Nonlinear gradient mappings and stochastic optimization: A general framework with applications to heavy-tail noise

We introduce a general framework for nonlinear stochastic gradient descent (SGD) for the scenarios when gradient noise exhibits heavy tails. The proposed framework subsumes several popular nonlinearity choices, like clipped, normalized, signed or quantized gradient, but we also consider novel nonlinearity choices. We establish for the considered class of methods strong convergence guarantees assuming a strongly convex cost function with Lipschitz continuous gradients under very general assumptions on the gradient noise. Most notably, we show that, for a nonlinearity with bounded outputs and for the gradient noise that may not have finite moments of order greater than one, the nonlinear SGD's mean squared error (MSE), or equivalently, the expected cost function's optimality gap, converges to zero at rate~$O(1/t^\zeta)$, $\zeta \in (0,1)$. In contrast, for the same noise setting, the linear SGD generates a sequence with unbounded variances. Furthermore, for the nonlinearities that can be decoupled component wise, like, e.g., sign gradient or component-wise clipping, we show that the nonlinear SGD asymptotically (locally) achieves a $O(1/t)$ rate in the weak convergence sense and explicitly quantify the corresponding asymptotic variance. Experiments show that, while our framework is more general than existing studies of SGD under heavy-tail noise, several easy-to-implement nonlinearities from our framework are competitive with state of the art alternatives on real data sets with heavy tail noises.

Choosing the Best of Both Worlds: Diverse and Novel Recommendations through Multi-Objective Reinforcement Learning

Since the inception of Recommender Systems (RS), the accuracy of the recommendations in terms of relevance has been the golden criterion for evaluating the quality of RS algorithms. However, by focusing on item relevance, one pays a significant price in terms of other important metrics: users get stuck in a "filter bubble" and their array of options is significantly reduced, hence degrading the quality of the user experience and leading to churn. Recommendation, and in particular session-based/sequential recommendation, is a complex task with multiple - and often conflicting objectives - that existing state-of-the-art approaches fail to address. In this work, we take on the aforementioned challenge and introduce Scalarized Multi-Objective Reinforcement Learning (SMORL) for the RS setting, a novel Reinforcement Learning (RL) framework that can effectively address multi-objective recommendation tasks. The proposed SMORL agent augments standard recommendation models with additional RL layers that enforce it to simultaneously satisfy three principal objectives: accuracy, diversity, and novelty of recommendations. We integrate this framework with four state-of-the-art session-based recommendation models and compare it with a single-objective RL agent that only focuses on accuracy. Our experimental results on two real-world datasets reveal a substantial increase in aggregate diversity, a moderate increase in accuracy, reduced repetitiveness of recommendations, and demonstrate the importance of reinforcing diversity and novelty as complementary objectives.