Learning not to learn: Nature versus nurture in silico

1 INTRODUCTION . The ’ nature versus nurture ’ debate ( e.g. , Mutti et al. , 1996 ; Tabery , 2014 ) – the question of which aspects of behavior are ’ hard-coded ’ by evolution , and which are learned from experience – is one of the oldest and most controversial debates in biology . Evolutionary principles prescribe that hard-coded behavioral routines should be those for which there is no benefit in adaptation . This is believed to be the case for behaviors whose evolutionary advantage varies little among individuals of a species . Mating instincts or flight reflexes are general solutions that rarely present an evolutionary disadvantage . On the other hand , features of the environment that vary substantially for individuals of a species potentially ask for adaptive behavior ( Buss , 2015 ) . Naturally , the same principles should not only apply to biological but also to artificial agents . But how can a reinforcement learning agent differentiate between these two behavioral regimes ? A promising approach to automatically learn rules of adaptation that facilitate environment-specific specialization is meta-learning ( Schmidhuber , 1987 ; Thrun & Pratt , 1998 ) . At its core lies the idea of using generic optimization methods to learn inductive biases for a given ensemble of tasks . In this approach , the inductive bias usually has its own set of parameters ( e.g. , weights in a recurrent network ; Hochreiter et al. , 2001 ) that are optimized on the whole task ensemble , that is , on a long , ’ evolutionary ’ time scale . These parameters in turn control how a different set of parameters ( e.g. , activities in the network ) are updated on a much faster time scale . These rapidly adapting parameters then allow the system to adapt to a specific task at hand . Notably , the parameters of the system that are subject to ’ nature ’ – i.e. , those that shape the inductive bias and are common across tasks – and those that are subject to ’ nurture ’ are usually predefined from the start . In this work , we use the memory-based meta-learning approach for a different goal , namely to acquire a qualitative understanding of which aspects of behavior should be hard-coded and which should be adaptive . Our hypothesis is that meta-learning can not only learn efficient learning algorithms , but can also decide not to be adaptive at all , and to instead apply a generic heuristic to the whole ensemble of tasks . Phrased in the language of biology , meta-learning can decide whether to hard-code a behavior or to render it adaptive , based on the range of environments the individuals of a species could encounter . We study the dependence of the meta-learned algorithm on three central features of the metareinforcement learning problem : • Ecological uncertainty : How diverse is the range of tasks the agent could encounter ? • Task complexity : How long does it take to learn the optimal strategy for the task at hand ? Note that this could be different from the time it takes to execute the optimal strategy . • Expected lifetime : How much time can the agent spend on exploration and exploitation ? Using analytical and numerical analyses , we show that non-adaptive behaviors are optimal in two cases – when the optimal policy varies little across the tasks within the task ensemble and when the time it takes to learn the optimal policy is too long to allow a sufficient exploitation of the learned policy . Our results suggest that not only the design of the meta-task distribution , but also the lifetime of the agent can have strong effects on the meta-learned algorithm of RNN-based agents . In particular , we find highly nonlinear and potentially discontinuous effects of ecological uncertainty , task complexity and lifetime on the optimal algorithm . As a consequence , a meta-learned adaptation strategy that was optimized , e.g. , for a given lifetime may not generalize well to other lifetimes . This is essential for research questions that are interested in the conducted adaptation behavior , including curriculum design , safe exploration as well as human-in-the-loop applications . Our work may provide a principled way of examining the constraint-dependence of meta-learned inductive biases . The remainder of this paper is structured as follows : First , we review the background in memorybased meta-reinforcement learning and contrast the related literature . Afterwards , we analyze a Gaussian multi-arm bandit setting , which allows us to analytically disentangle the behavioral impact of ecological uncertainty , task complexity and lifetime . Our derivation of the lifetime-dependent Bayes optimal exploration reveals a highly non-linear interplay of these three factors . We show numerically that memory-based meta-learning reproduces our theoretical results and can learn not to learn . Furthermore , we extend our analysis to more complicated exploration problems . Throughout , we analyze the resulting recurrent dynamics of the network and the representations associated with learning and non-adaptive strategies . 2 RELATED WORK & BACKGROUND . Meta-learning or ’ learning to learn ’ ( e.g. , Schmidhuber , 1987 ; Thrun & Pratt , 1998 ; Hochreiter et al. , 2001 ; Duan et al. , 2016 ; Wang et al. , 2016 ; Finn et al. , 2017 ) has been proposed as a computational framework for acquiring task distribution-specific learning rules . During a costly outer loop optimization , an agent crafts a niche-specific adaptation strategy , which is applicable to an engineered task distribution . At inference time , the acquired inner loop learning algorithm is executed for a fixed amount of timesteps ( lifetime ) on a test task . This framework has successfully been applied to a range of applications such as the meta-learning of optimization updates ( Andrychowicz et al. , 2016 ; Flennerhag et al. , 2018 ; 2019 ) , agent ( Rabinowitz et al. , 2018 ) and world models ( Nagabandi et al. , 2018 ) and explicit models of memory ( Santoro et al. , 2016 ; Bartunov et al. , 2019 ) . Already , early work by Schmidhuber ( 1987 ) suggested an evolutionary perspective on recursively learning the rules of learning . This perspective holds the promise of explaining the emergence of mechanisms underlying both natural and artificial behaviors . Furthermore , a similarity between the hidden activations of LSTM-based meta-learners and the recurrent activity of neurons in the prefrontal cortex ( Wang et al. , 2018 ) has recently been suggested . Previous work has shown that LSTM-based meta-learning is capable of distilling a sequential integration algorithm akin to amortized Bayesian inference ( Ortega et al. , 2019 ; Rabinowitz , 2019 ; Mikulik et al. , 2020 ) . Here we investigate when the integration of information might not be the optimal strategy to meta-learn . We analytically characterize a task regime in which not adapting to sensory information is optimal . Furthermore , we study whether LSTM-based meta-learning is capable of inferring when to learn and when to execute a non-adaptive program . Rabinowitz ( 2019 ) previously studied the outer loop learning dynamics and found differences across several tasks , the origin of which is however not fully understood . Our work may provide an explanation for these different meta-learning dynamics and the dependence on the task distribution as well as the time horizon of adaptation . Our work is most closely related to Pardo et al . ( 2017 ) and Zintgraf et al . ( 2019 ) . Pardo et al . ( 2017 ) study the impact of fixed time limits and time-awareness on deep reinforcement learning agents . They propose using a timestamp as part of the state representation in order to avoid state-aliasing and the non-Markovianity resulting from a finite horizon treatment of an infinite horizon problem . Our setting differs in several aspects . First , we study the case of meta-reinforcement learning where the agent has to learn within a single lifetime . Second , we focus on a finite horizon perspective with limited adaptation . Zintgraf et al . ( 2019 ) , on the other hand , investigate meta reinforcement-learning for Bayes-adaptive Markov Decision Processes and introduce a novel architecture that disentangles task-specific belief representations from policy representations . Similarly to our work , Zintgraf et al . ( 2019 ) are interested in using the meta-learning framework to distill Bayes optimal exploration behavior . While their adaptation setup extends over multiple episodes , we focus on single lifetime adaption and analytically analyze when it is beneficial to learn in the first place . Finally , our work extends upon the efforts of computational ethology ( Stephens , 1991 ) and experimental evolution ( Dunlap & Stephens , 2009 ; 2016 ; Marcus et al. , 2018 ) , which aims to characterize the conditions under which behavioral plasticity may evolve . Their work shows that both environmental change and the predictability of the environment shape the selection pressure , which evolves adaptive traits . Our work is based on memory-based meta-learning with function approximation and aims to extend these original findings to task distributions for which no analytical solution may be available . 3 LEARNING NOT TO LEARN . To disentangle the influence of ecological uncertainty , task complexity , and lifetime on the nature of the meta-learned strategy , we first focus on a minimal two-arm Gaussian bandit task , which allows for an analytical solution . The agent experiences episodes consisting of T arm pulls , representing the lifetime of the agent . The statistics of the bandit are constant during each episode , but vary between episodes . To keep it simple , one of the two arms is deterministic and always returns a reward of 0 . The task distribution is represented by the variable expected reward of the other arm , which is sampled at the beginning of an episode , from a Gaussian distribution with mean -1 and standard deviation σp , i.e . µ ∼ N ( −1 , σ2p ) . The standard deviation σp controls the uncertainty of the ecological niche . For σp 1 , the deterministic arm is almost always the better option . For σp 1 , the chances of either arm being the best in the given episode are largely even . While the mean µ remains constant for the lifetime T of the agent , the reward obtained in a given trial is stochastic and is sampled from a second Gaussian , r ∼ N ( µ , σl ) . This trial-to-trial variability controls how many pulls the agent needs to estimate the mean reward of the stochastic arm . The standard deviation σl hence controls how quickly the agent can learn the optimal policy . We therefore use it as a proxy for task complexity . In this simple setting , the optimal meta-learned strategy can be calculated analytically . The optimal exploration strategy is to initially explore the stochastic arm for a given trial number n. Afterwards , it chooses the best arm based on its maximum a posteriori-estimate of the remaining episode return . The optimal amount of exploration trials n ? can then be derived analytically : 1 n ? = arg max n E [ T∑ t=1 rt|n , T , σl , σp ] = arg max n [ −n+ Eµ , r [ ( T − n ) × µ× p ( µ̂ > 0 ) ] ] , where µ̂ is the estimate of the mean reward of the stochatic arm after the n exploration trials . We find two distinct types of behavior ( left-hand side of figure 1 ) : A regime in which learning via exploration is effective and a second regime in which not learning is the optimal behavior . It may be optimal not to learn for two reasons : First , the ecological uncertainty may be so small that it is very unlikely that the stochastic first arm is better . Second , if the trial-to-trial variability is too large relative to the range 1Please refer to the supplementary material for a detailed derivation of this analytical result as well as the hyperparameters of the numerical experiments . of potential ecological niches , so that it may simply not be possible to integrate sufficient information given a limited lifespan . We make two observations : 1 . There exists a hard nonlinear threshold between learning and not learning behaviors described by the ratio of σl and σp . If σl is too large , the value of exploration ( or the reduction in uncertainty ) is too small to be profitable within the remaining lifetime of the agent . Instead , it is advantageous to hard-code a heuristic choice . 2 . The two regimes consistently exist across different lifetimes . As the lifetime grows , the learning regime becomes more and more prevalent . Given a sufficient amount of time , learning by exploring the uncertain arm is the best strategy . Is the common meta-learning framework capable of reproducing these different qualitative behaviors and performing Bayes optimal amortized inference across the entire spectrum of meta-task distributions ? Or differently put : Can memory-based meta-learning yield agents that do not only learn to learn but that also learn not to learn ? To answer this question , we train LSTM-based RL2 ( Wang et al. , 2016 ) agents with the standard synchronous actor-critic ( Mnih et al. , 2016 ) setup on the same grid of ecological uncertainties σp and ” task complexities ” σl . The input xt to the network at time t consists of the action of the previous timestep , a monotonically increasing timestamp within the current episode and crucially the reward of the previous timestep , xt = { at−1 , φ ( t ) , rt−1 } . The recurrent weight dynamics of the inner loop can then implement an internal learning algorithm that integrates previous experiences . After collecting a set of trajectories , we optimize the weights and initial condition of the hidden state with an outer loop gradient descent update to minimize the common actor-critic objective . We obtain the amount of meta-learned exploration by testing the RL2 agents on hold-out bandits for which we set σp = 0 and only vary σl . Thereby , it is ensured that the deterministic arm is the better arm . We can then define the number of exploration trials as the pulls from the suboptimal stochastic arm . We observe that meta-learning is capable of yielding agents that behave according to our derived theory of a Bayes optimal agent , which explicitly knows the given lifetime as well as uncertainties σl , σp ( figure 1 ) . Importantly , the meta-learned behavior also falls into two regimes : A regime in which the meta-learned strategy resembles a learning algorithm and a regime in which the recurrent dynamics encode a hard-coded choice of the deterministic arm . Furthermore , the edge between the two meta-learned regimes shifts with the agent ’ s lifetime as predicted by the Bayesian theory . As the lifetime increases , wider ecological niches at higher levels of task complexity become solvable and the strategy of learning profitable . In the Bayesian model , the edge between the two regimes is located at parameter values where the learning strategy and the non-learning strategy perform equally well . Because these two strategies are very distinct , we wondered whether the reward landscape for the memory-based meta-learner has two local maxima corresponding to the two strategies ( figure 2 ) . To test this , we trained N = 1000 networks with different initial conditions , for task parameters close to the edge , but in the regime where the theoretically optimal strategy would be to learn . We then evaluated for each network the number of explorative pulls of the stochastic arm , averaged across 100 episodes . The distribution of the number of explorative pulls across the 1000 networks shows i ) a peak at zero exploration and ii ) a broad tail of mean explorative pulls ( figure 2 ) , suggesting that there are indeed two classes of networks . One class never pulls the stochastic arm , i.e. , those networks adopt a non-learning strategy . The other class learns . For task parameters further away from the edge , this bimodality disappears . The two behavioral regimes are characterized by distinct recurrent dynamics of the trained LSTM agents . The two left-most columns of figure 3 display the policy entropy and hidden state statistics for a network trained on a σl , σp-combination associated with the regime in which learning is the optimal behavior . We differentiate between the case in which the deterministic arm is the better one ( µ < 0 ) and the case in which the second arm should be preferred ( µ > 0 ) . In both cases the agent first explores in order to identify the better arm . Moreover , the hidden dynamics appear to display two different attractors , which correspond to either of the arms being the better choice . The better arm can clearly be identified from the PCA-dimensionality reduced hidden state dynamics ( bottom row of figure 3 ) . The two right-most columns of figure 3 , on the other hand , depict the same statistics for a network that was meta-trained on the regime in which the optimal strategy is not to learn . Indeed , the agent always chooses the deterministic arm , regardless of whether it is the better choice . Accordingly , the network dynamics seem to fall into a single attractor . We examined how these strategies evolve over the course of meta-training and find that there are two phases : After an initial period of universal random behavior across all conditions , the distinct behavioral regimes emerge ( supplementary figure 9 ) . We note that this observation may be partially caused by the linear annealing of the entropy regularization coefficient in the actor-critic objective which we found to be crucial in training the networks . In summary , we observe that the meta-learned strategy shows a highly nonlinear , partially discontinuous dependence on task parameters . In transition regions between strategies , we find local maxima in the reward landscape that correspond to different learning strategies . In the simple bandit setting , these local maxima correspond to a learning and a non-learning strategy , respectively , hence providing a minimal model for a sharp nature-nurture trade-off . Next , we investigate whether these insights generalize to more complex domains by studying spatial reasoning .

Summary: .** The authors investigate the question of when the optimal behavior for an agent is to learn from experience versus when the optimal behavior is to apply the same (memorized) policy in every scenario. They begin by introducing a simple bandits environment wherein they derive the optimal policy and identify regimes in which it involves memorization vs. learning. Then they train an RL^2 agent and verify that it behaves as expected in these regimes. Next, they expand their approach to a slightly more complicated gridworld environment which does not have an analytic solution to the question. The agent behaves as expected in the gridworld environment.