David Balduzzi

Marta Garnelo, Wojciech Marian Czarnecki, Siqi Liu et al.

Strategic diversity is often essential in games: in multi-player games, for example, evaluating a player against a diverse set of strategies will yield a more accurate estimate of its performance. Furthermore, in games with non-transitivities diversity allows a player to cover several winning strategies. However, despite the significance of strategic diversity, training agents that exhibit diverse behaviour remains a challenge. In this paper we study how to construct diverse populations of agents by carefully structuring how individuals within a population interact. Our approach is based on interaction graphs, which control the flow of information between agents during training and can encourage agents to specialise on different strategies, leading to improved overall performance. We provide evidence for the importance of diversity in multi-agent training and analyse the effect of applying different interaction graphs on the training trajectories, diversity and performance of populations in a range of games. This is an extended version of the long abstract published at AAMAS.

Wojciech Marian Czarnecki, Gauthier Gidel, Brendan Tracey et al.

This paper investigates the geometrical properties of real world games (e.g. Tic-Tac-Toe, Go, StarCraft II). We hypothesise that their geometrical structure resemble a spinning top, with the upright axis representing transitive strength, and the radial axis, which corresponds to the number of cycles that exist at a particular transitive strength, representing the non-transitive dimension. We prove the existence of this geometry for a wide class of real world games, exposing their temporal nature. Additionally, we show that this unique structure also has consequences for learning - it clarifies why populations of strategies are necessary for training of agents, and how population size relates to the structure of the game. Finally, we empirically validate these claims by using a selection of nine real world two-player zero-sum symmetric games, showing 1) the spinning top structure is revealed and can be easily re-constructed by using a new method of Nash clustering to measure the interaction between transitive and cyclical strategy behaviour, and 2) the effect that population size has on the convergence in these games.

Edward Hughes, Thomas W. Anthony, Tom Eccles et al.

Zero-sum games have long guided artificial intelligence research, since they possess both a rich strategy space of best-responses and a clear evaluation metric. What's more, competition is a vital mechanism in many real-world multi-agent systems capable of generating intelligent innovations: Darwinian evolution, the market economy and the AlphaZero algorithm, to name a few. In two-player zero-sum games, the challenge is usually viewed as finding Nash equilibrium strategies, safeguarding against exploitation regardless of the opponent. While this captures the intricacies of chess or Go, it avoids the notion of cooperation with co-players, a hallmark of the major transitions leading from unicellular organisms to human civilization. Beyond two players, alliance formation often confers an advantage; however this requires trust, namely the promise of mutual cooperation in the face of incentives to defect. Successful play therefore requires adaptation to co-players rather than the pursuit of non-exploitability. Here we argue that a systematic study of many-player zero-sum games is a crucial element of artificial intelligence research. Using symmetric zero-sum matrix games, we demonstrate formally that alliance formation may be seen as a social dilemma, and empirically that naïve multi-agent reinforcement learning therefore fails to form alliances. We introduce a toy model of economic competition, and show how reinforcement learning may be augmented with a peer-to-peer contract mechanism to discover and enforce alliances. Finally, we generalize our agent model to incorporate temporally-extended contracts, presenting opportunities for further work.

Julien Perolat, Remi Munos, Jean-Baptiste Lespiau et al.

In this paper we investigate the Follow the Regularized Leader dynamics in sequential imperfect information games (IIG). We generalize existing results of Poincaré recurrence from normal-form games to zero-sum two-player imperfect information games and other sequential game settings. We then investigate how adapting the reward (by adding a regularization term) of the game can give strong convergence guarantees in monotone games. We continue by showing how this reward adaptation technique can be leveraged to build algorithms that converge exactly to the Nash equilibrium. Finally, we show how these insights can be directly used to build state-of-the-art model-free algorithms for zero-sum two-player Imperfect Information Games (IIG).

Gauthier Gidel, David Balduzzi, Wojciech Marian Czarnecki et al.

Adversarial training, a special case of multi-objective optimization, is an increasingly prevalent machine learning technique: some of its most notable applications include GAN-based generative modeling and self-play techniques in reinforcement learning which have been applied to complex games such as Go or Poker. In practice, a \emph{single} pair of networks is typically trained in order to find an approximate equilibrium of a highly nonconcave-nonconvex adversarial problem. However, while a classic result in game theory states such an equilibrium exists in concave-convex games, there is no analogous guarantee if the payoff is nonconcave-nonconvex. Our main contribution is to provide an approximate minimax theorem for a large class of games where the players pick neural networks including WGAN, StarCraft II, and Blotto Game. Our findings rely on the fact that despite being nonconcave-nonconvex with respect to the neural networks parameters, these games are concave-convex with respect to the actual models (e.g., functions or distributions) represented by these neural networks.

David Balduzzi, Wojciech M Czarnecki, Thomas W Anthony et al.

With the success of modern machine learning, it is becoming increasingly important to understand and control how learning algorithms interact. Unfortunately, negative results from game theory show there is little hope of understanding or controlling general n-player games. We therefore introduce smooth markets (SM-games), a class of n-player games with pairwise zero sum interactions. SM-games codify a common design pattern in machine learning that includes (some) GANs, adversarial training, and other recent algorithms. We show that SM-games are amenable to analysis and optimization using first-order methods.

Yan Wu, Jeff Donahue, David Balduzzi et al.

Training generative adversarial networks requires balancing of delicate adversarial dynamics. Even with careful tuning, training may diverge or end up in a bad equilibrium with dropped modes. In this work, we improve CS-GAN with natural gradient-based latent optimisation and show that it improves adversarial dynamics by enhancing interactions between the discriminator and the generator. Our experiments demonstrate that latent optimisation can significantly improve GAN training, obtaining state-of-the-art performance for the ImageNet ($128 \times 128$) dataset. Our model achieves an Inception Score (IS) of $148$ and an Fréchet Inception Distance (FID) of $3.4$, an improvement of $17\%$ and $32\%$ in IS and FID respectively, compared with the baseline BigGAN-deep model with the same architecture and number of parameters.

Alistair Letcher, David Balduzzi, Sebastien Racaniere et al.

Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new tools to understand and control the dynamics in n-player differentiable games. The key result is to decompose the game Jacobian into two components. The first, symmetric component, is related to potential games, which reduce to gradient descent on an implicit function. The second, antisymmetric component, relates to Hamiltonian games, a new class of games that obey a conservation law akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in differentiable games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- while at the same time being applicable to, and having guarantees in, much more general cases.

David Balduzzi, Marta Garnelo, Yoram Bachrach et al.

Zero-sum games such as chess and poker are, abstractly, functions that evaluate pairs of agents, for example labeling them `winner' and `loser'. If the game is approximately transitive, then self-play generates sequences of agents of increasing strength. However, nontransitive games, such as rock-paper-scissors, can exhibit strategic cycles, and there is no longer a clear objective -- we want agents to increase in strength, but against whom is unclear. In this paper, we introduce a geometric framework for formulating agent objectives in zero-sum games, in order to construct adaptive sequences of objectives that yield open-ended learning. The framework allows us to reason about population performance in nontransitive games, and enables the development of a new algorithm (rectified Nash response, PSRO_rN) that uses game-theoretic niching to construct diverse populations of effective agents, producing a stronger set of agents than existing algorithms. We apply PSRO_rN to two highly nontransitive resource allocation games and find that PSRO_rN consistently outperforms the existing alternatives.

Alistair Letcher, Jakob Foerster, David Balduzzi et al.

A growing number of learning methods are actually differentiable games whose players optimise multiple, interdependent objectives in parallel -- from GANs and intrinsic curiosity to multi-agent RL. Opponent shaping is a powerful approach to improve learning dynamics in these games, accounting for player influence on others' updates. Learning with Opponent-Learning Awareness (LOLA) is a recent algorithm that exploits this response and leads to cooperation in settings like the Iterated Prisoner's Dilemma. Although experimentally successful, we show that LOLA agents can exhibit 'arrogant' behaviour directly at odds with convergence. In fact, remarkably few algorithms have theoretical guarantees applying across all (n-player, non-convex) games. In this paper we present Stable Opponent Shaping (SOS), a new method that interpolates between LOLA and a stable variant named LookAhead. We prove that LookAhead converges locally to equilibria and avoids strict saddles in all differentiable games. SOS inherits these essential guarantees, while also shaping the learning of opponents and consistently either matching or outperforming LOLA experimentally.

David Balduzzi, Karl Tuyls, Julien Perolat et al.

Progress in machine learning is measured by careful evaluation on problems of outstanding common interest. However, the proliferation of benchmark suites and environments, adversarial attacks, and other complications has diluted the basic evaluation model by overwhelming researchers with choices. Deliberate or accidental cherry picking is increasingly likely, and designing well-balanced evaluation suites requires increasing effort. In this paper we take a step back and propose Nash averaging. The approach builds on a detailed analysis of the algebraic structure of evaluation in two basic scenarios: agent-vs-agent and agent-vs-task. The key strength of Nash averaging is that it automatically adapts to redundancies in evaluation data, so that results are not biased by the incorporation of easy tasks or weak agents. Nash averaging thus encourages maximally inclusive evaluation -- since there is no harm (computational cost aside) from including all available tasks and agents.

David Balduzzi, Sebastien Racaniere, James Martens et al.

The cornerstone underpinning deep learning is the guarantee that gradient descent on an objective converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, where there are multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new techniques to understand and control the dynamics in general games. The key result is to decompose the second-order dynamics into two components. The first is related to potential games, which reduce to gradient descent on an implicit function; the second relates to Hamiltonian games, a new class of games that obey a conservation law, akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in general games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- whilst at the same time being applicable to -- and having guarantees in -- much more general games.

David Balduzzi, Marcus Frean, Lennox Leary et al.

A long-standing obstacle to progress in deep learning is the problem of vanishing and exploding gradients. Although, the problem has largely been overcome via carefully constructed initializations and batch normalization, architectures incorporating skip-connections such as highway and resnets perform much better than standard feedforward architectures despite well-chosen initialization and batch normalization. In this paper, we identify the shattered gradients problem. Specifically, we show that the correlation between gradients in standard feedforward networks decays exponentially with depth resulting in gradients that resemble white noise whereas, in contrast, the gradients in architectures with skip-connections are far more resistant to shattering, decaying sublinearly. Detailed empirical evidence is presented in support of the analysis, on both fully-connected networks and convnets. Finally, we present a new "looks linear" (LL) initialization that prevents shattering, with preliminary experiments showing the new initialization allows to train very deep networks without the addition of skip-connections.

David Balduzzi

As artificial agents proliferate, it is becoming increasingly important to ensure that their interactions with one another are well-behaved. In this paper, we formalize a common-sense notion of when algorithms are well-behaved: an algorithm is safe if it does no harm. Motivated by recent progress in deep learning, we focus on the specific case where agents update their actions according to gradient descent. The paper shows that that gradient descent converges to a Nash equilibrium in safe games. The main contribution is to define strongly-typed agents and show they are guaranteed to interact safely, thereby providing sufficient conditions to guarantee safe interactions. A series of examples show that strong-typing generalizes certain key features of convexity, is closely related to blind source separation, and introduces a new perspective on classical multilinear games based on tensor decomposition.

David Balduzzi, Brian McWilliams, Tony Butler-Yeoman

Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the first convergence guarantee applicable to modern convnets, which furthermore matches a lower bound for convex nonsmooth functions. The key technical tool is the neural Taylor approximation -- a straightforward application of Taylor expansions to neural networks -- and the associated Taylor loss. Experiments on a range of optimizers, layers, and tasks provide evidence that the analysis accurately captures the dynamics of neural optimization. The second half of the paper applies the Taylor approximation to isolate the main difficulty in training rectifier nets -- that gradients are shattered -- and investigates the hypothesis that, by exploring the space of activation configurations more thoroughly, adaptive optimizers such as RMSProp and Adam are able to converge to better solutions.

Muhammad Ghifary, W. Bastiaan Kleijn, Mengjie Zhang et al.

In this paper, we propose a novel unsupervised domain adaptation algorithm based on deep learning for visual object recognition. Specifically, we design a new model called Deep Reconstruction-Classification Network (DRCN), which jointly learns a shared encoding representation for two tasks: i) supervised classification of labeled source data, and ii) unsupervised reconstruction of unlabeled target data.In this way, the learnt representation not only preserves discriminability, but also encodes useful information from the target domain. Our new DRCN model can be optimized by using backpropagation similarly as the standard neural networks. We evaluate the performance of DRCN on a series of cross-domain object recognition tasks, where DRCN provides a considerable improvement (up to ~8% in accuracy) over the prior state-of-the-art algorithms. Interestingly, we also observe that the reconstruction pipeline of DRCN transforms images from the source domain into images whose appearance resembles the target dataset. This suggests that DRCN's performance is due to constructing a single composite representation that encodes information about both the structure of target images and the classification of source images. Finally, we provide a formal analysis to justify the algorithm's objective in domain adaptation context.

David Balduzzi

Methods from convex optimization are widely used as building blocks for deep learning algorithms. However, the reasons for their empirical success are unclear, since modern convolutional networks (convnets), incorporating rectifier units and max-pooling, are neither smooth nor convex. Standard guarantees therefore do not apply. This paper provides the first convergence rates for gradient descent on rectifier convnets. The proof utilizes the particular structure of rectifier networks which consists in binary active/inactive gates applied on top of an underlying linear network. The approach generalizes to max-pooling, dropout and maxout. In other words, to precisely the neural networks that perform best empirically. The key step is to introduce gated games, an extension of convex games with similar convergence properties that capture the gating function of rectifiers. The main result is that rectifier convnets converge to a critical point at a rate controlled by the gated-regret of the units in the network. Corollaries of the main result include: (i) a game-theoretic description of the representations learned by a neural network; (ii) a logarithmic-regret algorithm for training neural nets; and (iii) a formal setting for analyzing conditional computation in neural nets that can be applied to recently developed models of attention.

Nicolás Della Penna, Mark D. Reid, David Balduzzi

Motivated by clinical trials, we study bandits with observable non-compliance. At each step, the learner chooses an arm, after, instead of observing only the reward, it also observes the action that took place. We show that such noncompliance can be helpful or hurtful to the learner in general. Unfortunately, naively incorporating compliance information into bandit algorithms loses guarantees on sublinear regret. We present hybrid algorithms that maintain regret bounds up to a multiplicative factor and can incorporate compliance information. Simulations based on real data from the International Stoke Trial show the practical potential of these algorithms.

David Balduzzi, Muhammad Ghifary

Recurrent neural networks are increasing popular models for sequential learning. Unfortunately, although the most effective RNN architectures are perhaps excessively complicated, extensive searches have not found simpler alternatives. This paper imports ideas from physics and functional programming into RNN design to provide guiding principles. From physics, we introduce type constraints, analogous to the constraints that forbids adding meters to seconds. From functional programming, we require that strongly-typed architectures factorize into stateless learnware and state-dependent firmware, reducing the impact of side-effects. The features learned by strongly-typed nets have a simple semantic interpretation via dynamic average-pooling on one-dimensional convolutions. We also show that strongly-typed gradients are better behaved than in classical architectures, and characterize the representational power of strongly-typed nets. Finally, experiments show that, despite being more constrained, strongly-typed architectures achieve lower training and comparable generalization error to classical architectures.

Muhammad Ghifary, David Balduzzi, W. Bastiaan Kleijn et al.

This paper addresses classification tasks on a particular target domain in which labeled training data are only available from source domains different from (but related to) the target. Two closely related frameworks, domain adaptation and domain generalization, are concerned with such tasks, where the only difference between those frameworks is the availability of the unlabeled target data: domain adaptation can leverage unlabeled target information, while domain generalization cannot. We propose Scatter Component Analyis (SCA), a fast representation learning algorithm that can be applied to both domain adaptation and domain generalization. SCA is based on a simple geometrical measure, i.e., scatter, which operates on reproducing kernel Hilbert space. SCA finds a representation that trades between maximizing the separability of classes, minimizing the mismatch between domains, and maximizing the separability of data; each of which is quantified through scatter. The optimization problem of SCA can be reduced to a generalized eigenvalue problem, which results in a fast and exact solution. Comprehensive experiments on benchmark cross-domain object recognition datasets verify that SCA performs much faster than several state-of-the-art algorithms and also provides state-of-the-art classification accuracy in both domain adaptation and domain generalization. We also show that scatter can be used to establish a theoretical generalization bound in the case of domain adaptation.

David Balduzzi

Deep learning is currently the subject of intensive study. However, fundamental concepts such as representations are not formally defined -- researchers "know them when they see them" -- and there is no common language for describing and analyzing algorithms. This essay proposes an abstract framework that identifies the essential features of current practice and may provide a foundation for future developments. The backbone of almost all deep learning algorithms is backpropagation, which is simply a gradient computation distributed over a neural network. The main ingredients of the framework are thus, unsurprisingly: (i) game theory, to formalize distributed optimization; and (ii) communication protocols, to track the flow of zeroth and first-order information. The framework allows natural definitions of semantics (as the meaning encoded in functions), representations (as functions whose semantics is chosen to optimized a criterion) and grammars (as communication protocols equipped with first-order convergence guarantees). Much of the essay is spent discussing examples taken from the literature. The ultimate aim is to develop a graphical language for describing the structure of deep learning algorithms that backgrounds the details of the optimization procedure and foregrounds how the components interact. Inspiration is taken from probabilistic graphical models and factor graphs, which capture the essential structural features of multivariate distributions.

David Balduzzi, Muhammad Ghifary

This paper proposes GProp, a deep reinforcement learning algorithm for continuous policies with compatible function approximation. The algorithm is based on two innovations. Firstly, we present a temporal-difference based method for learning the gradient of the value-function. Secondly, we present the deviator-actor-critic (DAC) model, which comprises three neural networks that estimate the value function, its gradient, and determine the actor's policy respectively. We evaluate GProp on two challenging tasks: a contextual bandit problem constructed from nonparametric regression datasets that is designed to probe the ability of reinforcement learning algorithms to accurately estimate gradients; and the octopus arm, a challenging reinforcement learning benchmark. GProp is competitive with fully supervised methods on the bandit task and achieves the best performance to date on the octopus arm.

David Balduzzi

Methods from convex optimization such as accelerated gradient descent are widely used as building blocks for deep learning algorithms. However, the reasons for their empirical success are unclear, since neural networks are not convex and standard guarantees do not apply. This paper develops the first rigorous link between online convex optimization and error backpropagation on convolutional networks. The first step is to introduce circadian games, a mild generalization of convex games with similar convergence properties. The main result is that error backpropagation on a convolutional network is equivalent to playing out a circadian game. It follows immediately that the waking-regret of players in the game (the units in the neural network) controls the overall rate of convergence of the network. Finally, we explore some implications of the results: (i) we describe the representations learned by a neural network game-theoretically, (ii) propose a learning setting at the level of individual units that can be plugged into deep architectures, and (iii) propose a new approach to adaptive model selection by applying bandit algorithms to choose which players to wake on each round.

Muhammad Ghifary, W. Bastiaan Kleijn, Mengjie Zhang et al.

The problem of domain generalization is to take knowledge acquired from a number of related domains where training data is available, and to then successfully apply it to previously unseen domains. We propose a new feature learning algorithm, Multi-Task Autoencoder (MTAE), that provides good generalization performance for cross-domain object recognition. Our algorithm extends the standard denoising autoencoder framework by substituting artificially induced corruption with naturally occurring inter-domain variability in the appearance of objects. Instead of reconstructing images from noisy versions, MTAE learns to transform the original image into analogs in multiple related domains. It thereby learns features that are robust to variations across domains. The learnt features are then used as inputs to a classifier. We evaluated the performance of the algorithm on benchmark image recognition datasets, where the task is to learn features from multiple datasets and to then predict the image label from unseen datasets. We found that (denoising) MTAE outperforms alternative autoencoder-based models as well as the current state-of-the-art algorithms for domain generalization.

David Balduzzi, Hastagiri Vanchinathan, Joachim Buhmann

Error backpropagation is an extremely effective algorithm for assigning credit in artificial neural networks. However, weight updates under Backprop depend on lengthy recursive computations and require separate output and error messages -- features not shared by biological neurons, that are perhaps unnecessary. In this paper, we revisit Backprop and the credit assignment problem. We first decompose Backprop into a collection of interacting learning algorithms; provide regret bounds on the performance of these sub-algorithms; and factorize Backprop's error signals. Using these results, we derive a new credit assignment algorithm for nonparametric regression, Kickback, that is significantly simpler than Backprop. Finally, we provide a sufficient condition for Kickback to follow error gradients, and show that Kickback matches Backprop's performance on real-world regression benchmarks.

David Balduzzi

The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that predicts optimally. An analogous result is shown for universal induction.

David Balduzzi

We investigate cortical learning from the perspective of mechanism design. First, we show that discretizing standard models of neurons and synaptic plasticity leads to rational agents maximizing simple scoring rules. Second, our main result is that the scoring rules are proper, implying that neurons faithfully encode expected utilities in their synaptic weights and encode high-scoring outcomes in their spikes. Third, with this foundation in hand, we propose a biologically plausible mechanism whereby neurons backpropagate incentives which allows them to optimize their usefulness to the rest of cortex. Finally, experiments show that networks that backpropagate incentives can learn simple tasks.

David Balduzzi

Despite its size and complexity, the human cortex exhibits striking anatomical regularities, suggesting there may simple meta-algorithms underlying cortical learning and computation. We expect such meta-algorithms to be of interest since they need to operate quickly, scalably and effectively with little-to-no specialized assumptions. This note focuses on a specific question: How can neurons use vast quantities of unlabeled data to speed up learning from the comparatively rare labels provided by reward systems? As a partial answer, we propose randomized co-training as a biologically plausible meta-algorithm satisfying the above requirements. As evidence, we describe a biologically-inspired algorithm, Correlated Nystrom Views (XNV) that achieves state-of-the-art performance in semi-supervised learning, and sketch work in progress on a neuronal implementation.

Brian McWilliams, David Balduzzi, Joachim M. Buhmann

This paper presents Correlated Nystrom Views (XNV), a fast semi-supervised algorithm for regression and classification. The algorithm draws on two main ideas. First, it generates two views consisting of computationally inexpensive random features. Second, XNV applies multiview regression using Canonical Correlation Analysis (CCA) on unlabeled data to bias the regression towards useful features. It has been shown that, if the views contains accurate estimators, CCA regression can substantially reduce variance with a minimal increase in bias. Random views are justified by recent theoretical and empirical work showing that regression with random features closely approximates kernel regression, implying that random views can be expected to contain accurate estimators. We show that XNV consistently outperforms a state-of-the-art algorithm for semi-supervised learning: substantially improving predictive performance and reducing the variability of performance on a wide variety of real-world datasets, whilst also reducing runtime by orders of magnitude.

Krikamol Muandet, David Balduzzi, Bernhard Schölkopf

This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the dissimilarity across domains, whilst preserving the functional relationship between input and output variables. A learning-theoretic analysis shows that reducing dissimilarity improves the expected generalization ability of classifiers on new domains, motivating the proposed algorithm. Experimental results on synthetic and real-world datasets demonstrate that DICA successfully learns invariant features and improves classifier performance in practice.

David Balduzzi

The bias/variance tradeoff is fundamental to learning: increasing a model's complexity can improve its fit on training data, but potentially worsens performance on future samples. Remarkably, however, the human brain effortlessly handles a wide-range of complex pattern recognition tasks. On the basis of these conflicting observations, it has been argued that useful biases in the form of "generic mechanisms for representation" must be hardwired into cortex (Geman et al). This note describes a useful bias that encourages cooperative learning which is both biologically plausible and rigorously justified.

David Balduzzi, Michel Besserve

This paper suggests a learning-theoretic perspective on how synaptic plasticity benefits global brain functioning. We introduce a model, the selectron, that (i) arises as the fast time constant limit of leaky integrate-and-fire neurons equipped with spiking timing dependent plasticity (STDP) and (ii) is amenable to theoretical analysis. We show that the selectron encodes reward estimates into spikes and that an error bound on spikes is controlled by a spiking margin and the sum of synaptic weights. Moreover, the efficacy of spikes (their usefulness to other reward maximizing selectrons) also depends on total synaptic strength. Finally, based on our analysis, we propose a regularized version of STDP, and show the regularization improves the robustness of neuronal learning when faced with multiple stimuli.

Pedro A. Ortega, Jordi Grau-Moya, Tim Genewein et al.

We propose a novel Bayesian approach to solve stochastic optimization problems that involve finding extrema of noisy, nonlinear functions. Previous work has focused on representing possible functions explicitly, which leads to a two-step procedure of first, doing inference over the function space and second, finding the extrema of these functions. Here we skip the representation step and directly model the distribution over extrema. To this end, we devise a non-parametric conjugate prior based on a kernel regressor. The resulting posterior distribution directly captures the uncertainty over the maximum of the unknown function. We illustrate the effectiveness of our model by optimizing a noisy, high-dimensional, non-convex objective function.

David Balduzzi, Pedro A Ortega, Michel Besserve

This paper investigates how neurons can use metabolic cost to facilitate learning at a population level. Although decision-making by individual neurons has been extensively studied, questions regarding how neurons should behave to cooperate effectively remain largely unaddressed. Under assumptions that capture a few basic features of cortical neurons, we show that constraining reward maximization by metabolic cost aligns the information content of actions with their expected reward. Thus, metabolic cost provides a mechanism whereby neurons encode expected reward into their outputs. Further, aside from reducing energy expenditures, imposing a tight metabolic constraint also increases the accuracy of empirical estimates of rewards, increasing the robustness of distributed learning. Finally, we present two implementations of metabolically constrained learning that confirm our theoretical finding. These results suggest that metabolic cost may be an organizing principle underlying the neural code, and may also provide a useful guide to the design and analysis of other cooperating populations.