An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems

Q: What is the relationship between F and Ps?

The relationship between F and Ps is quantified by Lemma 2.1, which states that if E[Ps(X T )] - Ps <= T, then E[F (X T )] - F <= uD 2 0 + T. This lemma provides a rough relation between F and Ps, indicating that if u is small, Ps and F will be arbitrarily close. Lemma 2.2 further shows that if F has a quadratic lower bound, Ps can be bounded by F 1 + u G, allowing for the choice of u G to make F and Ps close in the relative scale. The proof of these lemmas is provided in the supplementary material 5.

Q: What is the partial linearization of the objective function in Obvious stochastic mirror descent?

The partial linearization of the objective function in Obvious stochastic mirror descent is F(x_t) + + H(x). This approximation is derived from the first order Taylor approximation of F, where H is not linearized. The convexity and smoothness of F ensure that the expected value of this difference is zero. This concept traces back to the literature on composite minimization [1, 22]. The sequence of step-sizes {a_t, g_t} t>=1 is oblivious to the instance, meaning it is selected a priori and problem-independent. Unlike other mirror descent schemes [9, 3], this approach considers a general step-size that does not take any parameters of the problem into account.

Q: How does the smoothed stochastic oracle affect Lan and Levy's algorithms?

The smoothed stochastic oracle, with an over-estimated parameter, slows down Lan and Levy's algorithms significantly. The authors conducted experiments using prior knowledge on L and D, resulting in Fig. 4.1 showing the impact. Hyper-parameter tuning improves the algorithms' performance, especially in high dimensions. The sparse case is well handled by the algorithms, while the Levy adaptive algorithm maintains conservative step-sizes. Overall, the smoothed stochastic oracle provides an advantage to these methods, making them substantially faster than alternatives.

Q: How does the power method oracle perform in the sparse case?

In the sparse case, the power method oracle shows unstable first iterations and converges after a transition time, as predicted by theory. Figure 2 illustrates the performance using provided parameters L and D, and hyper-tuned step-sizes. However, the accuracy is worse compared to stochastic smoothing oracles. Further discussions and simulations results are available in the supplementary material. The power method oracle is computationally less expensive but sacrifices accuracy. This highlights the need for careful tuning of step-sizes and consideration of the trade-off between computational cost and accuracy in sparse data scenarios.

Question

1. What is the main limitation of the mirror prox algorithm?

2. How does regularization help in optimization problems?

3. What are the key differences between non-smooth and smooth cases in optimization?

4. What is the relationship between F and Ps?

Accepted Answer

The main limitation of the mirror prox algorithm is the requirement of a full spectral decomposition of a symmetric matrix at each iteration. This can be computationally expensive and may hinder its efficiency for large-scale problems. Alternative approaches, such as stochastic smoothing, have been developed to address this limitation and provide more efficient solutions for minimizing the maximum eigenvalue of a symmetric matrix over a convex set.

Accepted Answer

Regularization introduces a regularization term in optimization problems with limited information on regularity parameters. Techniques from complementary composite stochastic optimization are used to handle this term. An oblivious step-size scheme is developed to allow the algorithm to converge without prior knowledge of Lipschitz or smoothness constants. By considering u small, the regularized problem can be made arbitrarily close to the original one. The gap between the minimum value of the composite setting and the minimum value of F is smaller than D20/T. This approach can be extended to relative scale precision targets, resulting in significant computational savings.

Accepted Answer

In the non-smooth case, the stochastic oracle needs to have a M-bounded second moment, i.e., E x G(X, x) 2 <= M 2. This requires prior knowledge on D 0 and M for optimal step-size tuning. In the smooth case, the objective function F needs to be L-smooth and the stochastic noise should have a finite variance s 2. This allows the use of the accelerated framework in [14] to achieve optimal convergence. The relative scale setting imposes a quadratic lower bound for F and a relative constraint on the stochastic oracle. Adaptive methods like D-Adaptation and Adagrad are used to handle uncertainties in the non-smooth and smooth cases, respectively. The convergence rates differ between the two cases, with the non-smooth case achieving EQUATION and the smooth case achieving EQUATION. Over or under-estimation of parameters can make algorithms slow and inefficient, highlighting the importance of adaptive methods in optimization.

Accepted Answer

The relationship between F and Ps is quantified by Lemma 2.1, which states that if E[Ps(X T )] - Ps <= T, then E[F (X T )] - F <= uD 2 0 + T. This lemma provides a rough relation between F and Ps, indicating that if u is small, Ps and F will be arbitrarily close. Lemma 2.2 further shows that if F has a quadratic lower bound, Ps can be bounded by F 1 + u G, allowing for the choice of u G to make F and Ps close in the relative scale. The proof of these lemmas is provided in the supplementary material 5.

Accepted Answer

The partial linearization of the objective function in Obvious stochastic mirror descent is F(x_t) + <F(x_t), x - x_t> + H(x). This approximation is derived from the first order Taylor approximation of F, where H is not linearized. The convexity and smoothness of F ensure that the expected value of this difference is zero. This concept traces back to the literature on composite minimization [1, 22]. The sequence of step-sizes {a_t, g_t} t>=1 is oblivious to the instance, meaning it is selected a priori and problem-independent. Unlike other mirror descent schemes [9, 3], this approach considers a general step-size that does not take any parameters of the problem into account.

Accepted Answer

Stochastic mirror descent for convex Lipschitz function is an algorithm used for non-smooth cases. It adapts the smooth composite setting studied in [3] to the nonsmooth case. The algorithm involves a number of iterations, starting point, step-sizes, and a regularizer term. Theorem 3.1 states that if Algorithm 1 runs under the oblivious step-size and convex Lipschitz function assumptions, it converges. The convergence result is derived from [17, 13, 3] and extended for the relative scale case. Lemma 2.1 and Lemma 2.2 are used in the proof to obtain the first convergence result for the non-smooth case.

Accepted Answer

In the relative scale case, the transition time T0 is defined as the supremum of t greater than or equal to 1, where gt at is less than or equal to 2L u. It represents the time at which the oblivious step-size a t / g t exceeds the threshold u / [2L]. This transition time is crucial in analyzing the convergence rate of the algorithm and plays a significant role in determining the efficiency of the optimization process.

Accepted Answer

In the smooth setting, we use the acceleration framework from [3] with step-size (16). By introducing a transition time T0, we have shown the convergence of the accelerated algorithm. Theorem 3.3 states that if Algorithm 2 runs under the oblivious step-size (16), convex smooth functions assumptions are satisfied, and Algorithm 2 Accelerated Stochastic Mirror Descent Require: Iterations T >= 0, starting point X1, step-sizes (a t, g t), regularizer term u > 0 for 1 <= t <= T, then the algorithm converges. The feasible set X has diameter bounded by D. The transition time analysis can co-exist with the original analysis [3], and the proof can be found in the supplementary material 8. Numerical experiments show that minimizing the maximum eigenvalue over a hypercube is possible with both methods, as the feasible set is bounded and the objective function is bounded over this set. Gaussian noise levels affect algorithm performance, and the degree of polynomials used for step-sizes affects performance in practice.

Accepted Answer

The smoothed stochastic oracle, with an over-estimated parameter, slows down Lan and Levy's algorithms significantly. The authors conducted experiments using prior knowledge on L and D, resulting in Fig. 4.1 showing the impact. Hyper-parameter tuning improves the algorithms' performance, especially in high dimensions. The sparse case is well handled by the algorithms, while the Levy adaptive algorithm maintains conservative step-sizes. Overall, the smoothed stochastic oracle provides an advantage to these methods, making them substantially faster than alternatives.

Accepted Answer

In the sparse case, the power method oracle shows unstable first iterations and converges after a transition time, as predicted by theory. Figure 2 illustrates the performance using provided parameters L and D, and hyper-tuned step-sizes. However, the accuracy is worse compared to stochastic smoothing oracles. Further discussions and simulations results are available in the supplementary material. The power method oracle is computationally less expensive but sacrifices accuracy. This highlights the need for careful tuning of step-sizes and consideration of the trade-off between computational cost and accuracy in sparse data scenarios.

An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems

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Most frequently asked questions

1. What is the main limitation of the mirror prox algorithm?

2. How does regularization help in optimization problems?

3. What are the key differences between non-smooth and smooth cases in optimization?

4. What is the relationship between F and Ps?

5. What is the partial linearization of the objective function in Obvious stochastic mirror descent?

6. What is stochastic mirror descent for convex Lipschitz function?

7. What is the transition time T0 in the relative scale case?

8. Accelerated stochastic mirror descent smooth setting?

9. How does the smoothed stochastic oracle affect Lan and Levy's algorithms?

10. How does the power method oracle perform in the sparse case?

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