WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES

By Kazuoki Azuma. 1967

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Citations in the corpus

Adaptive Gradient-Based Meta-Learning Methods (2019, methodology): “[14] only prove the first inequality; the second follows via the same argument but applying the symmetric version of the Azuma-Hoeffding inequality [4].”
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling (2010, positive): “To show that the statement holds with high probability when X is compact and‖ĝi(t)‖∗ ≤ L, it is sufficient to establish that the sequence 〈gi(t)− ĝi(t), xi(t)−x∗〉 is a bounded difference martingale and apply Azuma’s inequality [31].”
Solving variational inequalities with Stochastic Mirror-Prox algorithm (2008, background): “For Euclidean spaces E 1 , ..., E m , E = E 1 × ... × E m denotes their Euclidean direct product, so that a vector from E is a collection u = [ u 1 ; ... ; u m ] (“MATLAB notation”) of vectors u ℓ ∈ E ℓ , and u, v E = ℓ u ℓ , v ℓ E ℓ . instead h i Sometimes we allow ourselves to write ( u 1 , ...,…”
Escaping From Saddle Points - Online Stochastic Gradient for Tensor Decomposition (2015, methodology): “Azuma, K. (1967). Weighted sums of certain dependent random variables.Tohoku Mathematical Journal, Second Series , 19(3):357–367. Bengio, Y. (2009). Learning deep architectures for AI. Foundations and trends R © in Machine Learning, 2(1):1–127. Cardoso, J.-F. (1989). Source separation using higher o”
ESTIMATION OF NONLINEAR ERROR CORRECTION MODELS (2010, unknown): “An application of a Hoe¤ding type inequality for martingale di¤erence sequence (Azuma 1967) yields that for 1i 2 ni and some constant C1”
Random Graphs and Complex Networks (2016, methodology): “The Azuma-Hoeffding inequality was first proved by Azuma [18] and Hoeffding [135].”
Time-uniform, nonparametric, nonasymptotic confidence sequences (2018, methodology): “Where does Definition 1 come from? The jumping-off point is the martingale method for concentration inequalities ([4, 24, 51]; [54], Section 2.”
PAC-Bayesian Inequalities for Martingales (2011, background): “Background We first remind Hoeffding-Azuma inequality [1], [2].”
Relative deviation learning bounds and generalization with unbounded loss functions (2013, methodology): “unrelated to the statistical nature of the problem considered but it is convenient since when the loss functions are uniformly bounded, standard tools such as Hoeffding’s inequality (Hoeffding, 1963; Azuma, 1967), McDiarmid’s inequality (McDiarmid, 1989), or Talagrand’s concentration inequality (Tal”
Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces (2008, background): “, N , be Borel functions on Ω such that ψi is Fi-measurable, let α ∈ [1, 2], and let μi, νi > 0 be deterministic reals.”
Regret Testing: Learning to Play Nash Equilibrium Without Knowing You Have an Opponent (2006, background): “Since the payoffs lie between zero and one (see assumption (1)), it suffices that λ1, λ2 ≤ ε/4.”
Reward-Free Model-Based Reinforcement Learning with Linear Function Approximation (2021, unknown): “5 (Azuma-Hoeffding’s inequality, [4]).”
Probability on Trees and Networks (2017, background): “An extension (though not as sharp) of Chernoff-Cramér’s theorem is a large deviation inequality due to Hoeffding (1963) and rediscovered by Azuma (1967). As an upper bound, the Hoeffding-Azuma inequality is just as sharp to the first two orders in the exponent as the Chernoff-Cramér theorem for simp”
Hoeffding's inequality for uniformly ergodic Markov chains (2002, methodology): “1 (Here, Ex(·) ≡ E[ · |X0 = x].)”
Adaptive computation and machine learning (1998, result): “discussed in this chapter was presented and analyzed by Balcan et al. [2008]. The randomized algorithm as well as much of the results presented in section 9.”