In this work, we consider data association problems involving multi-object tracking (MOT). In particular, we address the challenges arising from object occlusions. We propose a framework called approximate dynamic programming track (ADPTrack), which applies dynamic programming principles to improve an existing method called the base heuristic. Given a set of tracks and the next target frame, the base heuristic extends the tracks by matching them to the objects of this target frame directly. In contrast, ADPTrack first processes a few subsequent frames and applies the base heuristic starting from the next target frame to obtain tentative tracks. It then leverages the tentative tracks to match the objects of the target frame. This tends to reduce the occlusion-based errors and leads to an improvement over the base heuristic. When tested on the MOT17 video dataset, the proposed method demonstrates a 0.7% improvement in the association accuracy (IDF1 metric) over a state-of-the-art method that is used as the base heuristic. It also obtains improvements with respect to all the other standard metrics. Empirically, we found that the improvements are particularly pronounced in scenarios where the video data is obtained by fixed-position cameras.

In this paper we consider a transformer with an $n$-gram structure, such as the one underlying ChatGPT. The transformer provides next word probabilities, which can be used to generate word sequences. We consider methods for computing word sequences that are highly likely, based on these probabilities. Computing the optimal (i.e., most likely) word sequence starting with a given initial state is an intractable problem, so we propose methods to compute highly likely sequences of $N$ words in time that is a low order polynomial in $N$ and in the vocabulary size of the $n$-gram. These methods are based on the rollout approach from approximate dynamic programming, a form of single policy iteration, which can improve the performance of any given heuristic policy. In our case we use a greedy heuristic that generates as next word one that has the highest probability. We show with analysis, examples, and computational experimentation that our methods are capable of generating highly likely sequences with a modest increase in computation over the greedy heuristic. While our analysis and experiments are focused on Markov chains of the type arising in transformer and ChatGPT-like models, our methods apply to general finite-state Markov chains, and related inference applications of Hidden Markov Models (HMM), where Viterbi decoding is used extensively.

In this paper, we focus on the autonomous multiagent taxi routing problem for a large urban environment where the location and number of future ride requests are unknown a-priori, but follow an estimated empirical distribution. Recent theory has shown that if a base policy is stable then a rollout-based algorithm with such a base policy produces a near-optimal stable policy. Although, rollout-based approaches are well-suited for learning cooperative multiagent policies with considerations for future demand, applying such methods to a large urban environment can be computationally expensive. Large environments tend to have a large volume of requests, and hence require a large fleet of taxis to guarantee stability. In this paper, we aim to address the computational bottleneck of multiagent (one-at-a-time) rollout, where the computational complexity grows linearly in the number of agents. We propose an approximate one-at-a-time rollout-based two-phase algorithm that reduces the computational cost, while still achieving a stable near-optimal policy. Our approach partitions the graph into sectors based on the predicted demand and an user-defined maximum number of agents that can be planned for using the one-at-a-time rollout approach. The algorithm then applies instantaneous assignment (IA) for re-balancing taxis across sectors and a sector-wide one-at-a-time rollout algorithm that is executed in parallel for each sector. We characterize the number of taxis $m$ that is sufficient for IA base policy to be stable, and derive a necessary condition on $m$ as time goes to infinity. Our numerical results show that our approach achieves stability for an $m$ that satisfies the theoretical conditions. We also empirically demonstrate that our proposed two-phase algorithm has comparable performance to the one-at-a-time rollout over the entire map, but with significantly lower runtimes.

We derive a learning framework to generate routing/pickup policies for a fleet of autonomous vehicles tasked with servicing stochastically appearing requests on a city map. We focus on policies that 1) give rise to coordination amongst the vehicles, thereby reducing wait times for servicing requests, 2) are non-myopic, and consider a-priori potential future requests, 3) can adapt to changes in the underlying demand distribution. Specifically, we are interested in policies that are adaptive to fluctuations of actual demand conditions in urban environments, such as on-peak vs. off-peak hours. We achieve this through a combination of (i) an online play algorithm that improves the performance of an offline-trained policy, and (ii) an offline approximation scheme that allows for adapting to changes in the underlying demand model. In particular, we achieve adaptivity of our learned policy to different demand distributions by quantifying a region of validity using the q-valid radius of a Wasserstein Ambiguity Set. We propose a mechanism for switching the originally trained offline approximation when the current demand is outside the original validity region. In this case, we propose to use an offline architecture, trained on a historical demand model that is closer to the current demand in terms of Wasserstein distance. We learn routing and pickup policies over real taxicab requests in San Francisco with high variability between on-peak and off-peak hours, demonstrating the ability of our method to adapt to real fluctuation in demand distributions. Our numerical results demonstrate that our method outperforms alternative rollout-based reinforcement learning schemes, as well as other classical methods from operations research.

We provide a unifying approximate dynamic programming framework that applies to a broad variety of problems involving sequential estimation. We consider first the construction of surrogate cost functions for the purposes of optimization, and we focus on the special case of Bayesian optimization, using the rollout algorithm and some of its variations. We then discuss the more general case of sequential estimation of a random vector using optimal measurement selection, and its application to problems of stochastic and adaptive control. We distinguish between adaptive control of deterministic and stochastic systems: the former are better suited for the use of rollout, while the latter are well suited for the use of rollout with certainty equivalence approximations. As an example of the deterministic case, we discuss sequential decoding problems, and a rollout algorithm for the approximate solution of the Wordle and Mastermind puzzles, recently developed in the paper [BBB22].

In this paper, we discuss a Reinforcement Learning (RL) approach towards a class of sequential decision problems, exemplified for the popular Wordle puzzle that appears daily in the New York Times. Wordle involves a list of 5-letter mystery words, which is a subset of a larger list of guess words. A word is selected at random from the mystery list, and the objective is to find that word by sequentially selecting no more than six words from the guess list. Each guess word selection provides information about the letters contained in the hidden mystery word according to a given set of rules, which involves color coding of letters shared by the guess word and the mystery word. We will adopt a more general point of view, by considering a broad class of problems that include Wordle as a special case. In particular, the problems that we consider include sequential search situations, where the objective is to guess correctly an unknown object from a given finite set of objects (the set of mystery words in the Wordle context), by using a sequence of decisions from a finite set (the set of guess words in Wordle), which result in a sequence of corresponding observations (the information outcomes of the guesses in Wordle). We aim to minimize some cost function, such as the expected number of observations required to determine the unknown object. Within the search context just described, some basic information theory concepts are relevant, which have already been applied to Wordle, and are important for our methodology.

In this paper we aim to provide analysis and insights (often based on visualization), which explain the beneficial effects of on-line decision making on top of off-line training. In particular, through a unifying abstract mathematical framework, we show that the principal AlphaZero/TD-Gammon ideas of approximation in value space and rollout apply very broadly to deterministic and stochastic optimal control problems, involving both discrete and continuous search spaces. Moreover, these ideas can be effectively integrated with other important methodologies such as model predictive control, adaptive control, decentralized control, discrete and Bayesian optimization, neural network-based value and policy approximations, and heuristic algorithms for discrete optimization.

Dimitri Bertsekas† Abstract In this paper we propose an on-line policy iteration (PI) algorithm for finite-state infinite horizon discounted dynamic programming, whereby the policy improvement operation is done on-line, only for the states that are encountered during operation of the system. This allows the continuous updating/improvement of the current policy, thus resulting in a form of on-line PI that incorporates the improved controls into the current policy as new states and controls are generated. The algorithm converges in a finite number of stages to a type of locally optimal policy, and suggests the possibility of variants of PI and multiagent PI where the policy improvement is simplified. Moreover, the algorithm can be used with on-line replanning, and is also well-suited for on-line PI algorithms with value and policy approximations. The common characteristic of these variants is that, in addition to being suitable for on-line implementation, they are simplified in two ways: (a) They perform policy improvement operations only for the states that are encountered during the on-line operation of the system.

In this paper we consider infinite horizon discounted dynamic programming problems with finite state and control spaces, partial state observations, and a multiagent structure. We discuss and compare algorithms that simultaneously or sequentially optimize the agents' controls by using multistep lookahead, truncated rollout with a known base policy, and a terminal cost function approximation. Our methods specifically address the computational challenges of partially observable multiagent problems. In particular: 1) We consider rollout algorithms that dramatically reduce required computation while preserving the key cost improvement property of the standard rollout method. The per-step computational requirements for our methods are on the order of $O(Cm)$ as compared with $O(C^m)$ for standard rollout, where $C$ is the maximum cardinality of the constraint set for the control component of each agent, and $m$ is the number of agents. 2) We show that our methods can be applied to challenging problems with a graph structure, including a class of robot repair problems whereby multiple robots collaboratively inspect and repair a system under partial information. 3) We provide a simulation study that compares our methods with existing methods, and demonstrate that our methods can handle larger and more complex partially observable multiagent problems (state space size $10^{37}$ and control space size $10^{7}$, respectively). Finally, we incorporate our multiagent rollout algorithms as building blocks in an approximate policy iteration scheme, where successive rollout policies are approximated by using neural network classifiers. While this scheme requires a strictly off-line implementation, it works well in our computational experiments and produces additional significant performance improvement over the single online rollout iteration method.

We consider infinite horizon dynamic programming problems, where the control at each stage consists of several distinct decisions, each one made by one of several agents. In an earlier work we introduced a policy iteration algorithm, where the policy improvement is done one-agent-at-a-time in a given order, with knowledge of the choices of the preceding agents in the order. As a result, the amount of computation for each policy improvement grows linearly with the number of agents, as opposed to exponentially for the standard all-agents-at-once method. For the case of a finite-state discounted problem, we showed convergence to an agent-by-agent optimal policy. In this paper, this result is extended to value iteration and optimistic versions of policy iteration, as well as to more general DP problems where the Bellman operator is a contraction mapping, such as stochastic shortest path problems with all policies being proper.