Stable Anisotropic Regularization

$\textbf{Response to weaknesses}$ $\textbf{1. The reasons for contradictions with prior research are unclear, weakening the persuasiveness of the main claim.}$ We agree that more work needs to be done to determine the relationship between isotropy and word similarity tasks. However, most word similarity benchmarks simply calculate the cosine similarity of word embeddings and do not involve any model fine-tuning. Since I-STAR is a regularization algorithm that adjusts isotropy during fine-tuning, I-STAR may not be the most appropriate method to analyze the relationship between word similarity and isotropy. However, a future direction of work could continue the pre-training task of masked language modeling with an added I-STAR penalty to adjust for isotropy. Instead, our paper focuses on the relationship between isotropy and fine-tuning performance, as I-STAR is a regularization algorithm requiring fine-tuning tasks. However, our paper evaluates the relationship between isotropy and a model’s ability to rate the similarity of two sentences using the Semantic Textual Similarity Benchmark (STS-B). STS-B consists of sentence pairs annotated with a similarity score from 1-5. Although a majority of previous works have argued that isotropy is beneficial for LLM performance, there are several recent studies that support the claim that adjusting isotropy in fine-tuning hurts downstream model performance. Namely, Rajaee & Pilehvar 2021 show that adjusting for isotropy by removing principal components hurts model performance. Further, recent work by Rudman et al. 2023 demonstrates that a dimension’s ability to store task-specific knowledge strongly correlates with high variance in a given dimension. $\textbf{2. The claim that the proposed method improves performance on downstream tasks seems a bit overstretched.}$ We agree that the performance increase of fine-tuning with I-STAR is modest. However, the primary aims of our paper are 1) to create a method for measuring isotropy that is differentiable and stable for mini-batch computations and 2) to analyze the relationship between isotropy and fine-tuning performance. In order to clarify the relationship between isotropy and performance, we have performed Pearson correlation analysis on the results of Figure 3. Initial analysis of our Pearson correlation analysis indicates that isotropy is indeed negatively correlated with performance in all cases (except ALBERT on COLA). Note that the small sample size of each entry (num_tuning_params x num_random_seeds = 35) may limit the generalizability of these results and cause some of the p values to be artificially high. Model & QQP & RTE & MRPC & COLA & STS-B BERT & $-0.573 / 0.001$ & $-0.418 / 0.012$ & $-0.115 / 0.511$ & $-0.227 / 0.190$ & $-0.677 / 0.001$ ALBERT & $-0.469 / 0.009$ & $-0.424 / 0.011$ & $-0.184 / 0.330$ & $0.195 / 0.261$ & $-0.624 / 0.004$ DistBERT & $-0.319 / 0.062$ & $-0.840 / 0.003$ & $-0.532 / 0.003$ & $-0.641 / 0.003$ & $-0.713 / 0.005$ $\textbf{Response to questions}$ $\textbf{1. If anisotropy is a natural consequence of SDG, what is the significance of deliberately adding anisotropy as a regularization term? }$ Thank you for a great question! Previous works have argued that anisotropy helps models better generalize to unseen examples and is important in fine-tuning (Zhou et al. 2018). Our work empirically supports this claim by demonstrating a relationship between isotropy and performance. We hypothesize that the significance of adding our anisotropy penalty term helps models learn fine-tuning tasks faster. In order to verify our hypothesis, we are actively running an experiment where we track how quickly models converge when training with no regularization and different \lambda values in I-STAR. $\textbf{2. What is the significance of adjusting the hyperparameter for shrinkage estimation when using IsoScore* as a regularization term?}$ Figure 7 in Section C of the Appendix addresses the significance of adjusting the shrinkage parameter $\zeta$ in I-STAR. In all cases, the optimal shrinkage parameter is in the interval [0.2,0.8]. Figure 7 shows that when $\zeta=0$ and we use (a differentiable version of) IsoScore as a regularization penalty model, performance decreases. Further, if we set $\zeta=0$, the isotropy penalty is computed directly from the shrinkage matrix. Namely, no mini-batch information is included in the loss function. When $\zeta=1$, performance is slightly lower than optimal $\zeta$, but significantly higher than when $\zeta=0$. Results from this experiment demonstrate the need for IsoScore* in order to perform isotropic regularization.

$\textbf{Response to weaknesses}$ $\textbf{1.}$ We will run significance testing for each of the model/task setups and display the significance testing results in the paper's main body. $\textbf{2.}$ In order to clarify the relationship between isotropy and performance, we have performed Pearson correlation analysis on the results of Figure 3. Initial analysis of our Pearson correlation analysis indicates that isotropy is indeed negatively correlated with performance in all cases (except ALBERT on COLA). Note that the small sample size of each entry (num_tuning_params x num_random_seeds = 35) may limit the generalizability of these results and cause some of the p values to be artificially high. Model & QQP & RTE & MRPC & COLA & STS-B BERT & $-0.573 / 0.001$ & $-0.418 / 0.012$ & $-0.115 / 0.511$ & $-0.227 / 0.190$ & $-0.677 / 0.001$ ALBERT & $-0.469 / 0.009$ & $-0.424 / 0.011$ & $-0.184 / 0.330$ & $0.195 / 0.261$ & $-0.624 / 0.004$ DistBERT & $-0.319 / 0.062$ & $-0.840 / 0.003$ & $-0.532 / 0.003$ & $-0.641 / 0.003$ & $-0.713 / 0.005$ $\textbf{Response to questions:}$ $\textbf{1. Pseudo-Code Clarification.}$ Steps 6 & 7 of Algorithm 1 are L2 norms. We will add this to the pseudo-code of the algorithm. $\textbf{2. RDA Clarification.}$ RDA is indeed a Regularized Discriminant Analysis, which we will introduce earlier in the revised version of our paper. $\textbf{3.}$ $\Sigma\_{s}$ $\textbf{computation cost and impact of}$ $\zeta$ Since we only need to collect a sample of 250k tokens, the partial forward pass on the dataset. Computing the isotropy penalty only on \Sigma_S is equivalent to setting $\zeta=1$. The downside to setting $\zeta=1$ is that no isotropy information about the mini-batches of data will be included in the backward pass. Section C of the Appendix shows the results of fine-tuning BERT, ALBERT, and DistilBERT on RTE using different values of \zeta. Figure 7 (Sec C) demonstrates that setting $\zeta=1$ results in a marginal decline in performance compared to incorporating mini-batch isotropy information during gradient descent. $\textbf{ 4. I do not clearly see Fig. 4 as supporting the claim that CosReg only alters the mean of activations.}$ Table 2 in Section F of the Appendix lists the optimal I-STAR hyperparameter values for all tasks and models. Apologies for the lack of clarity. Figure 4 shows the mean value for each dimension in the model. While most of the dimensions are centered around zero, a few dimensions have mean values very far away from the origin. Figure 4 demonstrates that CosReg simply alters the mean of model representations in a single basis dimension instead of altering the isotropy of representations. In each “base” case where no CosReg is performed (shown in green), one basis dimension has mean values far from the origin. Fine-tuning with tuning parameters with CosReg setting $\lambda=1$ has the effect of moving the mean value in that dimension to zero while using CosReg with $\lambda=-1$ pushes that outlier basis dimension even further away from the origin. For example, when fine-tuning with ALBERT on QNLI, the basis dimension with the largest mean changes from an average value of 11.72 (base) to 27.33 when using CosReg -1 and decreases to 2.84 when using CosReg +1. Altering the mean in this single direction with CosReg changes the cosine similarities from 0.003 ($\lambda=-1$) to 0.998 ($\lambda=1$) $\textbf{5. Adding}$ $\pm1 \ \lambda$ We have updated Figure 5 to add \lambda values of +1 and -1. $\textbf{6. Does anisotropy favor a roughly stable internal dimensionality in parameter space as model size increases?}$ That is a very interesting question! Based on your comment, there are two potential experiments to design. First, we can fix the task and fine-tune different model sizes for that task. For example, we can fine-tune BERT-base & BERT-large as well as all of the available ALBERT model sizes from ALBERT-base (12m parameters) to ALBERT-xxl (235M parameters) with different tuning parameter values in I-STAR and calculate the intrinsic dimension of the fine-tuned models. However, given that BERT, DistilBERT, and ALBERT all have a different number of parameters, an interesting follow-up experiment would be to run the experiments in Figure 5 for all of the tasks considered in this paper to verify if there is a desirable intrinsic dimension for model representations for each of the different datasets.

1. In order to clarify the relationship between isotropy and performance, we have performed Pearson correlation analysis on the results of Figure 3. Initial analysis of our Pearson correlation analysis indicates that isotropy is indeed negatively correlated with performance in all cases (except ALBERT on COLA). Note that the small sample size of each entry (num_tuning_params x num_random_seeds = 35) may limit the generalizability of these results and cause some of the p-values to be artificially high. Model & QQP & RTE & MRPC & COLA & STS-B BERT & $-0.573 / 0.001$ & $-0.418 / 0.012$ & $-0.115 / 0.511$ & $-0.227 / 0.190$ & $-0.677 / 0.001$ ALBERT & $-0.469 / 0.009$ & $-0.424 / 0.011$ & $-0.184 / 0.330$ & $0.195 / 0.261$ & $-0.624 / 0.004$ DistBERT & $-0.319 / 0.062$ & $-0.840 / 0.003$ & $-0.532 / 0.003$ & $-0.641 / 0.003$ & $-0.713 / 0.005$ 2. The updated version of our paper will contain the results of Figure 3 for CosReg and Figure 4 for IsoScore*.