Paper: Adaptive Stochastic Gradient Algorithm for Black-box Multi-Objective Learning (ICLR-2024)

Reviewer comment (Theory):
The proposed method assumes full access to the objective function, while in most applications (like in multi-task learning setup used in the experiment setup), full evaluation of objectives might not be possible or might be costly. The theoretical guarantees provided does not seem to cover this. However, the authors provide empirical evidence for the proposed method in this setting.

Author response:
It is challenging to obtain the full evaluation of the objective for many practice problems. Therefore, we can only approximate the loss function with a min-batch of data in our multi-task learning setup. It is interesting to analyze the effect of the stochasticity caused by mini-batch approximation to our algorithm theoretically, and we regarded it as a future work.

Reviewer comment (Theory):
The Pareto stationarity defined in Proposition 2.3. may not be a necessary condition for Pareto optimality when $\mathcal{X} \subset \mathbb{R}^d$ (Consider when there is a global minimum for all the objectives at a boundary of  $\mathcal{X}$)

Author response:
Thanks for your insightful comments. In the general nonconvex cases, our method focuses on finding the Pareto stationary solution. We have corrected the mentioned claim about Pareto stationarity in Section 2.1 of the updated version.

Reviewer comment (Experiments):
How doe the proposed method scale with the size of the parameter to be learned?

Author response:
We now provide the scalability analysis of our algorithm. For the proposed ASMG method, the computational cost of solving the weights in each iteration is negligible since the number of objectives is usually very small (i.e. $d>>m$). Therefore, the computational cost in each iteration is of order $\mathcal{O}(mNd)$. We provide an ablation study of our method on different dimensions, i.e. $\{d=200,500,1000\}$, and the results (Figure 7) are put in Appendix F. The results demonstrate the effectiveness of the proposed ASMG method on high-dimensional problems.

Reviewer comment (Experiments):
How sensitive is the proposed method to the stochasticity of the objectives (i.e. when the average over all data-points is estimated by the average of over a mini batch)?

Author response:
We conduct an experiment of the proposed method with different batch sizes on the Office-31 dataset with $d=512$. The average classification accuracy with different batch sizes is reported in the following table. The result shows that the proposed method is not very sensitive to the stochasticity of the objectives. We will complete this table in the revision.

\begin{array}{ccccc}
    \hline
    \text{Batch Size} & 32 & 64 & 128 &  256 \newline
    \hline
    \text{ASMG} & 81.78 & 81.47 & 81.96 & 81.48\newline
    \hline
\end{array}

Reviewer comment (Novelty):
The related work section only includes evolutionary methods and BO for multi-objective black-box optimization. The relevance to previous works on MOO/stochastic gradient approximation should also be clarified. A detailed discussion of technical difficulties in applying stochastic gradient approximation to black-box MOO may help better highlight the significance and technical novelty of this paper.

Author response:
Thank you for your thoughtful review and constructive comments.

---

>Q1. "The related work section only includes evolutionary methods and BO for multi-objective black-box optimization. The relevance to previous works on MOO/stochastic gradient approximation should also be clarified. A detailed discussion of technical difficulties in applying stochastic gradient approximation to black-box MOO may help better highlight the significance and technical novelty of this paper.'' 

**A1**:  We provide a discussion about the previous works on MOO and stochastic gradient approximation.

(i) For previous methods on MOO, the most relevant method to our approach is gradient-based MOO methods, as they also aim at finding the Pareto optimal solution or the Pareto stationary solution. A typical gradient-based method is the MGDA method, which also solves a max-min optimization problem to obtain the weights. However, **the proposed ASMG method is not a typical MGDA-type method**. The max-min optimization problem proposed in MGDA-type methods is related to the true gradient of the parameters. They add a regularization term $||d||^2$ to control the norm of the aggregated gradient. In our case, the update is conducted on a Gaussian distribution, and we need to jointly update the mean and covariance matrix. We use Kullback–Leibler divergence to regularize the distance between two distributions, i.e. $\theta$ and $\theta_t$. Therefore, the form of proposed max-min optimization (Eq. (4)) is **completely different** from MGDA-type methods (e.g. Eq. (2) in [4]), and the solution process is also different (Appendix A.1). Moreover, it is interesting that our novel max-min problem leads to a simple quadratic programming problem for the aggregation weights computation. 


(ii) The prior studies on stochastic gradient approximation have not specifically addressed the black-box MOO problem. Directly applying weighted sum aggregation of multi-objectives to a single objective is not effective, as observed in our experiments. To address this challenge, we propose an adaptive aggregation by introducing a novel max-min optimization problem (i.e. Eq. (4)).


The main technical difficulty of applying stochastic gradient approximation to black-box MOO is to propose an effective optimization method to update the distribution with a convergence guarantee and verify the performance in different experimental settings. The algorithm proposed in our work successfully resolves these technical difficulties, and possesses significant technical novelty in three key aspects:

(i) we introduce a novel max-min optimization problem (i.e. Eq. (4)) for the Gaussian-smoothed objectives to solve black-box MOO. This approach is crucial as previous MGDA-based methods are not applicable in the absence of the true gradients. The max-min optimization problem provides update rules for both the mean and covariance matrix, and notably, we uncover that it can derive weights as a solution to a quadratic programming problem;

(ii) we explore the connection of the optimal conditions between the original MOO and the corresponding Gaussian-smoothed MOO, and we prove the convergence rate for the proposed algorithm in both convex and non-convex scenarios. These original findings provide a theoretical guarantee for our algorithm;

(iii) we are the first to study such a black-box multi-task learning problem (i.e. Eq. (22)) as a black-box MOO. Our experiment on the black-box multi-task learning setting based on the CLIP model demonstrates the effectiveness of the proposed ASMG method in real-world applications.

Thanks for your valuable comments. We added these discussions in Section G of the updated version.

Reviewer comment (Experiments):
The experiment considers evolutionary algorithm as the only baseline, which seems insufficient. There are also other major branches of black-box MOO algorithms, e.g., multi-objective Bayesian optimization [Zhang & Golovin, 2020] or multi-objective bandits, that should be compared.

Author response:
>Q2. "The experiment considers evolutionary algorithm as the only baseline, which seems insufficient. There are also other major branches of black-box MOO algorithms, e.g., multi-objective Bayesian optimization [Zhang \& Golovin, 2020] or multi-objective bandits, that should be compared.''

**A2**: One motivation behind this paper is to develop a method applicable to some real-world black-box MOO scenarios, such as black-box multi-task prompt tuning. In such scenarios, the problem's dimension is relatively high, and most Bayesian optimization methods cannot handle it. The multi-objective Bayesian optimization [1] conducts experiments on the problems up to 24 dimensions. In contrast, our algorithm can handle high dimensional back-box MOO problems ($d\ge 1000$) on both synthetic and black-box multi-task learning (MTL) settings (Figure 7 and Table 1). Besides, the code of [1] has not been released and we did not find an official implementation of multi-objective bandits, making it challenging to include those baselines in our comparisons.

To enrich our experiments, we conducted additional experiments on two recently proposed methods: BES, a random search method [2], and MMES, an evolution strategy method [3]. The results of these two methods on both synthetic and black-box multi-task learning problems are put in Appendix F of the updated version. The results consistently demonstrate the effectiveness of the proposed ASMG method. The following tables show part of the result of these two baseline methods on the black-box multi-task learning settings. The first and the second tables show the experimental results on Office-31 and Office-home datasets, respectively.

\begin{array}{ccccc}
    \hline
    \text{Method} & \text{A} & \text{D} & \text{W} & \text{Avg} \newline
    \hline
    \text{BES (d=256)} &  72.65 & 80.60 & 73.52 & 75.59\newline
    \text{MMES (d=256)} & 75.90 & 83.33 & 76.67 & 78.63\newline
    \text{ASMG (d=256)} & 77.83 & 86.61 & 80.56 & \textbf{81.67}\newline
    \hline
    \text{BES (d=512)} & 75.73 & 82.51 & 74.81 & 77.68 \newline
    \text{MMES (d=512)} & 76.01 & 84.70 & 77.22 & 79.31\newline
    \text{ASMG (d=512)} & 78.63 & 87.43 & 78.33 & \textbf{81.47}\newline
    \hline
    \text{BES (d=1024)} & 72.14 & 79.51 & 71.67 & 74.44\newline
    \text{MMES (d=1024)} & 77.09 & 81.42 & 75.74 & 78.09\newline
    \text{ASMG (d=1024)} & 76.30 & 87.70 & 80.19 & \textbf{81.40}\newline
    \hline
\end{array}

\begin{array}{cccccc}
    \hline
    \text{Method} & \text{Ar} & \text{Cl} & \text{Pr} & \text{Rw} & \text{Avg} \newline
    \hline
    \text{BES (d=256)} & 68.94 & 45.97 & 81.53 & 79.42 & 68.97\newline
    \text{MMES (d=256)} & 71.85 & 49.26 & 82.10 & 81.37 & 71.14\newline
    \text{ASMG (d=256)} & 74.26 & 53.52 & 86.23 & 83.03 & \textbf{74.26}\newline
    \hline
    \text{BES (d=512)} & 69.39 & 48.18 & 82.94 & 80.29 & 70.20\newline
    \text{MMES (d=512)} & 70.40 & 50.45 & 85.10 & 82.56 & 72.13\newline
    \text{ASMG (d=512)} & 73.50 & 52.84 & 85.88 & 83.78 & \textbf{74.00}\newline
    \hline
    \text{BES (d=1024)} & 70.27 & 48.25 & 79.94 & 80.00 & 69.62\newline
    \text{MMES (d=1024)} & 71.03 & 49.19 & 84.29 & 81.95 & 71.61\newline
    \text{ASMG (d=1024)} & 73.18 & 51.82 & 85.84 & 83.21 & \textbf{73.51}\newline
    \hline
\end{array}

Reviewer comment (Theory):
The proposed method assumes that the covariance matrix of the search distribution is a diagonal matrix. As I understand, this assumption indicates that the different dimensions of x are independent. Can you justify this assumption? I wonder if there is any performance gap with/without using this assumption.

Author response:
>Q3. "The proposed method assumes that the covariance matrix of the search distribution is a diagonal matrix. As I understand, this assumption indicates that the different dimensions of x are independent. Can you justify this assumption? I wonder if there is any performance gap with/without using this assumption.''

**A3**: We employ the diagonal covariance update scheme to approximate the full covariance matrix update for fast computation. The diagonal update is best suited for separable functions (different dimensions of x are independent). For non-separable functions, it may not be as good as a full-covariance update scheme. However, the full-covariance update scheme needs to compute the inverse of the covariance matrix, which needs $O(d^3)$.  In practical tasks, e.g., prompt fine-tuning tasks,  the dimension of tasks is generally large. In our experiments, $d=\{256, 512,1024\}$. In such cases, the $O(d^3)$ computation cost is too expensive and prohibitable. Thus, the diagonal covariance update scheme with a $O(d)$ update instead of $O(d^3)$ is necessary.

In our experiments on practical prompt fine-tuning tasks, the underlying black-box functions generally are non-separable (each dimension of x is not independent).  Empirical results on these non-separable tasks demonstrate the performance improvement of our method compared with other baselines, including CMA-ES, BES [2], and MMES [3],  which shows the capabilities of our methods in handling the non-separable tasks.

Reviewer comment (Experiments):
The Monte-Carlo sampling approach for calculating the expectation may bring additional computational complexity to the proposed algorithm. Can you analyze the complexity or provide some empirical results on the real running time?

Author response:
>Q4. "The Monte-Carlo sampling approach for calculating the expectation may bring additional computational complexity to the proposed algorithm. Can you analyze the complexity or provide some empirical results on the real running time?''

**A4**: Since the ES-type approaches, e.g., CMAES, ES,  MMES[3], all take Monte-Carlo sampling in the parameter update. Our method does not incur additional time complexity costs. For our method, the computational cost of solving the weights in each iteration is negligible since the number of objectives is usually very small (i.e. $d>>m$). Therefore, the computational cost in each iteration is of order $\mathcal{O}(mNd)$. We report the running time (s) per iteration with various sample sizes $N$ in the black-box MTL settings with $d=512$ in the following two tables. The first and the second tables show the experimental results on Office-31 ($m=3$) and Office-home ($m=4$) datasets, respectively. All the methods achieve similar running times and the experimental results are consistent with our analysis. We added the discussion about computational complexity in Section 3.2 of the updated version.

\begin{array}{cccc}
    \hline
    \text{Method} & N=20 & N=40 & N=80 \newline
    \hline
    \text{ES} & 5.110 & 9.620 & 18.61\newline
    \text{CMA-ES} & 5.236 & 9.680 & 18.75\newline
    \text{ASMG} & 5.121 & 9.406 & 17.96\newline
    \hline
\end{array}

\begin{array}{cccc}
    \hline
    \text{Method} & N=20 & N=40 & N=80 \newline
    \hline
    \text{ES} & 7.473 &  14.80 & 29.48\newline
    \text{CMA-ES} & 7.647 & 14.60 & 28.88\newline
    \text{ASMG} & 7.553 & 14.56 & 28.67\newline
    \hline
\end{array}

Reviewer comment (Novelty):
More discussions about more related works, such as recent KDD papers about one important application of multi-objective optimization in learning to rank https://dl.acm.org/doi/abs/10.1145/3580305.3599870 and https://dl.acm.org/doi/abs/10.1145/3580305.3599482. One more example is the lack of discussion about the seminal work in MOO for Multi-task learning https://arxiv.org/pdf/1810.04650.pdf

Author response:
**A1**: Thank you for your suggestion, we've added the discussions about the two papers about the learn-to-rank (LTR) problem in the introduction section of the updated version. These works propose and evaluate various gradient-based MOO methods for solving the LTR problem. Different from those weighting approaches, our method is designed for black-box MOO and we obtain the weights by solving a max-min optimization problem for the Gaussian-smoothed MOO.

---

**A2**: We've added the discussions about [1] in the updated version. [1] is a seminal work that successfully applies the MGDA method to multi-task learning. However, [1] and various MGDA-based methods are inapplicable to black-box multi-task learning, and our method can handle such settings.

Reviewer comment (Theory):
How likely does the assumption that the covariance matrix Σ is a diagonal matrix hold?

Author response:
**A3**: In our algorithm, we employ the diagonal covariance update scheme to approximate the full covariance matrix, because we need to calculate the inversion of the covariance matrix to obtain the stochastic gradients and the computational cost of calculating the inversion of a non-diagonal $d$-dimensional matrix is very high (i.e. $\mathcal{O}(d^3)$). Therefore, due to the consideration of computational cost, we assume that the covariance matrix $\Sigma$ is a diagonal matrix. Then for the proposed method, the computational cost per iteration is of order $\mathcal{O}(mNd)$, which is much more affordable. In practice, We intercept the diagonal elements of the estimated gradient of $\Sigma$ to keep the covariance matrix a diagonal matrix.

Reviewer comment (Theory):
How likely the assumptions in Section 4, especially the convex case, hold? For example, in 6.1 SYNTHETIC PROBLEMS.

Author response:
**A4**: According to our assumptions in Theorem 4.4, the objective functions should be convex and are not necessary to be differentiable, and the Gaussian-smooth objective is $c$-strongly convex.  For the convex synthetic problem, such as $l_2$-norm problem, where each objective takes form of $||\boldsymbol{x}-C\||_2^2$ and $C$ is a constant, this assumption holds. The boundedness assumption of $||\mu_t-\mu^*||$ in Theorem 4.4 can be held when we conduct our algorithm in a bounded region.

Reviewer comment (Novelty):
In the introduction, lack of discussion on advantages of the Gaussian-smoothed MOO, compared with the traditional method.

Author response:
**A1**: The advantage of using the Gaussian-smoothed MOO method is that we can effectively estimate stochastic gradients, and enable a more accurate search direction for the distribution. This capability is valuable in solving high-dimensional black-box MOO problems and providing comprehensive theoretical analysis.  In contrast, the Bayesian optimization methods are good at dealing with low-dimensional expensive black-box problems and most multi-objective genetic algorithms lack convergence guarantees. For ES-based methods, such as ES, which optimizes only the mean and the variance is utilized purely to estimate the gradient. Those methods can not handle ill-conditional problems, as observed in our experiments on three synthetic problems. ES-based methods like CMA-ES, which optimizes mean and variance, are primarily designed for single-objective black-box optimization problems and may lack adaptations for handling MOO problems, while our methods are proposed for black-box MOO. We added these discussions in Section 3.1 of the updated version.

Reviewer comment (Experiments):
Lack of comparison with more recent state-of-the-art methods. The authors only compare their algorithm with the ES-based method.

Author response:
**A2**: Thanks for your suggestion. To improve the experimental section, we conducted additional experiments on two recently proposed methods: a random search method BES [1] and an evolution strategy method MMES [2]. The results of these two methods on both synthetic and black-box multi-task learning problems are put in Appendix F of the updated version. The results consistently demonstrate the effectiveness of the proposed ASMG method. The following tables show part of the result of these two baseline methods on the black-box multi-task learning settings. The first and the second tables show the experimental results on Office-31 and Office-home datasets, respectively.

\begin{array}{ccccc}
    \hline
    \text{Method} & \text{A} & \text{D} & \text{W} & \text{Avg} \newline
    \hline
    \text{BES (d=256)} &  72.65 & 80.60 & 73.52 & 75.59\newline
    \text{MMES (d=256)} & 75.90 & 83.33 & 76.67 & 78.63\newline
    \text{ASMG (d=256)} & 77.83 & 86.61 & 80.56 & \textbf{81.67}\newline
    \hline
    \text{BES (d=512)} & 75.73 & 82.51 & 74.81 & 77.68 \newline
    \text{MMES (d=512)} & 76.01 & 84.70 & 77.22 & 79.31\newline
    \text{ASMG (d=512)} & 78.63 & 87.43 & 78.33 & \textbf{81.47}\newline
    \hline
    \text{BES (d=1024)} & 72.14 & 79.51 & 71.67 & 74.44\newline
    \text{MMES (d=1024)} & 77.09 & 81.42 & 75.74 & 78.09\newline
    \text{ASMG (d=1024)} & 76.30 & 87.70 & 80.19 & \textbf{81.40}\newline
    \hline
\end{array}

\begin{array}{cccccc}
    \hline
    \text{Method} & \text{Ar} & \text{Cl} & \text{Pr} & \text{Rw} & \text{Avg} \newline
    \hline
    \text{BES (d=256)} & 68.94 & 45.97 & 81.53 & 79.42 & 68.97\newline
    \text{MMES (d=256)} & 71.85 & 49.26 & 82.10 & 81.37 & 71.14\newline
    \text{ASMG (d=256)} & 74.26 & 53.52 & 86.23 & 83.03 & \textbf{74.26}\newline
    \hline
    \text{BES (d=512)} & 69.39 & 48.18 & 82.94 & 80.29 & 70.20\newline
    \text{MMES (d=512)} & 70.40 & 50.45 & 85.10 & 82.56 & 72.13\newline
    \text{ASMG (d=512)} & 73.50 & 52.84 & 85.88 & 83.78 & \textbf{74.00}\newline
    \hline
    \text{BES (d=1024)} & 70.27 & 48.25 & 79.94 & 80.00 & 69.62\newline
    \text{MMES (d=1024)} & 71.03 & 49.19 & 84.29 & 81.95 & 71.61\newline
    \text{ASMG (d=1024)} & 73.18 & 51.82 & 85.84 & 83.21 & \textbf{73.51}\newline
    \hline
\end{array}

As we discussed in the introduction section, the Bayesian optimization methods are not suitable for high-dimensional problems, i.e. $d=1024$, and multi-objective genetic algorithms explore the entire Pareto optimal set, which is computationally expensive for deep multi-task learning problems. Due to these two reasons and the difference in the algorithmic structure, we did not include those algorithms as baselines in the experiments.

Reviewer comment (Evaluation):
Limited discussion of the algorithm's computational complexity and runtime.

Author response:
**A3**: We now provide the computational complexity analysis for the proposed ASMG methods, and the corresponding discussion is put in Section 3.2 of the updated version. 
For the ASMG method, the computational cost of solving the weights in each iteration is negligible since the number of objectives is usually very small (i.e. $d>>m$). Therefore, the computational cost in each iteration is of order $\mathcal{O}(mNd)$. The results of real runtime (s) per iteration with various sample sizes $N$ in the black-box MTL settings when $d=512$ are shown in the following two tables. The first and the second tables show the experimental results on Office-31 ($m=3$) and Office-home ($m=4$) datasets, respectively. The experimental results are consistent with our analysis, and the real runtime of our method is similar to that of the CMA-ES and ES methods.

\begin{array}{cccc}
    \hline
    \text{Method} & N=20 & N=40 & N=80 \newline
    \hline
    \text{ES} & 5.110 & 9.620 & 18.61\newline
    \text{CMA-ES} & 5.236 & 9.680 & 18.75\newline
    \text{ASMG} & 5.121 & 9.406 & 17.96\newline
    \hline
\end{array}

\begin{array}{cccc}
    \hline
    \text{Method} & N=20 & N=40 & N=80 \newline
    \hline
    \text{ES} & 7.473 &  14.80 & 29.48\newline
    \text{CMA-ES} & 7.647 & 14.60 & 28.88\newline
    \text{ASMG} & 7.553 & 14.56 & 28.67\newline
    \hline
\end{array}