to augment testing data with agnostic perturbations, and shows that classical network performs poorly on these augmented data. This toolbox could obviously be combined by adversarial attack like FGSM. Now, high performance under stress can be straightforwardly achieves by local smoothness typically obtained by considering state of the art of adversarial defence, but also, by global smoothness typically related to Lispchitz constant.

Most classical adversarial defence is adversarial training [START_REF] Shaham | Understanding adversarial training: Increasing local stability of supervised models through robust optimization[END_REF] i.e. training on data plus adversarial data. An emerging way is [START_REF] Wong | Provable defenses against adversarial examples via the convex outer adversarial polytope[END_REF] where network output is optimized to be absolutely stable around each training point allowing mathematical guarantee. A soft version of [START_REF] Wong | Provable defenses against adversarial examples via the convex outer adversarial polytope[END_REF] is [START_REF] Zhang | Theoretically principled trade-off between robustness and accuracy[END_REF] where the guarantee on robustness is relaxed to allow a training closer to classical training with a simpler trade off between robustness, and, (training) accuracy.

However, none of these methods is designed to increase global smoothness of the network. This global smoothness is more related to Lispchitz constant.

Lispchitz methods

A function f from R I in R J is said K Lipschitz for a norm ||.||I and ||.||J if ∀x, y ∈ R D , ||f (x) -f (y)||J ≤ K||x -y||I .
As continuous, piecewise infinitely derivable function, relu based deep network are K Lipschitz function. So, naively, deep network could be expected to be smooth function. However both K and I can be very large: K is estimated to by higher than 50000 for first Alexnet layers in L2 norm [START_REF] Szegedy | Intriguing properties of neural networks[END_REF] while I = 227×227×3. So, modifying each pixel value a an input image just by 1 can lead to a 7729350000 gap from the original likelihood produced by Alexnet on this image. Thus, to be really smooth, deep network should have a much more lower Lipschitz coefficient (LC).

Different penalties have been offered in literature to tackle this issue including L2 penalty, spectral penalty [START_REF] Peter L Bartlett | Spectrallynormalized margin bounds for neural networks[END_REF] or gradient penalty [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF]. This penalties are usually added to classical binary cross entropy loss. More formally, let x be input image, y the corresponding ground truth, f the network with θ the current weights, and, hence, f (x, θ) the probability produced by the network. Then, the total loss is l(x, y, θ) = BCE(f (x, θ), y) + λR(f, x, θ) with BCE the binary cross entropy and R the regularity function (e.g. ||θ|| or other). Weights θ of the network are then updated using stochastic gradient descent approach i.e. following the gradient ∇ θ l(x, y, θ).

As measuring the global smoothness of the network is not straightforward, most evaluations of these methods were focused on accuracy i.e. these regularisation losses have been used as a way to control overfitting. In this paper, the evaluation is instead focused on accuracy under stress. Indeed, as global smoothness should imply a relative local smoothness, evaluating accuracy under stress seems more relevant than overfitting reduction which is not directly related to robustness.

Applications

Soon after deep learning becomes the state of the art of image classification with Alexnet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], deep learning object detector appear with [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF] and immediately outperforms previous state of the art in object detection (like [START_REF] Felzenszwalb | A discriminatively trained, multiscale, deformable part model[END_REF]).

Since [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF], a large number of deep learning detectors have been designed. Most salient deep detectors are [START_REF] Shaoqing Ren | Faster R-CNN: Towards real-time object detection with region proposal networks[END_REF] which improves [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF] but using deep learning box proposal instead of static one, [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF] which directly predicts boxes from a dense feature map, then extended by [START_REF] Liu | Ssd: Single shot multibox detector[END_REF] which offers a better way to encode boxes score in feature map. Alternatively, detection can also be extracted from semantic segmentation map when object does not overlap. This segment before detect paradigm is widely used in remote sensing [START_REF] Audebert | Segmentbefore-detect: Vehicle detection and classification through semantic segmentation of aerial images[END_REF].

Segmentation is independently an other task where networks outperforms previous state of the art since [START_REF] Lagrange | Benchmarking classification of earthobservation data: From learning explicit features to convolutional networks[END_REF]. Typical structure for image segmentation is UNet [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF].

Both these tasks can found tremendous applications in remote sensing which are usually non critical applications. This way, such remote sensing applications may be the first real life industrial application of deep learning (excluding multimedia applications). This is why this paper focus on remote sensing applications (see datasets).

Batch centred Lispchitz regularization

This section describes the offered new Lispchitz regularization and difference with previous Lispchitz regularizations.

L2 weight penalty is the easiest way to lower LC, but, controlling of the regularisation is hard as all weights of all layers are considered equally. Spectral penalty takes advantages of the layered structure of the network. However, as pointed by [START_REF] Huster | Limitations of the lipschitz constant as a defense against adversarial examples[END_REF], product of each layer spectral norm is a very loose bound of LC. Finally, penalizing gradient norm (according to inputs not weights i.e. ||∇xf (x)||) has the advantage to estimate the real LC, but, only locally around each point on with the gradient is computed (and with the disadvantage of requiring second order derivatives).

Also, penalizing gradient norm is usually done on the loss: training tries to make the loss more Lispchitz. But this is problematic as loss is compute after softmax.

Instead in this paper, the objective is to make only the features more Lispchitz -neither the classifier (e.g. three last layers) neither the loss. So, network f is decomposed into an encoder g and a classifier h with weights θ being decomposed into φ, ψ, likelihood on input x is then g(h(x, φ), ψ) and loss on this sample (associated with y ground truth) is l(x, y, φ, ψ) = BCE(g(h(x, φ), ψ), y) + λ||∇xh(x, φ)|| where ||∇xh(x, φ)|| design the sum of ||∇xhi(x, φ)|| for all components i.

In practice, framework (like pytorch or tensorflow) allows to compute scalar function loss but does not allow to share the graph of derivative making it intractable to compute the sum over i of individual gradient. Yet, this paper offers to compute a batch centred Lispchitz regularization: ∇x h(x) -h ref is computed with h ref being the average of h(x) on the batch.

Experiments

Setting

Experiments have been conducted on a public remote sensing datasets of object detection and segmentation.

To evaluate performance under stress a toolbox similar to [START_REF] Hendrycks | Benchmarking neural network robustness to common corruptions and perturbations[END_REF] has been developed for detection context. Also, this toolbox contains adversarial noise (implemented by FGSM).

For detection, following [START_REF] Chan-Hon-Tong | Object detection in remote sensing images with center only[END_REF], task considered is center object detection. A distance of 1 meter is set to allow a predicted center to be matched with a ground truth center. Then, prediction and ground truth are matched like in classical detection. Performance is summarized by the g score (product of recall and precision). 3 datasets are used for detection: ISPRS POTSDAM1 , VEDAI [START_REF] Razakarivony | Vehicle detection in aerial imagery: A small target detection benchmark[END_REF], and the GDRSS DFC data fusion contest 2015 [START_REF] Lagrange | Benchmarking classification of earthobservation data: From learning explicit features to convolutional networks[END_REF].

Dataset are resized in order to provide a large range of resolution: VEDAI is used with little resizing such that a car is contained into a 32x32 box, DFC2015 is resized such that a car is contained into a 48x48 box, and, finally, POTSDAM is slightly resized such that a car is contained into a 64x64 box. Obviously, performance are expected to increase with the resolution (this will be the case).

For segmentation, performance is measured by the straightforward overall accuracy on the same datasets except VEDAI where no labelling is available.

Detector are based on SSD [START_REF] Liu | Ssd: Single shot multibox detector[END_REF] but adapted to center regression. Core of the detector are both VGG and Resnet like in [START_REF] Liu | Ssd: Single shot multibox detector[END_REF]. For segmentation, UNet is straightforwardly considered.

Results

All regularisation losses are evaluated with the same pipeline. However, on these datasets, L2 weight penalty has no impact on the performance except for large λ for which it makes all weights collapsing to 0 (leading to a very poor result) Spectral regularity exhibits a very strong numerical instability. Also, standard gradient penalty [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF] tends to degrade the performances (with and without stress). The fact that these methods to have such negative impact is quite surprising. For [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF] which is very close to our method, it may be due to the softmax (it penalizes after softmax and here it is penalize before independently from the label). Either, these methods are not adapted to this remote sensing context, or, this paper may have too naively implement these (despite λ which balances BCE loss and regularity loss is optimized on a grid). So, no conclusion will be stated for them. Now, an interesting result is comparison of performance with and without our regularity. Indeed, the main result of this paper is that the offered gradient regularisation mitigated the drop of performances caused by stress consistently across all datasets, tasks and networks. Raw performance are given as reference to measure this drop (see tabular 1). Most striking results is in table 2: native model are very sensitive to FGSM, and, performance under two FGSM steps are very low compared to performances on raw data. Yet, adding the offered regularization moderate this drop. Regularized algorithms eventually outperform their raw version by average 5% (g-score for detection, accuracy for segmentation). Off course, adversarial training may be even more efficient, but, here the regularisation is linked to global smoothness of the network, not just local.

VEDAI 32x32 DFC

An other striking results is in table 3: image are noised by a strong Gaussian noise (variance is 512). Again, performances drop, but, this drop is mitigated by regularisation. Let stress that, as, performance is measured in g-score in detection, it can even increase with noise by increasing precision and/or ratio between precision and recall. Typically, on VEDAI, performances often increase with agnostic noise (never observed for adversarial noise). This phenomenons is not seen in segmentation as accuracy is much more smooth metric. On this last experiment, it is not clear at all to known if common adversarial training and/or certified defense based polytop would have been effective in front of such large x axis is level of Gaussian noise. y axis is gscore.

Conclusion

This article focuses on object detection performance under stress, and, benchmark different kind of regularization theoretically expected to make the detector more Lipschitz. On three public remote sensing datasets, performances under stress on detection/segmentation are increased by using batch based gradient norm penalty. Such kind of regularisation are a way to advance on safety issue raise by deep learning detector.

However, as perspective, our result is far from closing this issue. First, such experiment should be reproduced in more large datasets, and, mainly, performance under stress still heavily decreases with stress level (even if this drop is much slower with regularisation than without). An other serious point is that, in our experiment, the more an algorithm is efficient on raw data, the more it tend to be sensitive to noise (typically performance without noise are higher on POTSDAM than on VEDAI, but, the drop of performance with noise is higher too on POTSDAM compared to VEDAI).

Figure 3 :

 3 Figure 3: Performances vs level of Gaussian noise.x axis is level of Gaussian noise. y axis is gscore.

Table 1 :

 1 Performances on raw data (without noise). On raw data, performance are quite similar with or without regularisation (a little less when adding regularisation), and, consistent with the literature. Performances are measured in g-score in detection for SSD (based on VGG and RESNET) -accuracy in segmentation for UNET.

	48x48 POTSDAM 64x64

Table 2 :

 2 Performances after 2 steps of FGSM. Adding regularization greatly moderate the performance drop caused by two step of FGSM attack compared to raw training. Performances are measured in g-score in detection for SSD (based on VGG and RESNET) -accuracy in segmentation for UNET (no available for VEDAI).

www2.isprs.org/commissions/comm3/wg4/2d-sem-label-potsdam.html

VEDAI DFC2015 POTSDAM

Figure 1: Performances vs steps of FGSM. x axis is number of steps of FGSM. y axis is gscore.

amplitude Gaussian noise while global smoothness offers some protection.

More detailed results are quantified in following figures (all raw results including variance estimation on several training are available on the github) which print performance for detection algorithms on the 3 datasets for Gaussian, black-white and adversarial noise (see figures 1, 2, 3). x axis is the level of noise: step of FGSM, percentage of dead pixel or Gaussian variance level (variance is proportionnal to square of level). y axis is the performance.

VEDAI DFC2015 POTSDAM

Figure 2: Performances vs percentage of dead pixels.

x axis is percentage of dead pixel. y axis is gscore.