defenses achieve their bounds by leveraging “tricks” associated with
particular model architectures (e.g., instance-based learners [Jia+22a] and
ensembles [LF21a, WLF22a]) as opposed to a detailed analysis of a prediction’s
stability [HL23a]. With limited exception [Jia+19b], today’s influence
estimators do not provide any meaningful guarantee of their accuracy. Rather,
most influence estimates should be viewed as only providing – at best –
guidance on an instance’s “possible influence.” Guaranteed or even
probabilistic bounds on an instance’s influence would enable influence
estimation to be applied in settings where more than a “heuristic
approximation” is required [HL23a].

Improved Scalability: Influence estimation is slow. Analyzing each training
instance’s influence on a single test instance can take several hours or more
[BBD20a, Kob+20a, Guo+21a, HL22a]. For influence estimation to be a practical
tool, it must be at least an order of magnitude faster. Heuristic influence
analysis speed-ups could prove very useful [Guo+21a, Sch+22a]. However, the
consequences (and limitations) of any empirical shortcuts need to be
thoroughly tested, verified, and understood. Similarly, limited existing work
has specialized influence methods to particular model classes [Jia+21b] or
data modalities [Yeh+22a]. While application-agnostic influence estimators are
useful, their flexibility limits their scalability and accuracy. Both of these
performance metrics may significantly improve via increased influence
estimator specialization.

Surrogate Influence and Influence Transferability: An underexplored
opportunity to improve influence analysis lies in the use of surrogate models
[Sha+18c, Jia+19c, Jia+21b, BHL22a]. For example, linear surrogates have
proven quite useful for model explainability [LL17a]. While using only a
model’s linear layer as a surrogate may be “overly reductive” [Yeh+22a], it
remains an open question whether other compact models remain an option. Any
surrogate method must be accompanied by rigorous empirical evaluation to
identify any risks and “blind spots” the surrogate may introduce [Rud19a].

Increased Evaluation Diversity: Influence analysis has the capability to
provide salient insights into why models behave as they do [FZ20a]. As an
example, [BF21a] demonstrate how influence analysis can identify potential
unfairness in an algorithmic decision. However, influence estimation
evaluation is too often superficial and focuses on a very small subset of
possible applications. For instance, most influence estimation evaluation
focuses primarily on contrived data cleaning and mislabeled training data
experiments [Woj+16a, KL17a, Kha+18a, GZ19a, Yeh+18a, Pru+20a, Che+21a,
Ter+21a, KS21a, SWS21a, BHL22a, Yeh+22a, KSH22a, KZ22a]. It is unclear how
these experiments translate into real-world or adversarial settings, with
recent work pointing to generalization fragility [BPF21a, Bae+22a]. We
question whether these data cleaning experiments – where specialized methods
already exist [Kri+16a, KW19a, Wan+19a] – adequately satisfy influence
analysis’s stated promise of providing “understanding [of] black-box
predictions” [KL17a].

Objective Over Subjective Evaluation Criteria: A common trope when evaluating
an influence analysis method is to provide a test example and display training
instances the estimator identified as most similar or dissimilar. These “eye
test” evaluations are generally applied to vision datasets [KL17a, Yeh+18a,
Jia+19b, Pru+20a, FZ20a] and to a limited extent other modalities. Such
experiments are unscientific. They provide limited meaningful insight given
the lack of a ground truth by which to judge the results. Most readers do not
have detailed enough knowledge of a dataset to know whether the selected
instances are especially representative. Rather, there may exist numerous
training instances that are much more similar to the target that the influence
estimator overlooked. Moreover, such visual assessments are known to be
susceptible to confirmation and expectancy biases [Mah77a, NDM13a, KDK13a].

Influence analysis evaluation should focus on experiments that are
quantifiable and verifiable w.r.t. a ground truth.

##  8 Conclusions

While influence analysis has received increased attention in recent years,
significant progress remains to be made. Influence estimation is
computationally expensive and can be prone to inaccuracy. Going forward, fast
certified influence estimators are needed. Nonetheless, despite these
shortcomings, existing applications already demonstrate influence estimation’s
capabilities and promise.

This work reviews numerous methods with different perspectives on – and even
definitions of – training data influence. It would be a mistake to view this
diversity of approaches as a negative. While no single influence analysis
method can be applied to all situations, most use cases should have at least
one method that fits well. An obvious consequence then is the need for
researchers and practitioners to understand the strengths and limitations of
the various methods so as to know which method best fits their individual use
case. This survey is intended to provide that insight from both empirical and
theoretical viewpoints.

## Acknowledgments

This work was supported by a grant from the Air Force Research Laboratory and
the Defense Advanced Research Projects Agency (DARPA) — agreement number
FA8750/̄16/̄C/̄0166, subcontract K001892/̄00/̄S05, as well as a second grant
from DARPA, agreement number HR00112090135.

[sorting=nyt]

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[sorting=ynt]

Training Data Influence Analysis
and Estimation: A Survey

Supplemental Materials

##  Appendix A Nomenclature

Table 1: General nomenclature reference ${[r]}$ |  Set $\\{1,\ldots,r\\}$ for arbitrary positive integer $r$
---|---
${A\stackrel{{\scriptstyle m}}{{\sim}}B}$ |  Set $A$ is a u.a.r. subset of size $m$ from set $B$
$2^{A}$ |  Power set of $A$
${\mathbbm{1}[a]}$ |  Indicator function where ${{\mathbbm{1}[a]}=1}$ if predicate $a$ is true and 0 otherwise
$\vec{0}$ |  Zero vector
$x$ |  Feature vector
$\mathcal{X}$ |  Feature domain where ${\mathcal{X}\subseteq^{d}}$ and ${\forall_{x}\,x\in\mathcal{X}}$
$d$ |  Feature dimension where ${d\coloneqq\lvert x\rvert}$
$y$ |  Dependent/target value, e.g., label
$\mathcal{Y}$ |  Dependent value domain, i.e., ${\forall_{y}\,y\in\mathcal{Y}}$. Generally ${\mathcal{Y}\subseteq}$
$z$ |  Feature vector-dependent value tuple where ${z\coloneqq(x,y)}$
$\mathcal{Z}$ |  Instance domain where ${\mathcal{Z}\coloneqq\mathcal{X}\times\mathcal{Y}}$ and ${\forall_{z}\,z\in\mathcal{Z}}$
$\mathcal{P}$ |  Instance data distribution where ${\mathcal{P}:\mathcal{Z}\rightarrow_{{\geq}0}}$
$\mathcal{D}$ |  Training set where ${\mathcal{D}\coloneqq\\{z_{i}\\}_{i=1}^{n}}$
$n$ |  Size of the training set where ${n\coloneqq\lvert\mathcal{D}\rvert}$
$D$ |  Arbitrary training subset where ${D\subseteq\mathcal{D}}$
$i$ |  Arbitrary training example index where ${i\in{[n]}}$
$z_{\text{te}}$ |  Arbitrary test instance where ${z_{\text{te}}\coloneqq(x_{\text{te}},y_{\text{te}})}$
$f$ |  Model where ${f:\mathcal{X}\rightarrow\mathcal{Y}}$
$\theta$ |  Model parameters where ${\theta\in^{p}}$
$p$ |  Parameter dimension where ${p\coloneqq\lvert\theta\rvert}$
$\ell$ |  Loss function where ${\ell:\mathcal{Y}\times\mathcal{Y}\rightarrow}$
${\mathcal{L}(z;\theta)}$ |  Empirical risk of example ${z=(x,y)}$ w.r.t. $\theta$, where ${{\mathcal{L}(z;\theta)}\coloneqq{\ell({f(x;\theta)},y)}}$
${\mathcal{I}\left(z_{i},z_{\text{te}}\right)}$ |  Exact pointwise influence of training instance $z_{i}$ on test instance $z_{\text{te}}$
${\widehat{\mathcal{I}}\left(z_{i},z_{\text{te}}\right)}$ |  Estimate of training instance $z_{i}$’s pointwise influence on test instance $z_{\text{te}}$
${\mathcal{I}\left(D,z_{\text{te}}\right)}$ |  Group influence of training subset ${D\subseteq\mathcal{D}}$ on test instance $z_{\text{te}}$
${\widehat{\mathcal{I}}\left(D,z_{\text{te}}\right)}$ |  Estimate of the group influence of training subset ${D\subseteq\mathcal{D}}$ on test instance $z_{\text{te}}$
Table 2: Training related nomenclature reference. $T$ |  Number of training iterations
---|---
$t$ |  Training iteration number where ${t\in\\{0,1,\ldots,T\\}}$. ${t=0}$ denotes initial conditions.
$\theta^{(t)}$ |  Model parameters at the end of iteration $t$.
$\theta^{(0)}$ |  Model parameters at the start of training
$\theta^{(T)}$ |  Model parameters at the end of training
$\theta^{*}$ |  Optimal model parameters
$\theta^{(T)}_{D}$ |  Final model parameters trained on training data subset ${D\subseteq\mathcal{D}}$
$\lambda$ |  $L_{2}$ regularization (i.e., weight decay) hyperparameter
$\mathcal{B}^{(t)}$ |  (Mini)batch used during training iteration $t$ where ${\mathcal{B}^{(t)}\subseteq\mathcal{D}}$
$b^{(t)}$ |  Batch size for iteration $t$, where ${b^{(t)}\coloneqq\lvert\mathcal{B}^{(t)}\rvert}$
$\eta^{(t)}$ |  Learning rate at training iteration $t$
$\Theta$ |  Serialized training parameters where ${\Theta\subseteq\\{\theta^{(0)},\ldots,\theta^{(T-1)}\\}}$
$\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t)})}$ |  Training instance $z_{i}$’s risk gradient for iteration $t$
$\nabla_{\theta}^{2}{\mathcal{L}(z_{i};\theta^{(t)})}$ |  Training instance $z_{i}$’s risk Hessian for iteration $t$
$H_{\theta}^{(t)}$ |  Empirical risk Hessian the entire training set where ${H_{\theta}^{(t)}\coloneqq\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}^{2}{\mathcal{L}(z_{i};\theta^{(t)})}}$
${(H_{\theta}^{(t)})^{-1}}$ |  Inverse of the empirical risk Hessian
Table 3: Influence (estimator) specific hyperparameters and nomenclature $\mathcal{D}^{\setminus z_{i}}$ |  Leave-one-out training set where instance $z_{i}$ is held out
---|---
$k$ |  $k$/̄nearest neighbors neighborhood size
${\text{Neigh}(x_{\text{te}};D)}$ |  $k$NN neighborhood for test feature vector $x_{\text{te}}$ from training set ${D\subseteq\mathcal{D}}$
1-2$K$ |  Number of submodels trained by the Downsampling estimator
$D^{k}$ |  Training set used by the $k$/̄th Downsampling submodel
$\theta^{(T)}_{D^{k}}$ |  Final model parameters used by the $k$/̄th Downsampling submodel
$K_{i}$ |  Number of Downsampling submodels trained using $z_{i}$, where ${K_{i}\coloneqq\sum_{k=1}^{K}{\mathbbm{1}[z_{i}\in D^{k}]}}$
$m$ |  Downsampling submodel training-set size where ${\forall_{k}\,\lvert D^{k}\rvert=m<n}$
1-2$\nu$ |  Shapley value characteristic function ${\nu:2^{A}\rightarrow}$ for arbitrary set $A$.
1-2$\epsilon_{i}$ |  Training instance $i$ weight perturbation
$\theta^{(t)}_{+\epsilon_{i}}$ |  Model parameters trained on a training set perturbed by $\epsilon_{i}$
1-2$\alpha_{i}$ |  Training instance $z_{i}$’s representer value, where ${\alpha_{i}\coloneqq-\frac{1}{2\lambda n}\frac{\partial{\ell(\widehat{y}_{i},y_{i})}}{\partial\widehat{y}}}$
$\ddot{\theta}^{(T)}$ |  Model $f$’s final linear layer parameters
$\dot{\theta}^{(T)}$ |  All model $f$’s parameters except the final layer, where ${\dot{\theta}^{(T)}:=\theta^{(T)}\setminus\ddot{\theta}^{(T)}}$
$\mathbf{f}_{i}$ |  Training instance $z_{i}$’s feature representation input into model $f$’s final linear layer
$\mathbf{f}_{\text{te}}$ |  Test instance $z_{\text{te}}$’s feature representation input into model $f$’s final linear layer
$\mathcal{K}$ |  Kernel (similarity) function between two (feature) vectors
1-2$\mathcal{T}$ |  Subset of the training iterations considered by TracInCP, where ${\mathcal{T}\subset{[T]}}$.
1-2$\widetilde{h}^{(t)}_{i}$ |  Training hypergradient where ${\widetilde{h}^{(t)}_{i}\coloneqq\frac{d\theta^{(t)}_{+\epsilon_{i}}}{d\epsilon_{i}}\in^{p}}$

##  Appendix B Influence Analysis Method Definition Reference

Table 4: Influence analysis method formal definitions including equation numbers & citations. Memorization [FZ20a, Pru+20a] |  ${\textsc{Mem}(z_{i})}\coloneqq{\mathcal{I}\left(z_{i},z_{i}\right)}$ (4)
---|---
1-2Cook’s Distance[Coo77a, Eq. (5)] |  ${\mathcal{I}_{\text{Cook}}\mathopen{}\left(z_{i}\right)\mathclose{}}\coloneqq\theta^{(T)}-{\theta^{(T)}_{\mathcal{D}^{\setminus i}}}$ (5)
1-2Leave-One-Out Influence [CW82a] |  ${\mathcal{I}_{\textsc{LOO}}\left(z_{i},z_{\text{te}}\right)}\coloneqq{\mathcal{L}(z_{\text{te}};{\theta^{(T)}_{\mathcal{D}^{\setminus i}}})}-{\mathcal{L}(z_{\text{te}};\theta^{(T)})}$ (7)
1-2Downsampling Influence Estimator [FZ20a] |  ${\widehat{\mathcal{I}}_{\textsc{Down}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\frac{1}{K-K_{i}}\sum_{\begin{subarray}{c}k\\\ z_{i}\notin D^{k}\end{subarray}}{\mathcal{L}(z_{\text{te}};\theta^{(T)}_{D^{k}})}-\frac{1}{K_{i}}\sum_{\begin{subarray}{c}k^{\prime}\\\ z_{i}\in D^{k^{\prime}}\end{subarray}}{\mathcal{L}(z_{\text{te}};\theta^{(T)}_{D^{k^{\prime}}})}$ (8)
1-2Consistency Score [Jia+21c] |  ${C_{\mathcal{D}}(z_{\text{te}})}\coloneqq\mathbb{E}_{m\,\sim\,{[n]}}[-\mathbb{E}_{D\,\stackrel{{\scriptstyle m}}{{\sim}}\,\mathcal{D}}[{\mathcal{L}(z_{\text{te}};\theta^{(T)}_{D})}]]$
1-2Shapley Value Pointwise Influence [Sha53a, GZ19a] |  ${\mathcal{I}_{\text{SV}}\left(z_{i},z_{\text{te}}\right)}\coloneqq c\sum_{D\subseteq\mathcal{D}^{\setminus z_{i}}}\frac{{\mathcal{L}(z_{\text{te}};\theta^{(T)}_{D})}-{\mathcal{L}(z_{\text{te}};\theta^{(T)}_{D\cup z_{i}})}}{\binom{n-1}{\lvert D\rvert}}$ (17)
1-2$k$/̄Nearest Neighbors Shapley Influence [Jia+19b] |  ${\mathcal{I}_{k\text{NN-SV}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\frac{{\mathbbm{1}[y_{n}=y_{\text{te}}]}}{n}+\sum_{j=i}^{n-1}\frac{{\mathbbm{1}[y_{j}=y_{\text{te}}]}-{\mathbbm{1}[y_{j+1}=y_{\text{te}}]}}{k}\frac{\min\\{k,j\\}}{j}$ (21)
1-2Influence Functions Estimator [KL17a] |  ${\widehat{\mathcal{I}}_{\textsc{IF}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\frac{1}{n}\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(T)})}^{\intercal}{(H_{\theta}^{(T)})^{-1}}\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(T)})}$ (27)
1-2Linear Model Representer Point Influence [Yeh+18a] |  ${\mathcal{I}_{\textsc{RP}}\left(z_{i},z_{\text{te}}\right)}=\alpha_{i}x_{i}^{\intercal}x_{\text{te}}\,|_{y_{\text{te}}}$ (40)
1-2Representer Point Influence Estimator [Yeh+18a] |  ${\widehat{\mathcal{I}}_{\textsc{RP}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\alpha_{i}\mathbf{f}_{i}^{\intercal}\mathbf{f}_{\text{te}}\,|_{y_{\text{te}}}$ (41)
1-2TracIn Ideal Pointwise Influence [Pru+20a] |  ${\mathcal{I}_{\text{TracIn{}}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\sum_{\begin{subarray}{c}t\\\ z_{i}=\mathcal{B}^{(t)}\end{subarray}}\left({\mathcal{L}(z_{\text{te}};\theta^{(t-1)})}-{\mathcal{L}(z_{\text{te}};\theta^{(t)})}\right)$ (44)
1-2TracIn Influence Estimator [Pru+20a] |  ${\widehat{\mathcal{I}}_{\text{TracIn{}}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\sum_{\begin{subarray}{c}t\\\ z_{i}\in\mathcal{B}^{(t)}\end{subarray}}\frac{\eta^{(t)}}{\lvert\mathcal{B}^{(t)}\rvert}\,{{\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}}^{\intercal}\,{\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(t-1)})}}}$ (48)
1-2TracInCP Influence Estimator [Pru+20a] |  ${\widehat{\mathcal{I}}_{\text{TracIn{}CP}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\sum_{t\in\mathcal{T}}\eta^{(t)}\,{{\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}}^{\intercal}\,{\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(t-1)})}}}$ (49)
1-2HyDRA Influence Estimator [Che+21a] |  ${\widehat{\mathcal{I}}_{\textsc{HyDRA}}\left(z_{i},z_{\text{te}}\right)}\coloneqq-\frac{1}{n}\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(T)})}^{\intercal}\,\widetilde{h}^{(T)}_{i}$ (55)
1-2$\theta$/̄Relative Influence Estimator [BBD20a] |  ${\widehat{\mathcal{I}}_{\text{RelatIF}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\frac{{\widehat{\mathcal{I}}_{\textsc{IF}}\left(z_{i},z_{\text{te}}\right)}}{\lVert{(H_{\theta}^{(T)})^{-1}}\,\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(T)})}\rVert}$ (59)
1-2GAS Renormalized Influence Estimator [HL22a] |  ${\widehat{\mathcal{I}}_{\textsc{GAS}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\sum_{t\in\mathcal{T}}\eta^{(t)}\,\frac{{{\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}}^{\intercal}\,{\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(t-1)})}}}}{\lVert{\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}}\rVert\,\lVert{\nabla_{\theta}{\mathcal{L}(z_{\text{te}};\theta^{(t-1)})}}\rVert}$ (61)
1-2Renormalized Influence Functions Estimator [HL22a] |  ${\widehat{\mathcal{I}}_{\text{RenormIF}}\left(z_{i},z_{\text{te}}\right)}\coloneqq\frac{{\widehat{\mathcal{I}}_{\textsc{IF}}\left(z_{i},z_{\text{te}}\right)}}{\lVert\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(T)})}\rVert}$ (62)
Table 5: Influence Analysis Method Abbreviations: Related methods are grouped together as in Figure 2. Each method includes its corresponding source reference. LOO |  Leave-One-Out Influence [CW82a, KL17a]
---|---
$k$NN LOO |  $k$/̄Nearest-Neighbors Leave-One-Out [Jia+21b]
LeafRefit |  Decision Forest Leaf Refitting [Sha+18c]
Influence Sketching |  Least Squares Influence Sketching [Woj+16a]
1-2Downsampling |  Downsampled Leave-One-Out [FZ20a]
C/̄score |  Consistency Score [Jia+21c]
Generative Downsampling |  Downsampling for Generative Density Models [vW21a]
1-2SV |  Shapley Value [Sha53a]
TMC/̄Shapley |  Truncated Monte Carlo Shapley [GZ19a]
G/̄Shapley |  Gradient Shapley [GZ19a]
$k$NN Shapley |  $k$/̄Nearest-Neighbors Shapley [Jia+19b]
Beta Shapley |  Beta Distribution-Weighted Shapley Value [KZ22a]
AME |  Average Marginal Effect [Lin+22a]
SHAP |  Shapley Additive Explanations [LL17a]
Neuron Shapley |  Shapley Value-Based Neural Explanations [GZ20a]
1-2IF |  Influence Functions [KL17a]
FastIF |  Fast Influence Functions [Guo+21a]
Scaled IF |  Scaled-Up Influence Functions [Sch+22a]
LeafInfluence |  Decision Forest Leaf Influence [Sha+18c]
Group IF |  Group Influence Functions [Koh+19a]
Second/̄Order IF |  Second-Order Group Influence Functions [BYF20a]
RelatIF |  Relative Influence (Functions) [BBD20a]
Renorm. IF |  Renormalized Influence Functions [HL22a]
1-2RP |  Representer Point [Yeh+18a]
RPS/̄LJE |  Representer Point Based on Local Jacobian Expansion [SWS21a]
TREX |  Tree-Ensemble Representer-Point Explanations [BHL22a]
1-2TracIn |  Traced Gradient Descent Influence [Pru+20a]
TracInCP |  TracIn Checkpoint [Pru+20a]
TracInRP |  TracIn Random Projection [Pru+20a]
TracIn/̄Last |  TracIn Last Layer Only [Pru+20a, Yeh+22a]
VAE/̄TracIn |  Variational Autoencoder TracIn [KC21a]
TracInAD |  TracIn Anomaly Detection [Thi+22a]
TracInWE |  TracIn Word Embeddings [Yeh+22a]
BoostIn |  Boosted (Tree) Influence [BHL22a]
GAS |  Gradient Aggregated Similarity [HL21a, HL22a]
1-2HyDRA |  Hypergradient Data Relevance Analysis [Che+21a]
SGD/̄Influence |  Stochastic Gradient Descent Influence [HNM19a]

##  Appendix C Unrolling Gradient Descent Hypergradients

This section provides a formal derivation of how to unroll HyDRA’s training
hypergradients. We provide this reference for the interested reader to
understand unrolling’s complexity.282828Similar complexity would be required
if TracIn [Pru+20a] were extended to support momentum or adaptive
optimization. Readers do not need to understand this section’s details to
understand how HyDRA relates to other influence analysis methods.

Recall that Eq. (54) does not describe how the exact contents of batch
$\mathcal{B}^{(T)}$ affect unrolling. Eq. (23) defines the effect that
infinitesimally perturbing the weight of training instance $z_{i}$ has on a
model’s empirical risk minimizer. Formally, the perturbed empirical risk is

${\mathcal{L}(\mathcal{D};\theta)}\coloneqq\frac{1}{n}\sum_{z\in\mathcal{D}}{\mathcal{L}(z;\theta)}+\epsilon_{i}{\mathcal{L}(z_{i};\theta)}\text{,}$
(63)

Observe that ${\epsilon_{i}=-\frac{1}{n}}$ is the same as removing instance
$z_{i}$ from the training set.

We now extend this idea to the effect of $\epsilon_{i}$ on a single minibatch.
For any iteration ${t\in{[T]}}$, Formally, batch $\mathcal{B}^{(t)}$’s risk
under a training set perturbation by $\epsilon_{i}$ is

${\mathcal{L}(\mathcal{B}^{(t)};\theta^{(t-1)})}\coloneqq\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}{\mathcal{L}(z;\theta^{(t-1)})}+{\mathbbm{1}[z_{i}\in\mathcal{B}^{(t)}]}\left(\frac{n\epsilon_{i}}{\lvert\mathcal{B}^{(t)}\rvert}\,{\mathcal{L}(z_{i};\theta^{(t-1)})}\right)\text{,}$
(64)

where indicator function ${\mathbbm{1}[z_{i}\in\mathcal{B}^{(t)}]}$ checks
whether instance $z_{i}$ is in batch $\mathcal{B}^{(t)}$. Observe that
${\mathcal{L}(z_{i};\theta^{(t-1)})}$ is scaled by $n$. Without this
multiplicative factor, then when ${\epsilon_{i}=-\frac{1}{n}}$, $z_{i}$’s
effect is not completely removed from the batch, i.e.,

$\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\,{\mathcal{L}(z_{i};\theta^{(t-1)})}-\frac{1}{n\lvert\mathcal{B}^{(t)}\rvert}\,{\mathcal{L}(z_{i};\theta^{(t-1)})}\neq
0\text{.}$ (65)

Eq. (64)’s gradient w.r.t. $\theta$ is

$\nabla_{\theta}{\mathcal{L}(\mathcal{B}^{(t)};\theta^{(t-1)})}=\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}\nabla_{\theta}{\mathcal{L}(z;\theta^{(t-1)})}+{\mathbbm{1}[z_{i}\in\mathcal{B}^{(t)}]}\left(\frac{n\epsilon_{i}}{\lvert\mathcal{B}^{(t)}\rvert}\,\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}\right)\text{.}$
(66)

HyDRA’s hypergradients specify that the derivative is taken w.r.t.
$\epsilon_{i}$. Consider the simpler case first where
${z_{i}\notin\mathcal{B}^{(t)}}$, then

$\displaystyle\frac{d}{d\epsilon_{i}}[\nabla_{\theta}{\mathcal{L}(\mathcal{B}^{(t)};\theta^{(t-1)}_{+\epsilon_{i}})}]$
$\displaystyle=\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}\frac{d}{d\epsilon_{i}}\,\frac{\partial}{\partial\theta}{\mathcal{L}(z;\theta^{(t-1)})}$
(67)
$\displaystyle=\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}\frac{\partial}{\partial\theta^{2}}{\mathcal{L}(z;\theta^{(t-1)})}\frac{d\theta^{(t)}}{d\epsilon_{i}}$
$\triangleright$ Chain rule (68)
$\displaystyle=\frac{1}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}\nabla_{\theta}^{2}{\mathcal{L}(z;\theta^{(t-1)})}\,\widetilde{h}^{(t-1)}_{i}\text{.}$
(69)

Note that ${\nabla_{\theta}^{2}{\mathcal{L}(z;\theta^{(t-1)})}}$ is training
instance $z$’s risk Hessian w.r.t. parameters $\theta^{(t-1)}$.

When instance $z_{i}$ is in $\mathcal{B}^{(t)}$, unrolling hypergradient
$\widetilde{h}^{(t)}_{i}$ requires an additional term. We derive that term
below with similar analysis as when ${z_{i}\notin\mathcal{B}^{(t)}}$ with the
addition of using the product rule. Observe that Eq. (71) below considers both
$z_{i}$’s risk gradient and Hessian.

$\displaystyle\frac{d}{d\epsilon_{i}}[\epsilon_{i}\,\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}]$
$\displaystyle=\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}+\epsilon_{i}\frac{d}{d\epsilon_{i}}\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}$
$\triangleright$ Product rule (70)
$\displaystyle=\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}+\epsilon_{i}\nabla_{\theta}^{2}{\mathcal{L}(z_{i};\theta^{(t-1)})}\widetilde{h}^{(t-1)}_{i}$
$\triangleright$ Chain rule (71)

Combining Eqs. (54), (69), and (71), the hypergradient update rule for vanilla
gradient descent without momentum is

$\displaystyle\widetilde{h}^{(t)}_{i}=\leavevmode\nobreak\ $
$\displaystyle(1-\eta^{(t)}\lambda)\,\widetilde{h}^{(t-1)}_{i}$
$\displaystyle-\frac{\eta^{(t)}}{\lvert\mathcal{B}^{(t)}\rvert}\sum_{z\in\mathcal{B}^{(t)}}\nabla_{\theta}^{2}{\mathcal{L}(z;\theta^{(t-1)})}\,\widetilde{h}^{(t-1)}_{i}$
$\displaystyle-\frac{\eta^{(t)}n}{\lvert\mathcal{B}^{(t)}\rvert}{\mathbbm{1}[z_{i}\in\mathcal{B}^{(t)}]}\left(\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}+\epsilon_{i}\nabla_{\theta}^{2}{\mathcal{L}(z_{i};\theta^{(t-1)})}\,\widetilde{h}^{(t-1)}_{i}\right)\text{.}$
(72)

In [Che+21a]’s [Che+21a] fast approximation of HyDRA, all Hessians (e.g.,
${\nabla_{\theta}^{2}{\mathcal{L}(z;\theta^{(t-1)})}}$) in Eq. (72) are
treated as zeros and the associated terms dropped. The resulting simplified
equation,

$\widetilde{h}^{(t)}_{i}=(1-\eta^{(t)}\lambda)\,\widetilde{h}^{(t-1)}_{i}-\frac{\eta^{(t)}n}{\lvert\mathcal{B}^{(t)}\rvert}{\mathbbm{1}[z_{i}\in\mathcal{B}^{(t)}]}\nabla_{\theta}{\mathcal{L}(z_{i};\theta^{(t-1)})}\text{,}$
(73)

is the basis of the fast-approximation update rule in Alg. 4.