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1. Exponential Lower Bounds on the Double Oracle Algorithm in Zero-Sum Games

   https://doi.org/10.24963/ijcai.2024/336  

   Zhang, Brian Hu; Sandholm, Tuomas (August 2024, International Joint Conferences on Artificial Intelligence Organization)

   The double oracle algorithm is a popular method of solving games, because it is able to reduce computing equilibria to computing a series of best responses. However, its theoretical properties are not well understood. In this paper, we provide exponential lower bounds on the performance of the double oracle algorithm in both partially-observable stochastic games (POSGs) and extensive-form games (EFGs). Our results depend on what is assumed about the tiebreaking scheme---that is, which meta-Nash equilibrium or best response is chosen, in the event that there are multiple to pick from. In particular, for EFGs, our lower bounds require adversarial tiebreaking, whereas for POSGs, our lower bounds apply regardless of how ties are broken. 

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2. XDO: A Double Oracle Algorithm for Extensive-Form Games

   McAleer, Stephen; Lanier, JB; Wang, Kevin; Baldi, Pierre; Fox, Roy (January 2021, Advances in neural information processing systems)

   Policy Space Response Oracles (PSRO) is a reinforcement learning (RL) algo- rithm for two-player zero-sum games that has been empirically shown to find approximate Nash equilibria in large games. Although PSRO is guaranteed to converge to an approximate Nash equilibrium and can handle continuous actions, it may take an exponential number of iterations as the number of information states (infostates) grows. We propose Extensive-Form Double Oracle (XDO), an extensive-form double oracle algorithm for two-player zero-sum games that is guar- anteed to converge to an approximate Nash equilibrium linearly in the number of infostates. Unlike PSRO, which mixes best responses at the root of the game, XDO mixes best responses at every infostate. We also introduce Neural XDO (NXDO), where the best response is learned through deep RL. In tabular experiments on Leduc poker, we find that XDO achieves an approximate Nash equilibrium in a number of iterations an order of magnitude smaller than PSRO. Experiments on a modified Leduc poker game and Oshi-Zumo show that tabular XDO achieves a lower exploitability than CFR with the same amount of computation. We also find that NXDO outperforms PSRO and NFSP on a sequential multidimensional continuous-action game. NXDO is the first deep RL method that can find an approximate Nash equilibrium in high-dimensional continuous-action sequential games. Experiment code is available at https://github.com/indylab/nxdo. 

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3. Cumulative Memory Lower Bounds for Randomized and Quantum Computation

   Beame, Paul; Kornerup, Niels (June 2023, arXivorg)

   Cumulative memory – the sum of space used per step over the duration of a computation – is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly(n) requires cumulative memory Ω(n^2), any classical matrix multiplication algorithm requires cumulative memory Ω(n^6/T), any quantum sorting circuit requires cumulative memory Ω(n^3/T), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω(k^3 n/T^2). (Full version of ICALP 2023 paper.) 

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4. Cumulative Memory Lower Bounds for Randomized and Quantum Computation

   Beame, Paul; Kornerup, Niels (June 2023, Electronic colloquium on computational complexity)

   Cumulative memory---the sum of space used per step over the duration of a computation---is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/\poly(n) requires cumulative memory \Omega(n^2), any classical matrix multiplication algorithm requires cumulative memory \Omega(n^6/T) , any quantum sorting circuit requires cumulative memory \Omega(n^3/T) , and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory \Omega(k^ 3 n/T^2) . 

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5. Cumulative Memory Lower Bounds for Randomized and Quantum Computation

   Beame, Paul; Kornerup, Niels (July 2023, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Cumulative memory – the sum of space used per step over the duration of a computation – is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly(n) requires cumulative memory Ω(n^2), any classical matrix multiplication algorithm requires cumulative memory Ω(n^6/T), any quantum sorting circuit requires cumulative memory Ω(n^3/T), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω(k^3 n/T^2). 

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