Recent successes of value-based multi-agent deep reinforcement learning employ optimism by limiting underestimation updates of value function estimator, through carefully controlled learning rate (, 2017) or reduced update probability (, 2018). To achieve full cooperation when learning independently, an agent must estimate the state values contingent on having optimal teammates; therefore, value overestimation is frequency injected to counteract negative effects caused by unobservable teammate sub-optimal policies and explorations. Aiming to solve this issue through automatic scheduling, this paper introduces a decentralized quantile estimator, which we found empirically to be more stable, sample efficient and more likely to converge to the joint optimal policy.
The quantile regression loss @cite_23 for estimating quantile at @math and error @math is defined using Huber loss @math with threshold @math which weighs overestimation by @math and underestimation by @math , @math is used for linear loss.
@cite_23
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We present @math -loss, @math , a tunable loss function for binary classification that bridges log-loss ( @math ) and @math - @math loss ( @math ). We prove that @math -loss has an equivalent margin-based form and is classification-calibrated, two desirable properties for a good surrogate loss function for the ideal yet intractable @math - @math loss. For logistic regression-based classification, we provide an upper bound on the difference between the empirical and expected risk for @math -loss by exploiting its Lipschitzianity along with recent results on the landscape features of empirical risk functions. Finally, we show that @math -loss with @math performs better than log-loss on MNIST for logistic regression. We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determine a symmetric loss in full from half of one of its partial losses, introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and "classification calibrated" losses. We also consider the question of the "best" surrogate binary loss. We introduce a precise notion of "best" and show there exist situations where two convex surrogate losses are incommensurable. We provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss. This characterisation suggests new ways of "surrogate tuning" as well as providing an explicit characterisation of when Bregman divergences on the unit interval are convex in their second argument. Finally, in an appendix we present some new algorithm-independent results on the relationship between properness, convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non-robust to misclassification noise. A general approach for estimating an unknown signal x0 ∈ ℝn from noisy, linear measurements y = Ax0 + z ∈ ℝm is via solving a so called regularized M-estimator: x := arg minx ℒ(y−Ax)+λf(x). Here, ℒ is a convex loss function, f is a convex (typically, non-smooth) regularizer, and, λ > 0 a regularizer parameter. We analyze the squared error performance ∥x − x0∥22 of such estimators in the high-dimensional proportional regime where m, n → ∞ and m n → δ. We let the design matrix A have entries iid Gaussian, and, impose minimal and rather mild regularity conditions on the loss function, on the regularizer, and, on the distributions of the noise and of the unknown signal. Under such a generic setting, we show that the squared error converges in probability to a nontrivial limit that is computed by solving four nonlinear equations on four scalar unknowns. We identify a new summary parameter, termed the expected Moreau envelope, which determines how the choice of the loss function and of the regularizer affects the error performance. The result opens the way for answering optimality questions regarding the choice of the loss function, the regularizer, the penalty parameter, etc. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods from such measurements typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math within a Euclidean ball of known radius.