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While the maximum entropy (MaxEnt) reinforcement learning (RL) framework -- often touted for its exploration and robustness capabilities -- is usually motivated from a probabilistic perspective, the use of deep probabilistic models has not gained much traction in practice due to their inherent complexity. In this work, we propose the adoption of latent variable policies within the MaxEnt framework, which we show can provably approximate any policy distribution, and additionally, naturally emerges under the use of world models with a latent belief state. We discuss why latent variable policies are difficult to train, how naive approaches can fail, then subsequently introduce a series of improvements centered around low-cost marginalization of the latent state, allowing us to make full use of the latent state at minimal additional cost. We instantiate our method under the actor-critic framework, marginalizing both the actor and critic. The resulting algorithm, referred to as Stochastic Marginal Actor-Critic (SMAC), is simple yet effective. We experimentally validate our method on continuous control tasks, showing that effective marginalization can lead to better exploration and more robust training. Our implementation is open sourced at //github.com/zdhNarsil/Stochastic-Marginal-Actor-Critic.

Sampling from an unnormalized target distribution is an essential problem with many applications in probabilistic inference. Stein Variational Gradient Descent (SVGD) has been shown to be a powerful method that iteratively updates a set of particles to approximate the distribution of interest. Furthermore, when analysing its asymptotic properties, SVGD reduces exactly to a single-objective optimization problem and can be viewed as a probabilistic version of this single-objective optimization problem. A natural question then arises: "Can we derive a probabilistic version of the multi-objective optimization?". To answer this question, we propose Stochastic Multiple Target Sampling Gradient Descent (MT-SGD), enabling us to sample from multiple unnormalized target distributions. Specifically, our MT-SGD conducts a flow of intermediate distributions gradually orienting to multiple target distributions, which allows the sampled particles to move to the joint high-likelihood region of the target distributions. Interestingly, the asymptotic analysis shows that our approach reduces exactly to the multiple-gradient descent algorithm for multi-objective optimization, as expected. Finally, we conduct comprehensive experiments to demonstrate the merit of our approach to multi-task learning.

Suppose we are given access to $n$ independent samples from distribution $\mu$ and we wish to output one of them with the goal of making the output distributed as close as possible to a target distribution $\nu$. In this work we show that the optimal total variation distance as a function of $n$ is given by $\tilde\Theta(\frac{D}{f'(n)})$ over the class of all pairs $\nu,\mu$ with a bounded $f$-divergence $D_f(\nu\|\mu)\leq D$. Previously, this question was studied only for the case when the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ is uniformly bounded. We then consider an application in the seemingly very different field of smoothed online learning, where we show that recent results on the minimax regret and the regret of oracle-efficient algorithms still hold even under relaxed constraints on the adversary (to have bounded $f$-divergence, as opposed to bounded Radon-Nikodym derivative). Finally, we also study efficacy of importance sampling for mean estimates uniform over a function class and compare importance sampling with rejection sampling.

We consider the differentially private estimation of multiple quantiles (MQ) of a distribution from a dataset, a key building block in modern data analysis. We apply the recent non-smoothed Inverse Sensitivity (IS) mechanism to this specific problem. We establish that the resulting method is closely related to the recently published ad hoc algorithm JointExp. In particular, they share the same computational complexity and a similar efficiency. We prove the statistical consistency of these two algorithms for continuous distributions. Furthermore, we demonstrate both theoretically and empirically that this method suffers from an important lack of performance in the case of peaked distributions, which can degrade up to a potentially catastrophic impact in the presence of atoms. Its smoothed version (i.e. by applying a max kernel to its output density) would solve this problem, but remains an open challenge to implement. As a proxy, we propose a simple and numerically efficient method called Heuristically Smoothed JointExp (HSJointExp), which is endowed with performance guarantees for a broad class of distributions and achieves results that are orders of magnitude better on problematic datasets.

A crucial assumption underlying the most current theory of machine learning is that the training distribution is identical to the testing distribution. However, this assumption may not hold in some real-world applications. In this paper, we propose an importance sampling based data variation robust loss (ISloss) for learning problems which minimizes the worst case of loss under the constraint of distribution deviation. The distribution deviation constraint can be converted to the constraint over a set of weight distributions centered on the uniform distribution derived from the importance sampling method. Furthermore, we reveal that there is a relationship between ISloss under the logarithmic transformation (LogISloss) and the p-norm loss. We apply the proposed LogISloss to the face verification problem on Racial Faces in the Wild dataset and show that the proposed method is robust under large distribution deviations.

A current assumption of most clustering methods is that the training data and future data are taken from the same distribution. However, this assumption may not hold in some real-world scenarios. In this paper, we propose an importance sampling based deterministic annealing approach (ISDA) for clustering problems which minimizes the worst case of expected distortions under the constraint of distribution deviation. The distribution deviation constraint can be converted to the constraint over a set of weight distributions centered on the uniform distribution derived from importance sampling. The objective of the proposed approach is to minimize the loss under maximum degradation hence the resulting problem is a constrained minimax optimization problem which can be reformulated to an unconstrained problem using the Lagrange method and be solved by the quasi-newton algorithm. Experiment results on synthetic datasets and a real-world load forecasting problem validate the effectiveness of the proposed ISDA. Furthermore, we show that fuzzy c-means is a special case of ISDA with the logarithmic distortion. This observation sheds a new light on the relationship between fuzzy c-means and deterministic annealing clustering algorithms and provides an interesting physical and information-theoretical interpretation for fuzzy exponent $m$.

Nearly all practical neural models for classification are trained using cross-entropy loss. Yet this ubiquitous choice is supported by little theoretical or empirical evidence. Recent work (Hui & Belkin, 2020) suggests that training using the (rescaled) square loss is often superior in terms of the classification accuracy. In this paper we propose the "squentropy" loss, which is the sum of two terms: the cross-entropy loss and the average square loss over the incorrect classes. We provide an extensive set of experiments on multi-class classification problems showing that the squentropy loss outperforms both the pure cross entropy and rescaled square losses in terms of the classification accuracy. We also demonstrate that it provides significantly better model calibration than either of these alternative losses and, furthermore, has less variance with respect to the random initialization. Additionally, in contrast to the square loss, squentropy loss can typically be trained using exactly the same optimization parameters, including the learning rate, as the standard cross-entropy loss, making it a true "plug-and-play" replacement. Finally, unlike the rescaled square loss, multiclass squentropy contains no parameters that need to be adjusted.

The nonconvex formulation of matrix completion problem has received significant attention in recent years due to its affordable complexity compared to the convex formulation. Gradient descent (GD) is the simplest yet efficient baseline algorithm for solving nonconvex optimization problems. The success of GD has been witnessed in many different problems in both theory and practice when it is combined with random initialization. However, previous works on matrix completion require either careful initialization or regularizers to prove the convergence of GD. In this work, we study the rank-1 symmetric matrix completion and prove that GD converges to the ground truth when small random initialization is used. We show that in logarithmic amount of iterations, the trajectory enters the region where local convergence occurs. We provide an upper bound on the initialization size that is sufficient to guarantee the convergence and show that a larger initialization can be used as more samples are available. We observe that implicit regularization effect of GD plays a critical role in the analysis, and for the entire trajectory, it prevents each entry from becoming much larger than the others.

Goal-conditioned reinforcement learning (GCRL) refers to learning general-purpose skills which aim to reach diverse goals. In particular, offline GCRL only requires purely pre-collected datasets to perform training tasks without additional interactions with the environment. Although offline GCRL has become increasingly prevalent and many previous works have demonstrated its empirical success, the theoretical understanding of efficient offline GCRL algorithms is not well established, especially when the state space is huge and the offline dataset only covers the policy we aim to learn. In this paper, we propose a novel provably efficient algorithm (the sample complexity is $\tilde{O}({\rm poly}(1/\epsilon))$ where $\epsilon$ is the desired suboptimality of the learned policy) with general function approximation. Our algorithm only requires nearly minimal assumptions of the dataset (single-policy concentrability) and the function class (realizability). Moreover, our algorithm consists of two uninterleaved optimization steps, which we refer to as $V$-learning and policy learning, and is computationally stable since it does not involve minimax optimization. To the best of our knowledge, this is the first algorithm with general function approximation and single-policy concentrability that is both statistically efficient and computationally stable.

The Q-learning algorithm is known to be affected by the maximization bias, i.e. the systematic overestimation of action values, an important issue that has recently received renewed attention. Double Q-learning has been proposed as an efficient algorithm to mitigate this bias. However, this comes at the price of an underestimation of action values, in addition to increased memory requirements and a slower convergence. In this paper, we introduce a new way to address the maximization bias in the form of a "self-correcting algorithm" for approximating the maximum of an expected value. Our method balances the overestimation of the single estimator used in conventional Q-learning and the underestimation of the double estimator used in Double Q-learning. Applying this strategy to Q-learning results in Self-correcting Q-learning. We show theoretically that this new algorithm enjoys the same convergence guarantees as Q-learning while being more accurate. Empirically, it performs better than Double Q-learning in domains with rewards of high variance, and it even attains faster convergence than Q-learning in domains with rewards of zero or low variance. These advantages transfer to a Deep Q Network implementation that we call Self-correcting DQN and which outperforms regular DQN and Double DQN on several tasks in the Atari 2600 domain.