arXiv subjects

Rahul Savani

Publications and source records attributed to Rahul Savani.

At least 37 records · Page 2Linked to original sources

arXiv2023

The Complexity of Gradient Descent: CLS = PPAD $\cap$ PLS

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain $[0,1]^2$ is PPAD $\cap$ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more "natural" counterpart to PPAD $\cap$ PLS and contains many interesting problems - is itself equal to PPAD $\cap$ PLS.

cs.CC↗

arXiv2022

Market Making with Scaled Beta Policies

This paper introduces a new representation for the actions of a market maker in an order-driven market. This representation uses scaled beta distributions, and generalises three approaches taken in the artificial intelligence for market making literature: single price-level selection, ladder strategies and "market making at the touch". Ladder strategies place uniform volume across an interval of contiguous prices. Scaled beta distribution based policies generalise these, allowing volume to be skewed across the price interval. We demonstrate that this flexibility is useful for inventory management, one of the key challenges faced by a market maker. In this paper, we conduct three main experiments: first, we compare our more flexible beta-based actions with the special case of ladder strategies; then, we investigate the performance of simple fixed distributions; and finally, we devise and evaluate a simple and intuitive dynamic control policy that adjusts actions in a continuous manner depending on the signed inventory that the market maker has acquired. All empirical evaluations use a high-fidelity limit order book simulator based on historical data with 50 levels on each side.

q-fin.TR↗

arXiv2022

Model-based gym environments for limit order book trading

Within the mathematical finance literature there is a rich catalogue of mathematical models for studying algorithmic trading problems -- such as market-making and optimal execution -- in limit order books. This paper introduces \mbtgym, a Python module that provides a suite of gym environments for training reinforcement learning (RL) agents to solve such model-based trading problems. The module is set up in an extensible way to allow the combination of different aspects of different models. It supports highly efficient implementations of vectorized environments to allow faster training of RL agents. In this paper, we motivate the challenge of using RL to solve such model-based limit order book problems in mathematical finance, we explain the design of our gym environment, and then demonstrate its use in solving standard and non-standard problems from the literature. Finally, we lay out a roadmap for further development of our module, which we provide as an open source repository on GitHub so that it can serve as a focal point for RL research in model-based algorithmic trading.

q-fin.TR↗

arXiv2022

Consensus Multiplicative Weights Update: Learning to Learn using Projector-based Game Signatures

Cheung and Piliouras (2020) recently showed that two variants of the Multiplicative Weights Update method - OMWU and MWU - display opposite convergence properties depending on whether the game is zero-sum or cooperative. Inspired by this work and the recent literature on learning to optimize for single functions, we introduce a new framework for learning last-iterate convergence to Nash Equilibria in games, where the update rule's coefficients (learning rates) along a trajectory are learnt by a reinforcement learning policy that is conditioned on the nature of the game: \textit{the game signature}. We construct the latter using a new decomposition of two-player games into eight components corresponding to commutative projection operators, generalizing and unifying recent game concepts studied in the literature. We compare the performance of various update rules when their coefficients are learnt, and show that the RL policy is able to exploit the game signature across a wide range of game types. In doing so, we introduce CMWU, a new algorithm that extends consensus optimization to the constrained case, has local convergence guarantees for zero-sum bimatrix games, and show that it enjoys competitive performance on both zero-sum games with constant coefficients and across a spectrum of games when its coefficients are learnt.

cs.GT↗

arXiv2022

Sample-based Approximation of Nash in Large Many-Player Games via Gradient Descent

Nash equilibrium is a central concept in game theory. Several Nash solvers exist, yet none scale to normal-form games with many actions and many players, especially those with payoff tensors too big to be stored in memory. In this work, we propose an approach that iteratively improves an approximation to a Nash equilibrium through joint play. It accomplishes this by tracing a previously established homotopy that defines a continuum of equilibria for the game regularized with decaying levels of entropy. This continuum asymptotically approaches the limiting logit equilibrium, proven by McKelvey and Palfrey (1995) to be unique in almost all games, thereby partially circumventing the well-known equilibrium selection problem of many-player games. To encourage iterates to remain near this path, we efficiently minimize average deviation incentive via stochastic gradient descent, intelligently sampling entries in the payoff tensor as needed. Monte Carlo estimates of the stochastic gradient from joint play are biased due to the appearance of a nonlinear max operator in the objective, so we introduce additional innovations to the algorithm to alleviate gradient bias. The descent process can also be viewed as repeatedly constructing and reacting to a polymatrix approximation to the game. In these ways, our proposed approach, average deviation incentive descent with adaptive sampling (ADIDAS), is most similar to three classical approaches, namely homotopy-type, Lyapunov, and iterative polymatrix solvers. The lack of local convergence guarantees for biased gradient descent prevents guaranteed convergence to Nash, however, we demonstrate through extensive experiments the ability of this approach to approximate a unique Nash in normal-form games with as many as seven players and twenty one actions (several billion outcomes) that are orders of magnitude larger than those possible with prior algorithms.

cs.GT↗

arXiv2021

Trading via Selective Classification

A binary classifier that tries to predict if the price of an asset will increase or decrease naturally gives rise to a trading strategy that follows the prediction and thus always has a position in the market. Selective classification extends a binary or many-class classifier to allow it to abstain from making a prediction for certain inputs, thereby allowing a trade-off between the accuracy of the resulting selective classifier against coverage of the input feature space. Selective classifiers give rise to trading strategies that do not take a trading position when the classifier abstains. We investigate the application of binary and ternary selective classification to trading strategy design. For ternary classification, in addition to classes for the price going up or down, we include a third class that corresponds to relatively small price moves in either direction, and gives the classifier another way to avoid making a directional prediction. We use a walk-forward train-validate-test approach to evaluate and compare binary and ternary, selective and non-selective classifiers across several different feature sets based on four classification approaches: logistic regression, random forests, feed-forward, and recurrent neural networks. We then turn these classifiers into trading strategies for which we perform backtests on commodity futures markets. Our empirical results demonstrate the potential of selective classification for trading.

q-fin.TR↗

arXiv2021

Reachability Switching Games

We study the problem of deciding the winner of reachability switching games for zero-, one-, and two-player variants. Switching games provide a deterministic analogue of stochastic games. We show that the zero-player case is NL-hard, the one-player case is NP-complete, and that the two-player case is PSPACE-hard and in EXPTIME. For the zero-player case, we also show P-hardness for a succinctly-represented model that maintains the upper bound of NP $\cap$ coNP. For the one- and two-player cases, our results hold in both the natural, explicit model and succinctly-represented model. Our results show that the switching variant of a game is harder in complexity-theoretic terms than the corresponding stochastic version.

cs.FL↗

arXiv2021

A faster algorithm for finding Tarski fixed points

Dang et al. have given an algorithm that can find a Tarski fixed point in a $k$-dimensional lattice of width $n$ using $O(\log^{k} n)$ queries. Multiple authors have conjectured that this algorithm is optimal [Dang et al., Etessami et al.], and indeed this has been proven for two-dimensional instances [Etessami et al.]. We show that these conjectures are false in dimension three or higher by giving an $O(\log^2 n)$ query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for $k$-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an $O(\log^{2 \lceil k/3 \rceil} n)$ query algorithm for the $k$-dimensional problem.

cs.DS↗

arXiv2021

A deep learning approach to identify unhealthy advertisements in street view images

While outdoor advertisements are common features within towns and cities, they may reinforce social inequalities in health. Vulnerable populations in deprived areas may have greater exposure to fast food, gambling and alcohol advertisements encouraging their consumption. Understanding who is exposed and evaluating potential policy restrictions requires a substantial manual data collection effort. To address this problem we develop a deep learning workflow to automatically extract and classify unhealthy advertisements from street-level images. We introduce the Liverpool 360 Street View (LIV360SV) dataset for evaluating our workflow. The dataset contains 25,349, 360 degree, street-level images collected via cycling with a GoPro Fusion camera, recorded Jan 14th - 18th 2020. 10,106 advertisements were identified and classified as food (1335), alcohol (217), gambling (149) and other (8405) (e.g., cars and broadband). We find evidence of social inequalities with a larger proportion of food advertisements located within deprived areas and those frequented by students. Our project presents a novel implementation for the incidental classification of street view images for identifying unhealthy advertisements, providing a means through which to identify areas that can benefit from tougher advertisement restriction policies for tackling social inequalities.

cs.CY↗

arXiv2020

Robust Market Making via Adversarial Reinforcement Learning

We show that adversarial reinforcement learning (ARL) can be used to produce market marking agents that are robust to adversarial and adaptively-chosen market conditions. To apply ARL, we turn the well-studied single-agent model of Avellaneda and Stoikov [2008] into a discrete-time zero-sum game between a market maker and adversary. The adversary acts as a proxy for other market participants that would like to profit at the market maker's expense. We empirically compare two conventional single-agent RL agents with ARL, and show that our ARL approach leads to: 1) the emergence of risk-averse behaviour without constraints or domain-specific penalties; 2) significant improvements in performance across a set of standard metrics, evaluated with or without an adversary in the test environment, and; 3) improved robustness to model uncertainty. We empirically demonstrate that our ARL method consistently converges, and we prove for several special cases that the profiles that we converge to correspond to Nash equilibria in a simplified single-stage game.

q-fin.TR↗

arXiv2020

A Natural Actor-Critic Algorithm with Downside Risk Constraints

Existing work on risk-sensitive reinforcement learning - both for symmetric and downside risk measures - has typically used direct Monte-Carlo estimation of policy gradients. While this approach yields unbiased gradient estimates, it also suffers from high variance and decreased sample efficiency compared to temporal-difference methods. In this paper, we study prediction and control with aversion to downside risk which we gauge by the lower partial moment of the return. We introduce a new Bellman equation that upper bounds the lower partial moment, circumventing its non-linearity. We prove that this proxy for the lower partial moment is a contraction, and provide intuition into the stability of the algorithm by variance decomposition. This allows sample-efficient, on-line estimation of partial moments. For risk-sensitive control, we instantiate Reward Constrained Policy Optimization, a recent actor-critic method for finding constrained policies, with our proxy for the lower partial moment. We extend the method to use natural policy gradients and demonstrate the effectiveness of our approach on three benchmark problems for risk-sensitive reinforcement learning.

cs.LG↗

arXiv2020

One-Clock Priced Timed Games are PSPACE-hard

The main result of this paper is that computing the value of a one-clock priced timed game (OCPTG) is PSPACE-hard. Along the way, we provide a family of OCPTGs that have an exponential number of event points. Both results hold even in very restricted classes of games such as DAGs with treewidth three. Finally, we provide a number of positive results, including polynomial-time algorithms for even more restricted classes of OCPTGs such as trees.

cs.GT↗

arXiv2020

The Automated Inspection of Opaque Liquid Vaccines

In the pharmaceutical industry the screening of opaque vaccines containing suspensions is currently a manual task carried out by trained human visual inspectors. We show that deep learning can be used to effectively automate this process. A moving contrast is required to distinguish anomalies from other particles, reflections and dust resting on a vial's surface. We train 3D-ConvNets to predict the likelihood of 20-frame video samples containing anomalies. Our unaugmented dataset consists of hand-labelled samples, recorded using vials provided by the HAL Allergy Group, a pharmaceutical company. We trained ten randomly initialized 3D-ConvNets to provide a benchmark, observing mean AUROC scores of 0.94 and 0.93 for positive samples (containing anomalies) and negative (anomaly-free) samples, respectively. Using Frame-Completion Generative Adversarial Networks we: (i) introduce an algorithm for computing saliency maps, which we use to verify that the 3D-ConvNets are indeed identifying anomalies; (ii) propose a novel self-training approach using the saliency maps to determine if multiple networks agree on the location of anomalies. Our self-training approach allows us to augment our data set by labelling 217,888 additional samples. 3D-ConvNets trained with our augmented dataset improve on the results we get when we train only on the unaugmented dataset.

cs.CV↗

arXiv2019

Negative Update Intervals in Deep Multi-Agent Reinforcement Learning

In Multi-Agent Reinforcement Learning (MA-RL), independent cooperative learners must overcome a number of pathologies to learn optimal joint policies. Addressing one pathology often leaves approaches vulnerable towards others. For instance, hysteretic Q-learning addresses miscoordination while leaving agents vulnerable towards misleading stochastic rewards. Other methods, such as leniency, have proven more robust when dealing with multiple pathologies simultaneously. However, leniency has predominately been studied within the context of strategic form games (bimatrix games) and fully observable Markov games consisting of a small number of probabilistic state transitions. This raises the question of whether these findings scale to more complex domains. For this purpose we implement a temporally extend version of the Climb Game, within which agents must overcome multiple pathologies simultaneously, including relative overgeneralisation, stochasticity, the alter-exploration and moving target problems, while learning from a large observation space. We find that existing lenient and hysteretic approaches fail to consistently learn near optimal joint-policies in this environment. To address these pathologies we introduce Negative Update Intervals-DDQN (NUI-DDQN), a Deep MA-RL algorithm which discards episodes yielding cumulative rewards outside the range of expanding intervals. NUI-DDQN consistently gravitates towards optimal joint-policies in our environment, overcoming the outlined pathologies.

cs.MA↗

arXiv2019

Evolving Indoor Navigational Strategies Using Gated Recurrent Units In NEAT

Simultaneous Localisation and Mapping (SLAM) algorithms are expensive to run on smaller robotic platforms such as Micro-Aerial Vehicles. Bug algorithms are an alternative that use relatively little processing power, and avoid high memory consumption by not building an explicit map of the environment. Bug Algorithms achieve relatively good performance in simulated and robotic maze solving domains. However, because they are hand-designed, a natural question is whether they are globally optimal control policies. In this work we explore the performance of Neuroevolution - specifically NEAT - at evolving control policies for simulated differential drive robots carrying out generalised maze navigation. We extend NEAT to include Gated Recurrent Units (GRUs) to help deal with long term dependencies. We show that both NEAT and our NEAT-GRU can repeatably generate controllers that outperform I-Bug (an algorithm particularly well-suited for use in real robots) on a test set of 209 indoor maze like environments. We show that NEAT-GRU is superior to NEAT in this task but also that out of the 2 systems, only NEAT-GRU can continuously evolve successful controllers for a much harder task in which no bearing information about the target is provided to the agent.

cs.NE↗

arXiv2018

Unique End of Potential Line

This paper studies the complexity of problems in PPAD $\cap$ PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions. We define the complexity class UEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to End-of-Potential-Line. This problem merges the canonical PPAD-complete problem End-of-Line, with the canonical PLS-complete problem Sink-of-Dag, and so EOPL captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function. Promise-UEOPL is a promise-subclass of EOPL in which the line in the End-of-Potential-Line instance is guaranteed to be unique via a promise. We turn this into a non-promise class UEOPL, by adding an extra solution type to EOPL that captures any pair of points that are provably on two different lines. We show that UEOPL $\subseteq$ EOPL $\subseteq$ CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an $\ell_p$-norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL. All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete.

cs.CC↗

arXiv2018

The Complexity of All-switches Strategy Improvement

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of Vöge and Jurdziński; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube.

cs.DS↗

arXiv2018

Beyond Local Nash Equilibria for Adversarial Networks

Save for some special cases, current training methods for Generative Adversarial Networks (GANs) are at best guaranteed to converge to a `local Nash equilibrium` (LNE). Such LNEs, however, can be arbitrarily far from an actual Nash equilibrium (NE), which implies that there are no guarantees on the quality of the found generator or classifier. This paper proposes to model GANs explicitly as finite games in mixed strategies, thereby ensuring that every LNE is an NE. With this formulation, we propose a solution method that is proven to monotonically converge to a resource-bounded Nash equilibrium (RB-NE): by increasing computational resources we can find better solutions. We empirically demonstrate that our method is less prone to typical GAN problems such as mode collapse, and produces solutions that are less exploitable than those produced by GANs and MGANs, and closely resemble theoretical predictions about NEs.

cs.LG↗