There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.

We experiment with 8 implementations of MoCaD, i.e. two different calibrators combined with four different ensembling strategies as the same as in previous experiments. For Learned-Mixin, the entropy term weight is set to the value suggested by [1]. We run each experiment five times and report the mean scores and the standard deviations. For the Dirichlet calibrator, we use the same configurationasinFEVER. Experimental Results Table 2 shows the experimental result on image classification.

From the three-points identity of the Bregman divergence (Lemma 3.1 of [9]), KL (x y) KL ( x y) = KL (x x) + ln x ln y,x x (12) The first term in (12) can be bounded by KL (x x) = By the Hölder's inequality, the second term in (12) is bounded as ln x ln y,x x ln x ln y Lemma 5. Consider a block diagonal matrix We prove the lemma via induction on N . This completes the induction proof.Lemma 6. We introduce one more notation before presenting the proof. This leads us to the initialization-dependent convergence rate of Algorithm 1, which we re-state and prove as follows. In addition, if we initialize the players' policies to be uniform policies, i.e., The rest of the proof follows by putting all the aforementioned results together.

The following lemma is standard in the literature, see e.g.

Part of this work was done while interning at Apple.