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During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Fairness in clustering has been considered extensively in the past; however, the trade-off between the two objectives -- e.g., can we sacrifice just a little in the quality of the clustering to significantly increase fairness, or vice-versa? -- has rarely been addressed. We introduce novel algorithms for tracing the complete trade-off curve, or Pareto front, between quality and fairness in clustering problems; that is, computing all clusterings that are not dominated in both objectives by other clusterings. Unlike previous work that deals with specific objectives for quality and fairness, we deal with all objectives for fairness and quality in two general classes encompassing most of the special cases addressed in previous work. Our algorithm must take exponential time in the worst case as the Pareto front itself can be exponential. Even when the Pareto front is polynomial, our algorithm may take exponential time, and we prove that this is inevitable unless P = NP. However, we also present a new polynomial-time algorithm for computing the entire Pareto front when the cluster centers are fixed, and for perhaps the most natural fairness objective: minimizing the sum, over all clusters, of the imbalance between the two groups in each cluster.

In repeated interaction problems with adaptive agents, our objective often requires anticipating and optimizing over the space of possible agent responses. We show that many problems of this form can be cast as instances of online (nonlinear) control which satisfy \textit{local controllability}, with convex losses over a bounded state space which encodes agent behavior, and we introduce a unified algorithmic framework for tractable regret minimization in such cases. When the instance dynamics are known but otherwise arbitrary, we obtain oracle-efficient $O(\sqrt{T})$ regret by reduction to online convex optimization, which can be made computationally efficient if dynamics are locally \textit{action-linear}. In the presence of adversarial disturbances to the state, we give tight bounds in terms of either the cumulative or per-round disturbance magnitude (for \textit{strongly} or \textit{weakly} locally controllable dynamics, respectively). Additionally, we give sublinear regret results for the cases of unknown locally action-linear dynamics as well as for the bandit feedback setting. Finally, we demonstrate applications of our framework to well-studied problems including performative prediction, recommendations for adaptive agents, adaptive pricing of real-valued goods, and repeated gameplay against no-regret learners, directly yielding extensions beyond prior results in each case., Comment: COLT 2024

How intelligence arises from the brain is a central problem in science. A crucial aspect of intelligence is dealing with uncertainty -- developing good predictions about one's environment, and converting these predictions into decisions. The brain itself seems to be noisy at many levels, from chemical processes which drive development and neuronal activity to trial variability of responses to stimuli. One hypothesis is that the noise inherent to the brain's mechanisms is used to sample from a model of the world and generate predictions. To test this hypothesis, we study the emergence of statistical learning in NEMO, a biologically plausible computational model of the brain based on stylized neurons and synapses, plasticity, and inhibition, and giving rise to assemblies -- a group of neurons whose coordinated firing is tantamount to recalling a location, concept, memory, or other primitive item of cognition. We show in theory and simulation that connections between assemblies record statistics, and ambient noise can be harnessed to make probabilistic choices between assemblies. This allows NEMO to create internal models such as Markov chains entirely from the presentation of sequences of stimuli. Our results provide a foundation for biologically plausible probabilistic computation, and add theoretical support to the hypothesis that noise is a useful component of the brain's mechanism for cognition., Comment: 22 pages, 8 figures

Story Visualization (SV) is a challenging generative vision task, that requires both visual quality and consistency between different frames in generated image sequences. Previous approaches either employ some kind of memory mechanism to maintain context throughout an auto-regressive generation of the image sequence, or model the generation of the characters and their background separately, to improve the rendering of characters. On the contrary, we embrace a completely parallel transformer-based approach, exclusively relying on Cross-Attention with past and future captions to achieve consistency. Additionally, we propose a Character Guidance technique to focus on the generation of characters in an implicit manner, by forming a combination of text-conditional and character-conditional logits in the logit space. We also employ a caption-augmentation technique, carried out by a Large Language Model (LLM), to enhance the robustness of our approach. The combination of these methods culminates into state-of-the-art (SOTA) results over various metrics in the most prominent SV benchmark (Pororo-SV), attained with constraint resources while achieving superior computational complexity compared to previous arts. The validity of our quantitative results is supported by a human survey.

What are the root causes of hallucinations in large language models (LLMs)? We use Communication Complexity to prove that the Transformer layer is incapable of composing functions (e.g., identify a grandparent of a person in a genealogy) if the domains of the functions are large enough; we show through examples that this inability is already empirically present when the domains are quite small. We also point out that several mathematical tasks that are at the core of the so-called compositional tasks thought to be hard for LLMs are unlikely to be solvable by Transformers, for large enough instances and assuming that certain well accepted conjectures in the field of Computational Complexity are true.

Consider the one-dimentional Poisson equation \(-u''=f\) on the interval \([-\pi,\pi]\), where \(f\) is an non-negative integrable function, with Robin boundary conditions \(-u'(-\pi)+\alpha u(-\pi)=u'(\pi)+\alpha u(\pi)=0\), where \(\alpha>0\) is a constant. In this paper we prove inequalities for the convex integral means of solutions of this problem, using the polarization of functions and its properties. We find solutions with maximal convex integral means and prove their uniqueness.

The problem of continual learning in the domain of reinforcement learning, often called non-stationary reinforcement learning, has been identified as an important challenge to the application of reinforcement learning. We prove a worst-case complexity result, which we believe captures this challenge: Modifying the probabilities or the reward of a single state-action pair in a reinforcement learning problem requires an amount of time almost as large as the number of states in order to keep the value function up to date, unless the strong exponential time hypothesis (SETH) is false; SETH is a widely accepted strengthening of the P $\neq$ NP conjecture. Recall that the number of states in current applications of reinforcement learning is typically astronomical. In contrast, we show that just $\textit{adding}$ a new state-action pair is considerably easier to implement.

We present a simulated biologically plausible language organ, made up of stylized but realistic neurons, synapses, brain areas, plasticity, and a simplified model of sensory perception. We show through experiments that this model succeeds in an important early step in language acquisition: the learning of nouns, verbs, and their meanings, from the grounded input of only a modest number of sentences. Learning in this system is achieved through Hebbian plasticity, and without backpropagation. Our model goes beyond a parser previously designed in a similar environment, with the critical addition of a biologically plausible account for how language can be acquired in the infant's brain, not just processed by a mature brain., Comment: 9 pages, 4 figures

Even as machine learning exceeds human-level performance on many applications, the generality, robustness, and rapidity of the brain's learning capabilities remain unmatched. How cognition arises from neural activity is a central open question in neuroscience, inextricable from the study of intelligence itself. A simple formal model of neural activity was proposed in Papadimitriou [2020] and has been subsequently shown, through both mathematical proofs and simulations, to be capable of implementing certain simple cognitive operations via the creation and manipulation of assemblies of neurons. However, many intelligent behaviors rely on the ability to recognize, store, and manipulate temporal sequences of stimuli (planning, language, navigation, to list a few). Here we show that, in the same model, time can be captured naturally as precedence through synaptic weights and plasticity, and, as a result, a range of computations on sequences of assemblies can be carried out. In particular, repeated presentation of a sequence of stimuli leads to the memorization of the sequence through corresponding neural assemblies: upon future presentation of any stimulus in the sequence, the corresponding assembly and its subsequent ones will be activated, one after the other, until the end of the sequence. Finally, we show that any finite state machine can be learned in a similar way, through the presentation of appropriate patterns of sequences. Through an extension of this mechanism, the model can be shown to be capable of universal computation. We support our analysis with a number of experiments to probe the limits of learning in this model in key ways. Taken together, these results provide a concrete hypothesis for the basis of the brain's remarkable abilities to compute and learn, with sequences playing a vital role., Comment: 31 pages, 11 figures

The brain is a complex system comprising a myriad of interacting elements, posing significant challenges in understanding its structure, function, and dynamics. Network science has emerged as a powerful tool for studying such intricate systems, offering a framework for integrating multiscale data and complexity. Here, we discuss the application of network science in the study of the brain, addressing topics such as network models and metrics, the connectome, and the role of dynamics in neural networks. We explore the challenges and opportunities in integrating multiple data streams for understanding the neural transitions from development to healthy function to disease, and discuss the potential for collaboration between network science and neuroscience communities. We underscore the importance of fostering interdisciplinary opportunities through funding initiatives, workshops, and conferences, as well as supporting students and postdoctoral fellows with interests in both disciplines. By uniting the network science and neuroscience communities, we can develop novel network-based methods tailored to neural circuits, paving the way towards a deeper understanding of the brain and its functions., Comment: 19 pages, 1 figure, 1 box

Goodness--of--fit tests for the distribution of the composed error term in a Stochastic Frontier Model (SFM) are suggested. The focus is on the case of a normal/gamma SFM and the heavy--tailed stable/gamma SFM. In the first case the moment generating function is used as tool while in the latter case the characteristic function of the error term is employed. In both cases our test statistics are formulated as weighted integrals of properly standardized data. The new normal/gamma test is consistent, and is shown to have an intrinsic relation to moment--based tests. The finite--sample behavior of resampling versions of both tests is investigated by Monte Carlo simulation, while several real--data applications are also included.

We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey's theorem, the Sunflower theorem, the Erd\H{o}s-Ko-Rado lemma, and K\"onig's lemma. Whether the first two of these four problems are PLC-complete is an important open question which we pursue; in contrast, we show that the latter two are PPP-complete. Finally, we reframe PPP as an optimization problem, and define a hierarchy of such problems related to Tur\'an's theorem.

Kakutani's Fixed Point theorem is a fundamental theorem in topology with numerous applications in game theory and economics. Computational formulations of Kakutani exist only in special cases and are too restrictive to be useful in reductions. In this paper, we provide a general computational formulation of Kakutani's Fixed Point Theorem and we prove that it is PPAD-complete. As an application of our theorem we are able to characterize the computational complexity of the following fundamental problems: (1) Concave Games. Introduced by the celebrated works of Debreu and Rosen in the 1950s and 60s, concave $n$-person games have found many important applications in Economics and Game Theory. We characterize the computational complexity of finding an equilibrium in such games. We show that a general formulation of this problem belongs to PPAD, and that finding an equilibrium is PPAD-hard even for a rather restricted games of this kind: strongly-concave utilities that can be expressed as multivariate polynomials of a constant degree with axis aligned box constraints. (2) Walrasian Equilibrium. Using Kakutani's fixed point Arrow and Debreu we resolve an open problem related to Walras's theorem on the existence of price equilibria in general economies. There are many results about the PPAD-hardness of Walrasian equilibria, but the inclusion in PPAD is only known for piecewise linear utilities. We show that the problem with general convex utilities is in PPAD. Along the way we provide a Lipschitz continuous version of Berge's maximum theorem that may be of independent interest.

A computational system implemented exclusively through the spiking of neurons was recently shown capable of syntax, that is, of carrying out the dependency parsing of simple English sentences. We address two of the most important questions left open by that work: constituency (the identification of key parts of the sentence such as the verb phrase) and the processing of dependent sentences, especially center-embedded ones. We show that these two aspects of language can also be implemented by neurons and synapses in a way that is compatible with what is known, or widely believed, about the structure and function of the language organ. Surprisingly, the way we implement center embedding points to a new characterization of context-free languages., Comment: NALOMA 2022

Continual learning, or lifelong learning, is a formidable current challenge to machine learning. It requires the learner to solve a sequence of $k$ different learning tasks, one after the other, while retaining its aptitude for earlier tasks; the continual learner should scale better than the obvious solution of developing and maintaining a separate learner for each of the $k$ tasks. We embark on a complexity-theoretic study of continual learning in the PAC framework. We make novel uses of communication complexity to establish that any continual learner, even an improper one, needs memory that grows linearly with $k$, strongly suggesting that the problem is intractable. When logarithmically many passes over the learning tasks are allowed, we provide an algorithm based on multiplicative weights update whose memory requirement scales well; we also establish that improper learning is necessary for such performance. We conjecture that these results may lead to new promising approaches to continual learning.

Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for $\epsilon$-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to $\epsilon$-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any $\epsilon$ between zero and $0.09$. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics., Comment: 25 pages

We consider goodness-of-fit tests for the distribution of the composed error in Stochastic Frontier Models. The proposed test statistic utilizes the characteristic function of the composed error term, and is formulated as a weighted integral of properly standardized data. The new test statistic is shown to be consistent and computationally convenient. Simulation results are presented whereby resampling versions of the new tests are compared to classical goodness-of-fit methods., Comment: 15 pages, 6 tables

We revisit the planning problem in the blocks world, and we implement a known heuristic for this task. Importantly, our implementation is biologically plausible, in the sense that it is carried out exclusively through the spiking of neurons. Even though much has been accomplished in the blocks world over the past five decades, we believe that this is the first algorithm of its kind. The input is a sequence of symbols encoding an initial set of block stacks as well as a target set, and the output is a sequence of motion commands such as "put the top block in stack 1 on the table". The program is written in the Assembly Calculus, a recently proposed computational framework meant to model computation in the brain by bridging the gap between neural activity and cognitive function. Its elementary objects are assemblies of neurons (stable sets of neurons whose simultaneous firing signifies that the subject is thinking of an object, concept, word, etc.), its commands include project and merge, and its execution model is based on widely accepted tenets of neuroscience. A program in this framework essentially sets up a dynamical system of neurons and synapses that eventually, with high probability, accomplishes the task. The purpose of this work is to establish empirically that reasonably large programs in the Assembly Calculus can execute correctly and reliably; and that rather realistic -- if idealized -- higher cognitive functions, such as planning in the blocks world, can be implemented successfully by such programs.

An assembly is a large population of neurons whose synchronous firing is hypothesized to represent a memory, concept, word, and other cognitive categories. Assemblies are believed to provide a bridge between high-level cognitive phenomena and low-level neural activity. Recently, a computational system called the Assembly Calculus (AC), with a repertoire of biologically plausible operations on assemblies, has been shown capable of simulating arbitrary space-bounded computation, but also of simulating complex cognitive phenomena such as language, reasoning, and planning. However, the mechanism whereby assemblies can mediate learning has not been known. Here we present such a mechanism, and prove rigorously that, for simple classification problems defined on distributions of labeled assemblies, a new assembly representing each class can be reliably formed in response to a few stimuli from the class; this assembly is henceforth reliably recalled in response to new stimuli from the same class. Furthermore, such class assemblies will be distinguishable as long as the respective classes are reasonably separated -- for example, when they are clusters of similar assemblies. To prove these results, we draw on random graph theory with dynamic edge weights to estimate sequences of activated vertices, yielding strong generalizations of previous calculations and theorems in this field over the past five years. These theorems are backed up by experiments demonstrating the successful formation of assemblies which represent concept classes on synthetic data drawn from such distributions, and also on MNIST, which lends itself to classification through one assembly per digit. Seen as a learning algorithm, this mechanism is entirely online, generalizes from very few samples, and requires only mild supervision -- all key attributes of learning in a model of the brain., Comment: 35 pages, 7 figures, accepted to COLT 2022

We describe a parser of English effectuated by biologically plausible neurons and synapses, and implemented through the Assembly Calculus, a recently proposed computational framework for cognitive function. We demonstrate that this device is capable of correctly parsing reasonably nontrivial sentences. While our experiments entail rather simple sentences in English, our results suggest that the parser can be extended beyond what we have implemented, to several directions encompassing much of language. For example, we present a simple Russian version of the parser, and discuss how to handle recursion, embedding, and polysemy.

The subject-verb-object (SVO) word order prevalent in English is shared by about $42\%$ of world languages. Another $45\%$ of all languages follow the SOV order, $9\%$ the VSO order, and fewer languages use the three remaining permutations. None of the many extant explanations of this phenomenon take into account the difficulty of implementing these permutations in the brain. We propose a plausible model of sentence generation inspired by the recently proposed Assembly Calculus framework of brain function. Our model results in a natural explanation of the uneven frequencies. Estimating the parameters of this model yields predictions of the relative difficulty of dis-inhibiting one brain area from another. Our model is based on the standard syntax tree, a simple binary tree with three leaves. Each leaf corresponds to one of the three parts of a basic sentence. The leaves can be activated through lock and unlock operations and the sequence of activation of the leaves implements a specific word order. More generally, we also formulate and algorithmically solve the problems of implementing a permutation of the leaves of any binary tree, and of selecting the permutation that is easiest to implement on a given binary tree.

Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to prove, finding the equilibrium is PPAD-complete. Finally, when the treewidth of the directed network is appropriately bounded, we prove that polynomial-time algorithms are possible.

Despite their impressive performance in NLP, self-attention networks were recently proved to be limited for processing formal languages with hierarchical structure, such as $\mathsf{Dyck}_k$, the language consisting of well-nested parentheses of $k$ types. This suggested that natural language can be approximated well with models that are too weak for formal languages, or that the role of hierarchy and recursion in natural language might be limited. We qualify this implication by proving that self-attention networks can process $\mathsf{Dyck}_{k, D}$, the subset of $\mathsf{Dyck}_{k}$ with depth bounded by $D$, which arguably better captures the bounded hierarchical structure of natural language. Specifically, we construct a hard-attention network with $D+1$ layers and $O(\log k)$ memory size (per token per layer) that recognizes $\mathsf{Dyck}_{k, D}$, and a soft-attention network with two layers and $O(\log k)$ memory size that generates $\mathsf{Dyck}_{k, D}$. Experiments show that self-attention networks trained on $\mathsf{Dyck}_{k, D}$ generalize to longer inputs with near-perfect accuracy, and also verify the theoretical memory advantage of self-attention networks over recurrent networks., Comment: ACL 2021. 19 pages with extended appendix. Fixed a small typo in the formula at the end of page 5 (thank to Gabriel Faria). Code: https://github.com/princeton-nlp/dyck-transformer

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a "prophet" who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? We study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of $1/2$-competitive algorithms are known against optimum offline. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm which approximates the optimal online algorithm within a factor of $0.51$ -- beating the best-possible prophet inequality. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant $\alpha < 1$.

On-line firms deploy suites of software platforms, where each platform is designed to interact with users during a certain activity, such as browsing, chatting, socializing, emailing, driving, etc. The economic and incentive structure of this exchange, as well as its algorithmic nature, have not been explored to our knowledge. We model this interaction as a Stackelberg game between a Designer and one or more Agents. We model an Agent as a Markov chain whose states are activities; we assume that the Agent's utility is a linear function of the steady-state distribution of this chain. The Designer may design a platform for each of these activities/states; if a platform is adopted by the Agent, the transition probabilities of the Markov chain are affected, and so is the objective of the Agent. The Designer's utility is a linear function of the steady state probabilities of the accessible states minus the development cost of the platforms. The underlying optimization problem of the Agent -- how to choose the states for which to adopt the platform -- is an MDP. If this MDP has a simple yet plausible structure (the transition probabilities from one state to another only depend on the target state and the recurrent probability of the current state) the Agent's problem can be solved by a greedy algorithm. The Designer's optimization problem (designing a custom suite for the Agent so as to optimize, through the Agent's optimum reaction, the Designer's revenue), is NP-hard to approximate within any finite ratio; however, the special case, while still NP-hard, has an FPTAS. These results generalize from a single Agent to a distribution of Agents with finite support, as well as to the setting where the Designer must find the best response to the existing strategies of other Designers. We discuss other implications of our results and directions of future research., Comment: updated with more results

The history of computer science and brain sciences are intertwined. In his unfinished manuscript "The Computer and the Brain," von Neumann debates whether or not the brain can be thought of as a computing machine and identifies some of the similarities and differences between natural and artificial computation. Turing, in his 1950 article in Mind, argues that computing devices could ultimately emulate intelligence, leading to his proposed Turing test. Herbert Simon predicted in 1957 that most psychological theories would take the form of a computer program. In 1976, David Marr proposed that the function of the visual system could be abstracted and studied at computational and algorithmic levels that did not depend on the underlying physical substrate. In December 2014, a two-day workshop supported by the Computing Community Consortium (CCC) and the National Science Foundation's Computer and Information Science and Engineering Directorate (NSF CISE) was convened in Washington, DC, with the goal of bringing together computer scientists and brain researchers to explore these new opportunities and connections, and develop a new, modern dialogue between the two research communities. Specifically, our objectives were: 1. To articulate a conceptual framework for research at the interface of brain sciences and computing and to identify key problems in this interface, presented in a way that will attract both CISE and brain researchers into this space. 2. To inform and excite researchers within the CISE research community about brain research opportunities and to identify and explain strategic roles they can play in advancing this initiative. 3. To develop new connections, conversations and collaborations between brain sciences and CISE researchers that will lead to highly relevant and competitive proposals, high-impact research, and influential publications., Comment: A Computing Community Consortium (CCC) workshop report, 24 pages

Proof-of-work blockchains reward each miner for one completed block by an amount that is, in expectation, proportional to the number of hashes the miner contributed to the mining of the block. Is this proportional allocation rule optimal? And in what sense? And what other rules are possible? In particular, what are the desirable properties that any "good" allocation rule should satisfy? To answer these questions, we embark on an axiomatic theory of incentives in proof-of-work blockchains at the time scale of a single block. We consider desirable properties of allocation rules including: symmetry; budget balance (weak or strong); sybil-proofness; and various grades of collusion-proofness. We show that Bitcoin's proportional allocation rule is the unique allocation rule satisfying a certain system of properties, but this does not hold for slightly weaker sets of properties, or when the miners are not risk-neutral. We also point out that a rich class of allocation rules can be approximately implemented in a proof-of-work blockchain.

The use of monotonicity and Tarski's theorem in existence proofs of equilibria is very widespread in economics, while Tarski's theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the $d$-dimensional grid with sides of length $N$, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time $\log^d N$, and we show it requires at least $\log^2 N$ function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski's theorem, however, requires $d\cdot N$ steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in $O(\log N)$ steps. We also observe that computing (approximating) the value of Condon's (Shapley's) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a $\Omega(\log^d N)$ lower bound for small fixed dimension $d \geq 3$.

We introduce $\alpha$-Rank, a principled evolutionary dynamics methodology for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov-Conley chains (MCCs). The approach leverages continuous- and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, the type of interactions, and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). $\alpha$-Rank provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics. This is a consequence of the links we establish to the MCC solution concept when the underlying evolutionary model's ranking-intensity parameter, $\alpha$, is chosen to be large, which exactly forms the basis of $\alpha$-Rank. In contrast to the Nash equilibrium, which is a static concept based on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley's Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fixed points, recurrent sets, periodic orbits, and limit cycles. $\alpha$-Rank runs in polynomial time with respect to the total number of pure strategy profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable. We introduce proofs that not only provide a unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the $\alpha$-Rank methodology. We empirically validate the method in several domains including AlphaGo, AlphaZero, MuJoCo Soccer, and Poker.

The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. The winner of each battlefield is the colonel who puts more troops in it and the overall utility of each colonel is the sum of weights of the battlefields that s/he wins. Over the past century, the Colonel Blotto game has found applications in many different forms of competition from advertisements to politics to sports. Two main objectives have been proposed for this game in the literature: (i) maximizing the guaranteed expected payoff, and (ii) maximizing the probability of obtaining a minimum payoff $u$. The former corresponds to the conventional utility maximization and the latter concerns scenarios such as elections where the candidates' goal is to maximize the probability of getting at least half of the votes (rather than the expected number of votes). In this paper, we consider both of these objectives and show how it is possible to obtain (almost) optimal solutions that have few strategies in their support. One of the main technical challenges in obtaining bounded support strategies for the Colonel Blotto game is that the solution space becomes non-convex. This prevents us from using convex programming techniques in finding optimal strategies which are essentially the main tools that are used in the literature. However, we show through a set of structural results that the solution space can, interestingly, be partitioned into polynomially many disjoint convex polytopes that can be considered independently. Coupled with a number of other combinatorial observations, this leads to polynomial time approximation schemes for both of the aforementioned objectives.

We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in experiments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their $\ell$-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram., Comment: To appear in NIPS 2018

This paper studies the problem of passive grasp stability under an external disturbance, that is, the ability of a grasp to resist a disturbance through passive responses at the contacts. To obtain physically consistent results, such a model must account for friction phenomena at each contact; the difficulty is that friction forces depend in non-linear fashion on contact behavior (stick or slip). We develop the first polynomial-time algorithm which either solves such complex equilibrium constraints for two-dimensional grasps, or otherwise concludes that no solution exists. To achieve this, we show that the number of possible `slip states' (where each contact is labeled as either sticking or slipping) that must be considered is polynomial (in fact quadratic) in the number of contacts, and not exponential as previously thought. Our algorithm captures passive response behaviors at each contact, while accounting for constraints on friction forces such as the maximum dissipation principle., Comment: Robotics: Science and Systems June 26-30, 2018 (9 pages, 7 figures)

Price of anarchy quantifies the degradation of social welfare in games due to the lack of a centralized authority that can enforce the optimal outcome. At its antipodes, mechanism design studies how to ameliorate these effects by incentivizing socially desirable behavior and implementing the optimal state as equilibrium. In practice, the responsiveness to such measures depends on the wealth of each individual. This leads to a natural, but largely unexplored, question. Does optimal mechanism design entrench, or maybe even exacerbate, social inequality? We study this question in nonatomic congestion games, arguably one of the most thoroughly studied settings from the perspectives of price of anarchy as well as mechanism design. We introduce a new model that incorporates the wealth distribution of the population and captures the income elasticity of travel time. This allows us to argue about the equality of wealth distribution both before and after employing a mechanism. We start our analysis by establishing a broad qualitative result, showing that tolls always increase inequality in symmetric congestion games under any reasonable metric of inequality, e.g., the Gini index. Next, we introduce the iniquity index, a novel measure for quantifying the magnitude of these forces towards a more unbalanced wealth distribution and show it has good normative properties (robustness to scaling of income, no-regret learning). We analyze iniquity both in theoretical settings (Pigou's network under various wealth distributions) as well as experimental ones (based on a large scale field experiment in Singapore). Finally, we provide an algorithm for computing optimal tolls for any point of the trade-off of relative importance of efficiency and equality. We conclude with a discussion of our findings in the context of theories of justice as developed in contemporary social sciences., Comment: 32 pages, 16 figures

Regularized learning is a fundamental technique in online optimization, machine learning and many other fields of computer science. A natural question that arises in these settings is how regularized learning algorithms behave when faced against each other. We study a natural formulation of this problem by coupling regularized learning dynamics in zero-sum games. We show that the system's behavior is Poincar\'e recurrent, implying that almost every trajectory revisits any (arbitrarily small) neighborhood of its starting point infinitely often. This cycling behavior is robust to the agents' choice of regularization mechanism (each agent could be using a different regularizer), to positive-affine transformations of the agents' utilities, and it also persists in the case of networked competition, i.e., for zero-sum polymatrix games., Comment: 22 pages, 4 figures