The key assumption underlying linear Markov Decision Processes (MDPs) is that the learner has access to a known feature map $\phi(x, a)$ that maps state-action pairs to $d$-dimensional vectors, and that the rewards and transitions are linear functions in this representation. But where do these features come from? In the absence of expert domain knowledge, a tempting strategy is to use the ``kitchen sink" approach and hope that the true features are included in a much larger set of potential features. In this paper we revisit linear MDPs from the perspective of feature selection. In a $k$-sparse linear MDP, there is an unknown subset $S \subset [d]$ of size $k$ containing all the relevant features, and the goal is to learn a near-optimal policy in only poly$(k,\log d)$ interactions with the environment. Our main result is the first polynomial-time algorithm for this problem. In contrast, earlier works either made prohibitively strong assumptions that obviated the need for exploration, or required solving computationally intractable optimization problems. Along the way we introduce the notion of an emulator: a succinct approximate representation of the transitions that suffices for computing certain Bellman backups. Since linear MDPs are a non-parametric model, it is not even obvious whether polynomial-sized emulators exist. We show that they do exist and can be computed efficiently via convex programming. As a corollary of our main result, we give an algorithm for learning a near-optimal policy in block MDPs whose decoding function is a low-depth decision tree; the algorithm runs in quasi-polynomial time and takes a polynomial number of samples. This can be seen as a reinforcement learning analogue of classic results in computational learning theory. Furthermore, it gives a natural model where improving the sample complexity via representation learning is computationally feasible.
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Recent experiments have shown that, often, when training a neural network with gradient descent (GD) with a step size $\eta$, the operator norm of the Hessian of the loss grows until it approximately reaches $2/\eta$, after which it fluctuates around this value. The quantity $2/\eta$ has been called the "edge of stability" based on consideration of a local quadratic approximation of the loss. We perform a similar calculation to arrive at an "edge of stability" for Sharpness-Aware Minimization (SAM), a variant of GD which has been shown to improve its generalization. Unlike the case for GD, the resulting SAM-edge depends on the norm of the gradient. Using three deep learning training tasks, we see empirically that SAM operates on the edge of stability identified by this analysis.
A range family $\mathcal{R}$ is a family of subsets of $\mathbb{R}^d$, like all halfplanes, or all unit disks. Given a range family $\mathcal{R}$, we consider the $m$-uniform range capturing hypergraphs $\mathcal{H}(V,\mathcal{R},m)$ whose vertex-sets $V$ are finite sets of points in $\mathbb{R}^d$ with any $m$ vertices forming a hyperedge $e$ whenever $e = V \cap R$ for some $R \in \mathcal{R}$. Given additionally an integer $k \geq 2$, we seek to find the minimum $m = m_{\mathcal{R}}(k)$ such that every $\mathcal{H}(V,\mathcal{R},m)$ admits a polychromatic $k$-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, $m_{\mathcal{R}}(k) \geq k$ and the gold standard is an upper bound $m_{\mathcal{R}}(k) = O(k)$ that is linear in $k$. A $t$-shallow hitting set in $\mathcal{H}(V,\mathcal{R},m)$ is a subset $S \subseteq V$ such that $1 \leq |e \cap S| \leq t$ for each hyperedge $e$; i.e., every hyperedge is hit at least once but at most $t$ times by $S$. We show for several range families $\mathcal{R}$ the existence of $t$-shallow hitting sets in every $\mathcal{H}(V,\mathcal{R},m)$ with $t$ being a constant only depending on $\mathcal{R}$. This in particular proves that $m_{\mathcal{R}}(k) \leq tk = O(k)$ in such cases, improving previous polynomial bounds in $k$. Particularly, we prove this for the range families of all axis-aligned strips in $\mathbb{R}^d$, all bottomless and topless rectangles in $\mathbb{R}^2$, and for all unit-height axis-aligned rectangles in $\mathbb{R}^2$.
We consider the classical Shiryaev--Roberts martingale diffusion, $(R_t)_{t\ge0}$, restricted to the interval $[0,A]$, where $A>0$ is a preset absorbing boundary. We take yet another look at the well-known phenomenon of quasi-stationarity (time-invariant probabilistic behavior, conditional on no absorbtion hitherto) exhibited by the diffusion in the temporal limit, as $t\to+\infty$, for each $A>0$. We obtain new upper- and lower-bounds for the quasi-stationary distribution's probability density function (pdf), $q_{A}(x)$; the bounds vary in the trade-off between simplicity and tightness. The bounds imply directly the expected result that $q_{A}(x)$ converges to the pdf, $h(x)$, of the diffusion's stationary distribution, as $A\to+\infty$; the convergence is pointwise, for all $x\ge0$. The bounds also yield an explicit upperbound for the gap between $q_{A}(x)$ and $h(x)$ for a fixed $x$. By virtue of integration the bounds for the pdf $q_{A}(x)$ translate into new bounds for the corresponding cumulative distribution function (cdf), $Q_{A}(x)$. All of our results are established explicitly, using certain latest monotonicity properties of the modified Bessel $K$ function involved in the exact closed-form formula for $q_{A}(x)$ recently obtained by Polunchenko (2017). We conclude with a discussion of potential applications of our results in quickest change-point detection: our bounds allow for a very accurate performance analysis of the so-called randomized Shiryaev--Roberts--Pollak change-point detection procedure.
We study the data complexity of consistent query answering (CQA) on databases that may violate the primary key constraints. A repair is a maximal subset of the database satisfying the primary key constraints. For a Boolean query q, the problem CERTAINTY(q) takes a database as input, and asks whether or not each repair satisfies q. The computational complexity of CERTAINTY(q) has been established whenever q is a self-join-free Boolean conjunctive query, or a (not necessarily self-join-free) Boolean path query. In this paper, we take one more step towards a general classification for all Boolean conjunctive queries by considering the class of rooted tree queries. In particular, we show that for every rooted tree query q, CERTAINTY(q) is in FO, NL-hard $\cap$ LFP, or coNP-complete, and it is decidable (in polynomial time), given q, which of the three cases applies. We also extend our classification to larger classes of queries with simple primary keys. Our classification criteria rely on query homomorphisms and our polynomial-time fixpoint algorithm is based on a novel use of context-free grammar (CFG).
Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network
Generative Flow Networks (GFlowNets), a class of generative models over discrete and structured sample spaces, have been previously applied to the problem of inferring the marginal posterior distribution over the directed acyclic graph (DAG) of a Bayesian Network, given a dataset of observations. Based on recent advances extending this framework to non-discrete sample spaces, we propose in this paper to approximate the joint posterior over not only the structure of a Bayesian Network, but also the parameters of its conditional probability distributions. We use a single GFlowNet whose sampling policy follows a two-phase process: the DAG is first generated sequentially one edge at a time, and then the corresponding parameters are picked once the full structure is known. Since the parameters are included in the posterior distribution, this leaves more flexibility for the local probability models of the Bayesian Network, making our approach applicable even to non-linear models parametrized by neural networks. We show that our method, called JSP-GFN, offers an accurate approximation of the joint posterior, while comparing favorably against existing methods on both simulated and real data.
We consider the performance of a least-squares regression model, as judged by out-of-sample $R^2$. Shapley values give a fair attribution of the performance of a model to its input features, taking into account interdependencies between features. Evaluating the Shapley values exactly requires solving a number of regression problems that is exponential in the number of features, so a Monte Carlo-type approximation is typically used. We focus on the special case of least-squares regression models, where several tricks can be used to compute and evaluate regression models efficiently. These tricks give a substantial speed up, allowing many more Monte Carlo samples to be evaluated, achieving better accuracy. We refer to our method as least-squares Shapley performance attribution (LS-SPA), and describe our open-source implementation.
Fr\'echet regression has received considerable attention to model metric-space valued responses that are complex and non-Euclidean data, such as probability distributions and vectors on the unit sphere. However, existing Fr\'echet regression literature focuses on the classical setting where the predictor dimension is fixed, and the sample size goes to infinity. This paper proposes sparse Fr\'echet sufficient dimension reduction with graphical structure among high-dimensional Euclidean predictors. In particular, we propose a convex optimization problem that leverages the graphical information among predictors and avoids inverting the high-dimensional covariance matrix. We also provide the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the optimization problem. Theoretically, the proposed method achieves subspace estimation and variable selection consistency under suitable conditions. Extensive simulations and a real data analysis are carried out to illustrate the finite-sample performance of the proposed method.
Some of the most successful knowledge graph embedding (KGE) models for link prediction -- CP, RESCAL, TuckER, ComplEx -- can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits -- constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design. Furthermore, our models scale more gracefully than the original KGEs on graphs with millions of entities.
Existing contrastive learning methods rely on pairwise sample contrast $z_x^\top z_{x'}$ to learn data representations, but the learned features often lack clear interpretability from a human perspective. Theoretically, it lacks feature identifiability and different initialization may lead to totally different features. In this paper, we study a new method named tri-factor contrastive learning (triCL) that involves a 3-factor contrast in the form of $z_x^\top S z_{x'}$, where $S=\text{diag}(s_1,\dots,s_k)$ is a learnable diagonal matrix that automatically captures the importance of each feature. We show that by this simple extension, triCL can not only obtain identifiable features that eliminate randomness but also obtain more interpretable features that are ordered according to the importance matrix $S$. We show that features with high importance have nice interpretability by capturing common classwise features, and obtain superior performance when evaluated for image retrieval using a few features. The proposed triCL objective is general and can be applied to different contrastive learning methods like SimCLR and CLIP. We believe that it is a better alternative to existing 2-factor contrastive learning by improving its identifiability and interpretability with minimal overhead. Code is available at //github.com/PKU-ML/Tri-factor-Contrastive-Learning.
Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + \rho \frac{\nabla f(x_t)}{\lVert \nabla f(x_t) \rVert}$ of the current point $x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $\rho$ and/or no gradient normalization in $y_t$, which is detached from practice. To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant $\rho$ and gradient normalization in $y_t$) and explore their convergence properties on smooth functions with (non)convexity assumptions. Perhaps surprisingly, in many scenarios, we find out that SAM has limited capability to converge to global minima or stationary points. For smooth strongly convex functions, we show that while deterministic SAM enjoys tight global convergence rates of $\tilde \Theta(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an inevitable additive term $O(\rho^2)$, indicating convergence only up to neighborhoods of optima. In fact, such $O(\rho^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases; importantly, we prove by examples that such terms are unavoidable. Our results highlight vastly different characteristics of SAM with vs. without decaying perturbation size or gradient normalization, and suggest that the intuitions gained from one version may not apply to the other.
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