David Balduzzi

AAMAS Conference 2022 Conference Paper

D3C: Reducing the Price of Anarchy in Multi-Agent Learning

• Ian Gemp
• Kevin R. McKee
• Richard Everett
• Edgar Duéñez-Guzmán
• Yoram Bachrach
• David Balduzzi
• Andrea Tacchetti

In multiagent systems, the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group, but also for each individual. Price of anarchy is a technical, game-theoretic definition that quantifies the inefficiency arising in these scenarios—it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable, upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator, agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents do so by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable, D3C resembles the celebrated Win-Stay, Lose-Shift strategy from behavioral game theory, thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting, as is common in multiagent reinforcement learning, we show the upper bound can be reduced via evolutionary strategies, until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox, a prisoner’s dilemma, and several multiagent domains.

ICML Conference 2021 Conference Paper

From Poincaré Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization

• Julien Pérolat
• Rémi Munos
• Jean-Baptiste Lespiau
• Shayegan Omidshafiei
• Mark Rowland 0001
• Pedro A. Ortega
• Neil Burch
• Thomas W. Anthony 0001

In this paper we investigate the Follow the Regularized Leader dynamics in sequential imperfect information games (IIG). We generalize existing results of Poincar{é} 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).

AAMAS Conference 2021 Conference Paper

Pick Your Battles: Interaction Graphs as Population-Level Objectives for Strategic Diversity

• Marta Garnelo
• Wojciech Marian Czarnecki
• Siqi Liu
• Dhruva Tirumala
• Junhyuk Oh
• Gauthier Gidel
• Hado van Hasselt
• David Balduzzi

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.

ICML Conference 2020 Conference Paper

From Chaos to Order: Symmetry and Conservation Laws in Game Dynamics

• Sai Ganesh Nagarajan
• David Balduzzi
• Georgios Piliouras

Games are an increasingly useful tool for training and testing learning algorithms. Recent examples include GANs, AlphaZero and the AlphaStar league. However, multi-agent learning can be extremely difficult to predict and control. Learning dynamics even in simple games can yield chaotic behavior. In this paper, we present basic \emph{mechanism design} tools for constructing games with predictable and controllable dynamics. We show that arbitrarily large and complex network games, encoding both cooperation (team play) and competition (zero-sum interaction), exhibit conservation laws when agents use the standard regret-minimizing dynamics known as Follow-the-Regularized-Leader. These laws persist when different agents use different dynamics and encode long-range correlations between agents’ behavior, even though the agents may not interact directly. Moreover, we provide sufficient conditions under which the dynamics have multiple, linearly independent, conservation laws. Increasing the number of conservation laws results in more predictable dynamics, eventually making chaotic behavior formally impossible in some cases.

NeurIPS Conference 2020 Conference Paper

Real World Games Look Like Spinning Tops

• Wojciech M. Czarnecki
• Gauthier Gidel
• Brendan Tracey
• Karl Tuyls
• Shayegan Omidshafiei
• David Balduzzi
• Max Jaderberg

This paper investigates the geometrical properties of real world games (e. g. Tic-Tac-Toe, Go, StarCraft II). We hypothesise that their geometrical structure resembles a spinning top, with the upright axis representing transitive strength, and the radial axis representing the non-transitive dimension, which corresponds to the number of cycles that exist at a particular transitive strength. We prove the existence of this geometry for a wide class of real world games by 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 reconstructed 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 of learning in these games.

ICLR Conference 2020 Conference Paper

Smooth markets: A basic mechanism for organizing gradient-based learners

• David Balduzzi
• Wojciech Czarnecki 0001
• Thomas W. Anthony 0001
• Ian Gemp
• Edward Hughes 0001
• Joel Z. Leibo
• Georgios Piliouras
• Thore Graepel

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.

JMLR Journal 2019 Journal Article

Differentiable Game Mechanics

• Alistair Letcher
• David Balduzzi
• Sébastien Racanière
• James Martens
• Jakob Foerster
• Karl Tuyls
• Thore Graepel

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. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

ICML Conference 2019 Conference Paper

Open-ended learning in symmetric zero-sum games

• David Balduzzi
• Marta Garnelo
• Yoram Bachrach
• Wojciech Czarnecki 0001
• Julien Pérolat
• Max Jaderberg
• Thore Graepel

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.

NeurIPS Conference 2018 Conference Paper

Re-evaluating evaluation

• David Balduzzi
• Karl Tuyls
• Julien Perolat
• Thore Graepel

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.

ICML Conference 2018 Conference Paper

The Mechanics of n-Player Differentiable Games

• David Balduzzi
• Sébastien Racanière
• James Martens
• Jakob N. Foerster
• Karl Tuyls
• Thore Graepel

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 local Nash equilibria in GANs – whilst at the same time being applicable to – and having guarantees in – much more general games.

ICML Conference 2017 Conference Paper

Neural Taylor Approximations: Convergence and Exploration in Rectifier Networks

• 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.

ICML Conference 2017 Conference Paper

Strongly-Typed Agents are Guaranteed to Interact Safely

• 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.

ICML Conference 2017 Conference Paper

The Shattered Gradients Problem: If resnets are the answer, then what is the question?

• David Balduzzi
• Marcus Frean
• Lennox Leary
• J. P. Lewis 0001
• Kurt Wan-Duo Ma
• Brian McWilliams

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.

ICML Conference 2016 Conference Paper

Strongly-Typed Recurrent Neural Networks

• 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.

AAAI Conference 2015 Conference Paper

Kickback Cuts Backprop’s Red-Tape: Biologically Plausible Credit Assignment in Neural Networks

• 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.

UAI Conference 2015 Conference Paper

Learning the Structure of Sum-Product Networks via an SVD-based Algorithm

• Tameem Adel
• David Balduzzi
• Ali Ghodsi 0001

Sum-product networks (SPNs) are a recently developed class of deep probabilistic models where inference is tractable. We present two new structure learning algorithms for sum-product networks, in the generative and discriminative settings, that are based on recursively extracting rank-one submatrices from data. The proposed algorithms find the subSPNs that are the most coherent jointly in the instances and variables – that is, whose instances are most strongly correlated over the given variables. Experimental results show that SPNs learned using the proposed generative algorithm have better likelihood and inference results – and also much faster – than previous approaches. Finally, we apply the discriminative SPN structure learning algorithm to handwritten digit recognition tasks, where it achieves state-of-the-art performance for an SPN.

NeurIPS Conference 2013 Conference Paper

Correlated random features for fast semi-supervised learning

• Brian McWilliams
• David Balduzzi
• Joachim 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, multiview regression, using Canonical Correlation Analysis (CCA) on unlabeled data, biases the regression towards useful features. It has been shown that CCA regression can substantially reduce variance with a minimal increase in bias if the views contains accurate estimators. Recent theoretical and empirical work shows that regression with random features closely approximates kernel regression, implying that the accuracy requirement holds for random views. 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.

ICML Conference 2013 Conference Paper

Domain Generalization via Invariant Feature Representation

• 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.

NeurIPS Conference 2012 Conference Paper

A Nonparametric Conjugate Prior Distribution for the Maximizing Argument of a Noisy Function

• Pedro Ortega
• Jordi Grau-Moya
• Tim Genewein
• David Balduzzi
• Daniel Braun

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 where the natural parameter corresponds to a given kernel function and the sufficient statistic is composed of the observed function values. The resulting posterior distribution directly captures the uncertainty over the maximum of the unknown function.

NeurIPS Conference 2012 Conference Paper

Towards a learning-theoretic analysis of spike-timing dependent plasticity

• 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.

ICML Conference 2011 Conference Paper

Uncovering the Temporal Dynamics of Diffusion Networks

• Manuel Gomez Rodriguez
• David Balduzzi
• Bernhard Schölkopf