To Retain or to Adapt? Generalizing Continual Learning — part 9/15

Cleaned-text code points [125351, 141123). Bibliography omitted here; source.md includes it.

For bounded growth ( ψ ⁡ ( k ) = 1 \psi(k)=1 ), the variance converges, so the difference is 𝒪 ⁡ ( 1 / k ) \mathcal{O}(1/k) . For diffusive sequences ( ψ ⁡ ( k ) = k \psi(k)=k ), the variance grows linearly, making the difference constant ( 𝒪 ⁡ ( 1 ) \mathcal{O}(1) ). For linear growth ( ψ ⁡ ( k ) = k 2 \psi(k)=k^{2} ), the variance grows quadratically, so the difference grows linearly ( 𝒪 ⁡ ( k ) \mathcal{O}(k) ). ∎

###### Lemma 29 (Expected Asymptotic Instability) .

The expected Instability cost 𝔼 ⁡ [ ℐ k ] \mathbb{E}[\mathcal{I}^{k}] , which comprises the approximation bias and the stationary gradient noise penalty, scales linearly with the environment Growth Rate: 𝔼 ⁡ [ ℐ k ] ∈ Θ ⁡ ( ψ ⁡ ( k ) ) . \mathbb{E}[\mathcal{I}^{k}]\in\Theta(\psi(k)). (60)

Implication: In expectation over the task sequence, the JTL agent suffers from unbounded performance degradation if the environment grows indefinitely (i.e., Diffusive or Linear growth), even if it converges perfectly to its stationary distribution.

###### Proof.

Taking the expectation over the task sequence of the exact Instability cost derived in Theorem 24 yields: 𝔼 ⁡ [ ℐ k ] ≈ 𝔼 ⁡ [ ℒ k ​ ( θ ⋆ JTL , k ) − ℒ k ​ ( θ ⋆ k ) ] ⏟ Bias + η 4 ​ B ​ ∑ j = 1 k w j k ​ 𝔼 ​ [ ‖ ∇ ℒ j ​ ( θ ⋆ JTL , k ) ‖ ℛ k 2 ] ⏟ Variance . \mathbb{E}[\mathcal{I}^{k}]\approx\underbrace{\mathbb{E}\left[\mathcal{L}^{k}(\theta_{\star}^{\text{JTL},k})-\mathcal{L}^{k}(\theta_{\star}^{k})\right]}_{\text{Bias}}+\underbrace{\frac{\eta}{4B}\sum_{j=1}^{k}w_{j}^{k}\,\mathbb{E}\left[\|\nabla\mathcal{L}^{j}(\theta_{\star}^{\text{JTL},k})\|_{\mathcal{R}^{k}}^{2}\right]}_{\text{Variance}}.

Bias Term. For a quadratic loss, the bias term simplifies to the expected weighted squared distance between the target task optimum and the JTL centroid: 𝔼 ⁡ [ ℒ k ​ ( θ ⋆ JTL , k ) − ℒ k ​ ( θ ⋆ k ) ] = 1 2 ​ 𝔼 ​ [ ‖ θ ⋆ JTL , k − θ ⋆ k ‖ H k 2 ] . \mathbb{E}\left[\mathcal{L}^{k}(\theta_{\star}^{\text{JTL},k})-\mathcal{L}^{k}(\theta_{\star}^{k})\right]=\frac{1}{2}\mathbb{E}\left[\|\theta_{\star}^{\text{JTL},k}-\theta_{\star}^{k}\|_{H_{k}}^{2}\right]. Because θ ⋆ JTL , k \theta_{\star}^{\text{JTL},k} is a convex combination of tasks 1 ​ … ​ k 1\dots k , its expected squared distance to the new task θ ⋆ k \theta_{\star}^{k} scales exactly with the variance of the task distribution, which is defined as Θ ⁡ ( ψ ⁡ ( k ) ) \Theta(\psi(k)) .

Variance Term. The expected gradient noise contribution depends on task diversity. Evaluating the gradients at the centroid yields ∇ ℒ j ​ ( θ ⋆ JTL , k ) = H j ​ ( θ ⋆ JTL , k − θ ⋆ j ) \nabla\mathcal{L}^{j}(\theta_{\star}^{\text{JTL},k})=H_{j}(\theta_{\star}^{\text{JTL},k}-\theta_{\star}^{j}) . Substituting this into the variance term: Variance ∝ ∑ j = 1 k w j k ​ 𝔼 ​ [ ‖ H j ​ ( θ ⋆ JTL , k − θ ⋆ j ) ‖ ℛ k 2 ] . \text{Variance}\propto\sum_{j=1}^{k}w_{j}^{k}\,\,\mathbb{E}\left[\|H_{j}(\theta_{\star}^{\text{JTL},k}-\theta_{\star}^{j})\|_{\mathcal{R}^{k}}^{2}\right]. Assuming bounded Hessians ( Assumption 23 ), this term is proportional to the expected variance of the task optima around their mean. As established in Lemma 28 , this spatial variance scales exactly with the environment growth rate Θ ⁡ ( ψ ⁡ ( k ) ) \Theta(\psi(k)) .

Since both the expected bias and expected variance scale with ψ ⁡ ( k ) \psi(k) , the total expected instability scales as Θ ⁡ ( ψ ⁡ ( k ) ) \Theta(\psi(k)) . ∎

##### A.5.2 Residual Errors

The transient cost of learning task k k is governed by the displacement of the agent’s moments relative to the new stationary target. Let μ N k − 1 k − 1 \mu_{N_{k-1}}^{k-1} and Σ N k − 1 k − 1 \Sigma_{N_{k-1}}^{k-1} denote the agent’s actual mean and covariance at the end of task k − 1 k-1 (where the expectations inside μ \mu and Σ \Sigma are strictly over the parameter trajectory). The initialization errors for task k k are defined as: μ ~ 0 JTL , k \displaystyle\tilde{\mu}_{0}^{\text{JTL},k} : = μ N k − 1 k − 1 − θ ⋆ JTL , k , Σ ~ 0 JTL , k := Σ N k − 1 k − 1 − Σ ∞ JTL , k . \displaystyle:=\mu_{N_{k-1}}^{k-1}-\theta_{\star}^{\text{JTL},k},\qquad\tilde{\Sigma}_{0}^{\text{JTL},k}:=\Sigma_{N_{k-1}}^{k-1}-\Sigma_{\infty}^{\text{JTL},k}. (61) We decompose these errors into the Geometric Shift defined earlier, and a Residual Error stemming from the incomplete convergence of the previous task: μ ~ 0 JTL , k \displaystyle\tilde{\mu}_{0}^{\text{JTL},k} = Δ ​ μ k + ( μ N k − 1 k − 1 − θ ⋆ JTL , k − 1 ) ⏟ Residual ​ r μ k − 1 , Σ ~ 0 JTL , k = Δ ​ Σ k + ( Σ N k − 1 k − 1 − Σ ∞ JTL , k − 1 ) ⏟ Residual ​ r Σ k − 1 . \displaystyle=\Delta\mu^{k}+\underbrace{(\mu_{N_{k-1}}^{k-1}-\theta_{\star}^{\text{JTL},k-1})}_{\text{Residual }r^{k-1}_{\mu}},\qquad\tilde{\Sigma}_{0}^{\text{JTL},k}=\Delta\Sigma^{k}+\underbrace{(\Sigma_{N_{k-1}}^{k-1}-\Sigma_{\infty}^{\text{JTL},k-1})}_{\text{Residual }r^{k-1}_{\Sigma}}. (62) The residual is effectively a contracted version of the previous task’s initialization error. Since SGD on a quadratic objective is a contraction mapping for appropriate step sizes, let ρ μ k \rho_{\mu}^{k} and ρ Σ k \rho_{\Sigma}^{k} denote the contraction rates for the mean and covariance errors during task k k . Then: ‖ r μ k − 1 ‖ ≤ ρ μ k − 1 ⋅ ‖ μ ~ 0 JTL , k − 1 ‖ , ‖ r Σ k − 1 ‖ ≤ ρ Σ k − 1 ⋅ ‖ Σ ~ 0 JTL , k − 1 ‖ . \|r^{k-1}_{\mu}\|\leq\rho_{\mu}^{k-1}\cdot\|\tilde{\mu}_{0}^{\text{JTL},k-1}\|,\qquad\|r^{k-1}_{\Sigma}\|\leq\rho_{\Sigma}^{k-1}\cdot\|\tilde{\Sigma}_{0}^{\text{JTL},k-1}\|. The following lemma establishes that these contraction factors are strictly less than 1.

###### Lemma 30 (Contraction of Residuals) .

For a quadratic task with Hessian H H and step-size η < 2 / λ max ​ ( H ) \eta<2/\lambda_{\max}(H) , the parameter distribution errors contract at each step. After N k N_{k} steps: ρ μ k = ( 1 − η ​ λ min ​ ( H ) ) N k < 1 , ρ Σ k = ( 1 − η ​ λ min ​ ( H ) ) 2 ​ N k < 1 . \rho_{\mu}^{k}=(1-\eta\lambda_{\min}(H))^{N_{k}}<1,\qquad\rho_{\Sigma}^{k}=(1-\eta\lambda_{\min}(H))^{2N_{k}}<1. (63)

###### Proof.

The dynamics of the mean error vector μ ~ n = μ n − θ ⋆ \tilde{\mu}_{n}=\mu_{n}-\theta_{\star} under SGD with a quadratic objective follow the recursion μ ~ n + 1 = ( I − η ​ H ) ​ μ ~ n \tilde{\mu}_{n+1}=(I-\eta H)\tilde{\mu}_{n} . Unrolling this for N k N_{k} steps yields: μ ~ N k = ( I − η ​ H ) N k ​ μ ~ 0 . \tilde{\mu}_{N_{k}}=(I-\eta H)^{N_{k}}\tilde{\mu}_{0}. (64) Taking the Euclidean norm, we have ‖ μ ~ N k ‖ ≤ ‖ ( I − η ​ H ) N k ‖ 2 ​ ‖ μ ~ 0 ‖ \|\tilde{\mu}_{N_{k}}\|\leq\|(I-\eta H)^{N_{k}}\|_{2}\|\tilde{\mu}_{0}\| . The spectral norm of the propagator is determined by the eigenvalues of H H . For η < 2 / λ max ​ ( H ) \eta<2/\lambda_{\max}(H) , the eigenvalues of ( I − η ​ H ) (I-\eta H) lie strictly within ( − 1 , 1 ) (-1,1) . Assuming a non-oscillatory step size ( η < 1 / λ max ​ ( H ) \eta<1/\lambda_{\max}(H) ), the contraction is bottlenecked by the slowest decaying mode, corresponding to λ min ​ ( H ) \lambda_{\min}(H) . Thus, the residual mean error is bounded by: ‖ r μ k ‖ ≤ ( 1 − η ​ λ min ​ ( H ) ) N k ​ ‖ μ ~ 0 JTL , k ‖ . \|r_{\mu}^{k}\|\leq(1-\eta\lambda_{\min}(H))^{N_{k}}\|\tilde{\mu}_{0}^{\text{JTL},k}\|. (65)

Similarly, the covariance deviation Σ ~ n = Σ n − Σ ∞ \tilde{\Sigma}_{n}=\Sigma_{n}-\Sigma_{\infty} evolves as Σ ~ n + 1 = ( I − η ​ H ) ​ Σ ~ n ​ ( I − η ​ H ) \tilde{\Sigma}_{n+1}=(I-\eta H)\tilde{\Sigma}_{n}(I-\eta H) . After N k N_{k} steps: Σ ~ N k = ( I − η ​ H ) N k ​ Σ ~ 0 ​ ( I − η ​ H ) N k . \tilde{\Sigma}_{N_{k}}=(I-\eta H)^{N_{k}}\tilde{\Sigma}_{0}(I-\eta H)^{N_{k}}. (66) Taking the spectral norm (or the trace) leads to a contraction factor governed by the square of the propagator’s eigenvalues: ‖ r Σ k ‖ ≤ ‖ ( I − η ​ H ) N k ‖ 2 2 ​ ‖ Σ ~ 0 ‖ = ( 1 − η ​ λ min ​ ( H ) ) 2 ​ N k ​ ‖ Σ ~ 0 ‖ . \|r_{\Sigma}^{k}\|\leq\|(I-\eta H)^{N_{k}}\|_{2}^{2}\|\tilde{\Sigma}_{0}\|=(1-\eta\lambda_{\min}(H))^{2N_{k}}\|\tilde{\Sigma}_{0}\|. (67) This confirms that both first and second-moment errors contract exponentially with the task duration N k N_{k} . ∎

The decomposition into geometric shift and residuals leads to a recursive definition of the initialization error μ ~ 0 k \tilde{\mu}_{0}^{k} . The following analytical lemma bounds this recursion.

###### Lemma 31 (Asymptotic Dominance of Stable Recursions) .

Let { x k } k ≥ 0 \{x_{k}\}_{k\geq 0} be a sequence of non-negative real numbers satisfying the linear recurrence inequality: x k ≤ ρ ​ x k − 1 + u k , with ​ 0 ≤ ρ < 1 . x_{k}\leq\rho x_{k-1}+u_{k},\quad\text{with }0\leq\rho<1. (68) If the input sequence u k u_{k} scales asymptotically as a polynomial u k ∈ Θ ⁡ ( k p ) u_{k}\in\Theta(k^{p}) for any p ∈ ℝ p\in\mathbb{R} , then the sequence x k x_{k} is asymptotically dominated by the input: x k ∈ Θ ⁡ ( k p ) . x_{k}\in\Theta(k^{p}). (69) More precisely, lim sup k → ∞ x k u k ≤ 1 1 − ρ \limsup_{k\to\infty}\frac{x_{k}}{u_{k}}\leq\frac{1}{1-\rho} .

###### Proof.

Unrolling the recurrence yields the summation (variation of constants) formula: x k ≤ ρ k ​ x 0 + ∑ j = 0 k − 1 ρ j ​ u k − j . x_{k}\leq\rho^{k}x_{0}+\sum_{j=0}^{k-1}\rho^{j}u_{k-j}. (70) The term ρ k ​ x 0 \rho^{k}x_{0} decays exponentially and is negligible relative to the polynomial u k u_{k} . We focus on the convolution sum. Because u k ∈ Θ ⁡ ( k p ) u_{k}\in\Theta(k^{p}) , there exists a constant C C such that u k ≈ C ​ k p u_{k}\approx Ck^{p} for large k k . We examine the ratio of the sum to the input u k u_{k} : ∑ j = 0 k − 1 ρ j ​ u k − j u k = ∑ j = 0 k − 1 ρ j ​ ( u k − j u k ) . \frac{\sum_{j=0}^{k-1}\rho^{j}u_{k-j}}{u_{k}}=\sum_{j=0}^{k-1}\rho^{j}\left(\frac{u_{k-j}}{u_{k}}\right). (71) For any fixed j j , as k → ∞ k\to\infty , the ratio of the polynomial terms ( k − j ) p / k p (k-j)^{p}/k^{p} approaches 1 1 . Because ρ < 1 \rho<1 , the weights ρ j \rho^{j} decay exponentially, allowing us to apply the Dominated Convergence Theorem for series and bring the limit inside the sum: lim k → ∞ ∑ j = 0 k − 1 ρ j ​ ( u k − j u k ) = ∑ j = 0 ∞ ρ j ⋅ 1 = 1 1 − ρ . \lim_{k\to\infty}\sum_{j=0}^{k-1}\rho^{j}\left(\frac{u_{k-j}}{u_{k}}\right)=\sum_{j=0}^{\infty}\rho^{j}\cdot 1=\frac{1}{1-\rho}. (72) Thus, x k x_{k} behaves asymptotically like 1 1 − ρ ​ u k \frac{1}{1-\rho}u_{k} , preserving the polynomial order Θ ⁡ ( k p ) \Theta(k^{p}) . ∎

##### A.5.3 Asymptotic Transfer

###### Theorem 32 (Asymptotic Condition for Positive Transfer Efficiency) .

The Expected Transfer Efficiency over the task sequence is positive ( 𝔼 ⁡ [ TE k ] > 0 \mathbb{E}[\mathrm{TE}^{k}]>0 ) if the transient cost avoided by the JTL agent’s proximity to the target centroid dominates the expected instability penalties introduced by task conflict and bias. Specifically, for large k k and polynomial growth rate ψ ⁡ ( k ) \psi(k) , a sufficient condition is: τ ​ N ⋅ 𝔼 ⁡ [ ℐ k ] < C ITL k ⋅ 𝔼 ⁡ [ ‖ μ ~ 0 ITL ‖ 2 ] \tau N\cdot\mathbb{E}[\mathcal{I}^{k}]<C^{k}_{\text{ITL}}\cdot\mathbb{E}[\|\tilde{\mu}_{0}^{\text{ITL}}\|^{2}]

###### Proof.

The proof proceeds by decomposing the expected Transfer Efficiency and analyzing the asymptotic behavior of the transient costs.

Recall the definition of Transfer Efficiency: TE k = ( δ ITL k − δ JTL k ) − ℐ k . \text{TE}^{k}=(\delta^{k}_{\text{ITL}}-\delta^{k}_{\text{JTL}})-\mathcal{I}^{k}. The condition for expected positive transfer over tasks is 𝔼 ⁡ [ δ ITL k ] − 𝔼 ⁡ [ δ JTL k ] > 𝔼 ⁡ [ ℐ k ] \mathbb{E}[\delta^{k}_{\text{ITL}}]-\mathbb{E}[\delta^{k}_{\text{JTL}}]>\mathbb{E}[\mathcal{I}^{k}] .

We use the Transient Potential Law ( Theorem 25 ). Define the Spectral Dissipation Matrix as ξ N ​ ( H ) := I − ( I − η ​ H ) 2 ​ N \xi_{N}(H):=I-(I-\eta H)^{2N} . This matrix captures the fraction of the initial error energy dissipated after N N steps. Using Task Alignment ( Assumption 22 ), the transient costs simplify to: δ ITL k ≈ 1 τ ​ N ​ Tr ​ ( ξ N ​ ( H k ) ​ ( μ ~ 0 ITL ​ μ ~ 0 ITL ⊤ + Σ ~ 0 ITL ) ) , \displaystyle\delta^{k}_{\text{ITL}}\approx\frac{1}{\tau N}\text{Tr}\left(\xi_{N}(H_{k})(\tilde{\mu}^{\text{ITL}}_{0}{\tilde{\mu}^{\text{ITL}}_{0}}^{\top}+\tilde{\Sigma}_{0}^{\text{ITL}})\right), (73) δ JTL k ≈ 1 τ ​ N ​ Tr ​ ( ξ N ​ ( H ¯ k ) ​ ℛ k ​ ( μ ~ 0 JTL , k ​ μ ~ 0 JTL , k ⊤ + Σ ~ 0 JTL , k ) ) . \displaystyle\delta^{k}_{\text{JTL}}\approx\frac{1}{\tau N}\text{Tr}\left(\xi_{N}(\bar{H}_{k})\mathcal{R}^{k}\,(\tilde{\mu}^{\text{JTL},k}_{0}{\tilde{\mu}^{\text{JTL},k}_{0}}^{\top}+\tilde{\Sigma}_{0}^{\text{JTL},k})\right). (74) Let γ min k := λ min ​ ( ξ N ​ ( H k ) ) \gamma^{k}_{\min}:=\lambda_{\min}(\xi_{N}(H_{k})) and γ max k := λ max ​ ( ξ N ​ ( H ¯ k ) ) \gamma^{k}_{\max}:=\lambda_{\max}(\xi_{N}(\bar{H}_{k})) . We lower bound the ITL cost and upper bound the JTL cost: δ ITL k ≥ γ min k τ ​ N ​ ( ‖ μ ~ 0 ITL ‖ 2 + Tr ​ ( Σ ~ 0 ITL ) ) , \displaystyle\delta^{k}_{\text{ITL}}\geq\frac{\gamma^{k}_{\min}}{\tau N}\left(\|\tilde{\mu}_{0}^{\text{ITL}}\|^{2}+\text{Tr}(\tilde{\Sigma}_{0}^{\text{ITL}})\right), (75) δ JTL k ≤ λ max ​ ( ℛ k ) ⋅ γ max k τ ​ N ​ ( ‖ μ ~ 0 JTL , k ‖ 2 + Tr ​ ( Σ ~ 0 JTL , k ) ) . \displaystyle\delta^{k}_{\text{JTL}}\leq\lambda_{\max}(\mathcal{R}^{k})\cdot\frac{\gamma^{k}_{\max}}{\tau N}\left(\|\tilde{\mu}_{0}^{\text{JTL},k}\|^{2}+\text{Tr}(\tilde{\Sigma}_{0}^{\text{JTL},k})\right). (76) Denote the bounding constants by C ITL k = γ min k C^{k}_{\text{ITL}}=\gamma^{k}_{\min} and C JTL k = λ max ​ ( ℛ k ) ​ γ max k C^{k}_{\text{JTL}}=\lambda_{\max}(\mathcal{R}^{k})\,\gamma^{k}_{\max} . Taking the expectation over the task sequence on both sides, the condition for positive transfer translates to: τ ​ N ⋅ 𝔼 ⁡ [ ℐ k ] < C ITL k ​ ( 𝔼 ⁡ [ ‖ μ ~ 0 ITL ‖ 2 ] + 𝔼 ⁡ [ Tr ​ ( Σ ~ 0 ITL ) ] ) − C JTL k ​ ( 𝔼 ⁡ [ ‖ μ ~ 0 JTL , k ‖ 2 ] + 𝔼 ⁡ [ | Tr ​ ( Σ ~ 0 JTL , k ) | ] ) . \tau N\cdot\mathbb{E}[\mathcal{I}^{k}]<C^{k}_{\text{ITL}}\left(\mathbb{E}[\|\tilde{\mu}_{0}^{\text{ITL}}\|^{2}]+\mathbb{E}[\text{Tr}(\tilde{\Sigma}_{0}^{\text{ITL}})]\right)-C^{k}_{\text{JTL}}\left(\mathbb{E}[\|\tilde{\mu}_{0}^{\text{JTL},k}\|^{2}]+\mathbb{E}[|\text{Tr}(\tilde{\Sigma}_{0}^{\text{JTL},k})|]\right). We now analyze the asymptotic scaling of each expectation with respect to k k : • 𝔼 ⁡ [ ℐ k ] \mathbb{E}[\mathcal{I}^{k}] : By Lemma 29 , expected instability grows as Θ ⁡ ( ψ ⁡ ( k ) ) \Theta(\psi(k)) .

• 𝔼 ⁡ [ ‖ μ ~ 0 ITL ‖ 2 ] \mathbb{E}[\|\tilde{\mu}_{0}^{\text{ITL}}\|^{2}] : Because the ITL agent resets to a fixed origin prior, its error is the raw distance to the new task optimum. By Definition 26 , this scales as Θ ⁡ ( ψ ⁡ ( k ) ) \Theta(\psi(k)) .

• 𝔼 ⁡ [ Tr ​ ( Σ ~ 0 ITL ) ] \mathbb{E}[\text{Tr}(\tilde{\Sigma}_{0}^{\text{ITL}})] : The ITL covariance resets to a fixed prior variance, remaining constant Θ ⁡ ( 1 ) \Theta(1) relative to the task growth.

• 𝔼 ⁡ [ ‖ μ ~ 0 JTL , k ‖ 2 ] \mathbb{E}[\|\tilde{\mu}_{0}^{\text{JTL},k}\|^{2}] : We analyze the root-mean-square error e k := 𝔼 ⁡ [ ‖ μ ~ 0 JTL , k ‖ 2 ] e_{k}:=\sqrt{\mathbb{E}[\|\tilde{\mu}_{0}^{\text{JTL},k}\|^{2}]} . By Minkowski’s inequality applied to the decomposition μ ~ 0 = Δ ​ μ + r μ \tilde{\mu}_{0}=\Delta\mu+r_{\mu} , we obtain the recursive bound: e k ≤ 𝔼 ⁡ [ ‖ Δ ​ μ k ‖ 2 ] + 𝔼 ⁡ [ ‖ r μ k − 1 ‖ 2 ] ≤ d k + ρ μ ​ e k − 1 , e_{k}\leq\sqrt{\mathbb{E}[\|\Delta\mu^{k}\|^{2}]}+\sqrt{\mathbb{E}[\|r_{\mu}^{k-1}\|^{2}]}\leq d_{k}+\rho_{\mu}e_{k-1}, where d k := 𝔼 ⁡ [ ‖ Δ ​ μ k ‖ 2 ] d_{k}:=\sqrt{\mathbb{E}[\|\Delta\mu^{k}\|^{2}]} is the geometric drift. From Lemma 27 , the squared expected drift scales as Θ ⁡ ( ψ ⁡ ( k ) / k 2 ) \Theta(\psi(k)/k^{2}) , meaning d k ∈ Θ ⁡ ( ψ ⁡ ( k ) / k ) d_{k}\in\Theta(\sqrt{\psi(k)}/k) . Applying Lemma 31 , the error sequence e k e_{k} is dominated by d k d_{k} : 𝔼 ⁡ [ ‖ μ ~ 0 JTL , k ‖ 2 ] = e k 2 ∈ Θ ⁡ ( d k 2 ) = Θ ⁡ ( ψ ⁡ ( k ) k 2 ) . \mathbb{E}[\|\tilde{\mu}_{0}^{\text{JTL},k}\|^{2}]=e_{k}^{2}\in\Theta(d_{k}^{2})=\Theta\left(\frac{\psi(k)}{k^{2}}\right).