Regarding "plots are noisy and don't really support well the claim that the algorithm recovers the true Check the sharp jump in Figure 2 which is expected based on Theorem 3. Similarly, Figure 3 shows that Markov blanket can be recovered with sufficient number of observational data. NP-hard [Chickering, 1996, Learning Bayesian Networks Is NP-Complete]. Rank-2 is only used for clarity. Reviewer 2 has asked to present a case where Assumption 4 is violated. Assume that every variable can take 4 values.

We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$ uniformly random examples of an unknown function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$, our algorithm outputs a hypothesis $g:\{\pm 1\}^n \rightarrow \{\pm 1\}$ that is monotone and $(\mathrm{opt} + \varepsilon)$-close to $f$, where $\mathrm{opt}$ is the distance from $f$ to the closest monotone function. The running time of the algorithm (and consequently the size and evaluation time of the hypothesis) is also $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$, nearly matching the lower bound of Blais et al (RANDOM '15). We also give an algorithm for estimating up to additive error $\varepsilon$ the distance of an unknown function $f$ to monotone using a run-time of $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$. Previously, for both of these problems, sample-efficient algorithms were known, but these algorithms were not run-time efficient. Our work thus closes this gap in our knowledge between the run-time and sample complexity. This work builds upon the improper learning algorithm of Bshouty and Tamon (JACM '96) and the proper semiagnostic learning algorithm of Lange, Rubinfeld, and Vasilyan (FOCS '22), which obtains a non-monotone Boolean-valued hypothesis, then ``corrects'' it to monotone using query-efficient local computation algorithms on graphs. This black-box correction approach can achieve no error better than $2\mathrm{opt} + \varepsilon$ information-theoretically; we bypass this barrier by a) augmenting the improper learner with a convex optimization step, and b) learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the ``poset sorting'' problem of [LRV22] for functions over general posets with non-Boolean labels.

Foundation models like chatGPT have demonstrated remarkable performance on various tasks. However, for many questions, they may produce false answers that look accurate. How do we train the model to precisely understand the concepts? In this paper, we introduce succinct representations of concepts based on category theory. Such representation yields concept-wise invariance properties under various tasks, resulting a new learning algorithm that can provably and accurately learn complex concepts or fix misconceptions. Moreover, by recursively expanding the succinct representations, one can generate a hierarchical decomposition, and manually verify the concept by individually examining each part inside the decomposition.

The computational study of elections generally assumes that the preferences of the electorate come in as a list of votes. Depending on the context, it may be much more natural to represent the preferences of the electorate succinctly, as the distinct votes and their counts. Though the succinct representation may be exponentially smaller than the nonsuccinct, we find only one natural case where the complexity increases, in sharp contrast to the case where each voter has a weight, where the complexity usually increases.

We examine the computational complexity of testing and finding small plans in probabilistic planning domains with succinct representations. We find that many problems of interest are complete for a variety of complexity classes: NP, co-NP, PP, NP^PP, co-NP^PP, and PSPACE. Of these, the probabilistic classes PP and NP^PP are likely to be of special interest in the field of uncertainty in artificial intelligence and are deserving of additional study. These results suggest a fruitful direction of future algorithmic development.

Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.

Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.

We establish a new hardness result that shows that the difficulty of planning in factored Markov decision processes is representational rather than just computational. More precisely, we give a fixed family of factored MDPs with linear rewards whose optimal policies and value functions simply cannot be represented succinctly in any standard parametric form. Previous hardness results indicated that computing good policies from the MDP parameters was difficult, but left open the possibility of succinct function approximation for any fixed factored MDP. Our result applies even to policies which yield a polynomially poor approximation to the optimal value, and highlights interesting connections with the complexity class of Arthur-Merlin games.

We establish a new hardness result that shows that the difficulty of planning in factored Markov decision processes is representational rather than just computational. More precisely, we give a fixed family of factored MDPs with linear rewards whose optimal policies and value functions simply cannot be represented succinctly in any standard parametric form. Previous hardness results indicated that computing good policies from the MDP parameters was difficult, but left open the possibility of succinct function approximation for any fixed factored MDP. Our result applies even to policies which yield a polynomially poor approximation to the optimal value, and highlights interesting connections with the complexity class of Arthur-Merlin games.