Boundary-Profile Sensitivity in CL

• Boundary-Profile Sensitivity (BPS) is a model-free measure that quantifies how small perturbations in task boundaries affect the induced plasticity and stability profiles in streaming continual learning.
• It operationalizes the robustness of temporal taskifications by leveraging profile distributions computed via adjacent and long-range Wasserstein distances.
• Empirical studies reveal that shorter task windows yield higher BPS values and evaluation instability, emphasizing the need for robust task boundary selection in continual learning benchmarks.

Searching arXiv for the cited paper and closely related continual-learning context. Boundary-Profile Sensitivity (BPS) is a taskification-level, model-free measure in Streaming Continual Learning (CL) that quantifies how sensitive a temporal taskification $\tau$ is to small perturbations of its boundaries. In the formulation introduced in "Temporal Taskification in Streaming Continual Learning: A Source of Evaluation Instability" (Filat et al., 23 Apr 2026), BPS is defined on a continuous, time-ordered stream that has been partitioned into contiguous temporal tasks for the purpose of CL evaluation. The concept addresses the claim that temporal taskification is not a neutral preprocessing step, but a structural component of evaluation: different valid partitions of the same stream can induce different continual learning regimes and therefore different benchmark conclusions. BPS operationalizes the robustness of a given taskification before any CL model is trained, by measuring how strongly slight boundary shifts alter plasticity and stability profiles derived from task-level data distributions (Filat et al., 23 Apr 2026).

1. Streaming continual learning and temporal taskification

Streaming CL considers data that arrive as a time-ordered stream rather than as a pre-given collection of static tasks. The setting is formalized as a single stream

$\{x_t\}_{t=0}^{T-1},$

with a natural temporal order. In this setting, there is no inherent notion of task boundaries in the raw stream; tasks are constructed, not observed. To reuse standard CL protocols, the stream is temporally partitioned into contiguous tasks. A temporal taskification is defined as an ordered partition

$$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$

which induces task intervals

$I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$

Each interval $I_k^\tau$ gives rise to a task-level data distribution $P_k^\tau$, so the taskification $\tau$ is associated with the sequence

$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$

This operation is called temporal taskification (Filat et al., 23 Apr 2026).

The stated reasons for temporal taskification are to reuse standard CL metrics that are defined per task, to define training and evaluation phases, and to create a curriculum-like sequence of tasks from a raw stream. The key claim is that, for the same underlying stream $\{x_t\}$, different valid partitions $\tau$ produce different numbers of tasks $\{x_t\}_{t=0}^{T-1},$0, different adjacent transitions $\{x_t\}_{t=0}^{T-1},$1, and different long-range recurrence structure. Since standard CL metrics are computed over tasks and their order, evaluation depends on $\{x_t\}_{t=0}^{T-1},$2 (Filat et al., 23 Apr 2026).

This framing motivates two structural questions posed prior to training: when two taskifications of the same stream induce meaningfully different CL regimes, and how robust a fixed taskification is to small perturbations of its boundaries. Plasticity profiles, stability profiles, profile distance, and BPS are introduced to answer those questions (Filat et al., 23 Apr 2026).

2. Plasticity and stability profiles

Because the number of tasks varies across taskifications, the framework avoids direct comparison through fixed-length task-indexed vectors. Instead it uses profile distributions that are task-count invariant and model-free. They depend only on the induced $\{x_t\}_{t=0}^{T-1},$3 and a chosen distance $\{x_t\}_{t=0}^{T-1},$4 between task distributions; in the experiments, the distance is the 1-Wasserstein distance (Filat et al., 23 Apr 2026).

The plasticity profile captures local, adjacent changes. For a taskification $\{x_t\}_{t=0}^{T-1},$5 with distributions $\{x_t\}_{t=0}^{T-1},$6, the adjacent-task discrepancies are

$\{x_t\}_{t=0}^{T-1},$7

These are aggregated into an empirical distribution,

$\{x_t\}_{t=0}^{T-1},$8

the plasticity profile. Formally,

$\{x_t\}_{t=0}^{T-1},$9

Large values of $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$0 correspond to abrupt shifts and high plasticity demands, while $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$1 describes how often small versus large adjacent changes occur (Filat et al., 23 Apr 2026).

The stability profile captures longer-range relationships, excluding immediate neighbors by a minimum lag $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$2. It is defined from

$$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$3

with corresponding empirical distribution

$$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$4

Formally,

$$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$5

Small values of $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$6 for distant $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$7 indicate recurrence or long-range stability in the stream, while large values indicate drift (Filat et al., 23 Apr 2026).

Together, the two profiles summarize the induced CL regime before training. Plasticity profile $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$8 describes how hard each boundary is in terms of local shift, and stability profile $$\tau = (t_0,\dots,t_K), \qquad 0=t_0 < t_1 < \dots < t_K = T,$$9 describes how much the stream revisits previous distributions versus drifts away over time (Filat et al., 23 Apr 2026).

3. Profile distance and the definition of BPS

To compare two taskifications $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$0 and $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$1 of the same stream, the framework defines a distance between their plasticity profiles and a distance between their stability profiles: $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$2

$I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$3

where $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$4 is a distribution distance, such as Wasserstein-1 between the one-dimensional empirical distributions over distances. These are combined into an overall profile distance,

$I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$5

with $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$6 scaling coefficients; in the experiments, $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$7 (Filat et al., 23 Apr 2026).

If $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$8 is small, the two taskifications induce similar CL regimes; if large, they represent structurally different regimes. This quantity is task-count invariant and model-free (Filat et al., 23 Apr 2026).

BPS is defined from this profile distance by introducing a boundary neighborhood. For a taskification $I_k^\tau := [t_{k-1}, t_k), \qquad k=1,\dots,K.$9 and a small temporal radius $I_k^\tau$0, the boundary neighborhood

$I_k^\tau$1

is the set of all valid taskifications $I_k^\tau$2 obtained by perturbing each internal boundary $I_k^\tau$3 by at most $I_k^\tau$4 in time, while preserving order and the overall stream boundaries. In the experiments, each internal boundary is randomly shifted by up to $I_k^\tau$5 day, subject to validity constraints (Filat et al., 23 Apr 2026).

The Boundary-Profile Sensitivity of $I_k^\tau$6 at scale $I_k^\tau$7 is the mean profile distance over that boundary neighborhood: $I_k^\tau$8 In practice, the expectation is approximated by Monte Carlo averaging over several random perturbations: $I_k^\tau$9 BPS is thus defined per taskification rather than per individual boundary, although the paper notes that one could in principle inspect per-boundary contributions (Filat et al., 23 Apr 2026).

The interpretation is explicit. Low $P_k^\tau$0 indicates a structurally robust taskification: small boundary shifts leave the plasticity and stability profiles nearly unchanged. High $P_k^\tau$1 indicates a structurally fragile taskification: even slight movement of boundaries significantly changes adjacent-task transitions and/or long-range recurrence structure (Filat et al., 23 Apr 2026).

4. Structural meaning and mechanisms of fragility

BPS is presented as a pre-training diagnostic of the structural instability of the benchmark definition itself, rather than of any particular algorithm. Its significance lies in the fact that it detects benchmark fragility before any CL model is trained (Filat et al., 23 Apr 2026).

Three synthetic case studies are given to illustrate mechanisms by which small boundary moves can produce large profile changes. The first is an abrupt changepoint: $P_k^\tau$2 If a boundary crosses $P_k^\tau$3 under a small shift, the adjacent-task distance can change sharply, so the plasticity profile changes substantially (Filat et al., 23 Apr 2026).

The second is a narrow transient: $P_k^\tau$4 A small boundary shift can move a transient spike from one task to another or split it between tasks, disproportionately changing adjacent distances (Filat et al., 23 Apr 2026).

The third is phase-sensitive recurrence: $P_k^\tau$5 Slightly shifting all boundaries changes which phases of the cycle each task captures, substantially perturbing long-range distances $P_k^\tau$6 and therefore the stability profile (Filat et al., 23 Apr 2026).

These examples establish that BPS is sensitive to abrupt shifts, transient events, and cyclic structure. A plausible implication is that BPS is most informative when task boundaries are externally imposed on streams with latent temporal rhythms, changepoints, or recurrence patterns, since those are precisely the settings in which local perturbations can alter the induced CL regime without changing the underlying data stream.

5. Empirical study in network traffic forecasting

The empirical evaluation is conducted on CESNET-Timeseries24, described as 40 weeks of network traffic from a university ISP, using the 100 highest-density IPs with 10-minute aggregation. The task is multivariate time-series forecasting of avg_duration for each IP based on past multivariate context with 12 features (Filat et al., 23 Apr 2026).

The same model is used for all taskifications: a Transformer-based forecaster with 2 days of context, 32-dim input projection, 8 Transformer blocks, and 4 heads. Temporal taskifications are fixed-length windows of 9, 30, and 44 days, aligned so all lengths are $P_k^\tau$7 for weekday alignment. Within each task, the split is 80% train, 10% validation, and 10% test. The CL methods evaluated are continual finetuning, Experience Replay (ER), Elastic Weight Consolidation (EWC), and Learning without Forgetting (LwF). Metrics are Average MSE ($P_k^\tau$8), Backward transfer (BWT, adapted to regression; $P_k^\tau$9), and Forgetting (Fgt, adapted for regression; $\tau$0). BPS is implemented with Wasserstein-1 as the base distance, $\tau$1, and a boundary neighborhood generated by random perturbation of each boundary by up to $\tau$2 day, followed by Monte Carlo averaging (Filat et al., 23 Apr 2026).

The pairwise Wasserstein distance matrices between tasks show that the 9-day split is noisier, with high-frequency irregular patterns, whereas the 30-day and 44-day splits show smoother, more regular structure. After upsampling matrices to a common resolution, the 30-day and 44-day matrices are more similar to each other than either is to the 9-day matrix. This matches the profile distance results: $\tau$3 is largest, while $\tau$4 is smallest (Filat et al., 23 Apr 2026).

The paper reports plasticity, stability, and BPS as mean $\tau$5 std across 100 IPs. The values explicitly given are summarized below.

Window length	Plasticity	Stability	BPS
9 days	$\tau$6	$\tau$7	$\tau$8
30 days	$\tau$9	$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$0	$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$1
44 days	$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$2	$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$3	$(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$4

The trend is that, as window length increases, all three quantities decrease. BPS is highest for 9-day, intermediate for 30-day, and lowest for 44-day taskifications. When all task boundaries are shifted by two days and the computation is repeated, the ordering persists: 9-day remains most sensitive and 44-day least (Filat et al., 23 Apr 2026).

These findings support the interpretation that short windows induce more local variability and greater sensitivity to exact boundary placement, whereas longer windows smooth out short-lived fluctuations. At the same time, the paper notes that lower BPS does not guarantee best absolute performance: the 44-day split is structurally more robust but also yields higher MSE in many cases (Filat et al., 23 Apr 2026).

6. Relation to evaluation instability in continual learning

The stated motivation for BPS is its connection to evaluation instability in streaming CL benchmarks. When BPS is high, small changes in temporal boundaries lead to large changes in the induced CL regime. The strength and frequency of distribution shifts change, recurrence patterns change, and the difficulty of tasks and the stability-plasticity tradeoff experienced by a learner change. This explains why benchmark conclusions such as average error, forgetting, and backward transfer can vary significantly across valid taskifications of the same stream (Filat et al., 23 Apr 2026).

In the CESNET-Timeseries24 experiments, only the temporal taskification is changed, while the stream, model, data budget, and CL methods remain fixed. Under those conditions, the authors observe large changes in Average forecasting MSE, Forgetting, and Backward transfer. The 30-day split consistently yields lower MSE than the 44-day split for all methods. For some methods, including continual finetuning, the sign and magnitude of BWT change dramatically between splits; the paper gives as an example BWT $(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$5 for the 44-day split versus small positive values for shorter splits (Filat et al., 23 Apr 2026).

This use of BPS is diagnostic rather than causal in the algorithmic sense. It does not show that high BPS mechanically produces poor performance. Instead, it provides a structural explanation for why benchmark conclusions can shift when only the task boundaries are modified. High BPS is associated with noisier task-to-task distribution patterns, larger structural distances, and greater variability in CL metrics across taskifications; low BPS indicates a benchmark-stable regime in which conclusions are less likely to be artifacts of arbitrary temporal segmentation (Filat et al., 23 Apr 2026).

A common misconception would be to treat temporal partitioning as merely a reporting convenience. The framework explicitly rejects that interpretation. Temporal taskification is presented as a first-class evaluation variable because the benchmark itself changes structurally when the partition changes (Filat et al., 23 Apr 2026).

7. Scope, limitations, and methodological implications

BPS has several explicit strengths. It is model-free and pre-training, since it requires no CL training and uses only the stream together with a distance over task distributions. It is task-count invariant because it operates on distributions over pairwise distances rather than on fixed-length task-indexed vectors. It is also presented as domain-relevant in streaming settings such as network traffic, sensor data, and logs, where temporal structure exists but no canonical segmentation is given (Filat et al., 23 Apr 2026).

The limitations are equally explicit. Empirical validation is on a single domain, namely network traffic forecasting with CESNET-Timeseries24. The CL experiments use only four methods: finetuning, ER, EWC, and LwF. The taskification family is restricted to fixed-length temporal windows and local boundary perturbations. BPS is diagnostic, not prescriptive: it does not provide an automatic algorithm to optimize taskification or to design CL methods inherently robust to high-BPS scenarios. Finally, BPS depends on the choice of base distance $(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$6, the profile comparison distance $(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$7, the coefficients $(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$8, and the perturbation scale $(P_1^\tau, P_2^\tau, \dots, P_{K_\tau}^\tau).$9; different choices could alter numerical values and possibly relative comparisons (Filat et al., 23 Apr 2026).

The practical recommendations in the paper follow directly from those observations. Temporal taskification should not be fixed blindly. Where possible, taskifications with lower BPS should be preferred because they are more structurally robust. BPS and profile distances should be treated as cheap pre-training diagnostics, computed for candidate taskifications and reported alongside standard CL metrics. Multiple taskifications should be analyzed to determine how stable method rankings are across them and how structural differences align with performance variability (Filat et al., 23 Apr 2026).

This suggests a broader methodological shift in streaming CL evaluation. Rather than assuming that the learner and the data stream fully determine benchmark outcomes, the framework treats the imposed temporal segmentation as part of the experimental design. On that view, BPS is not a performance metric but a benchmark diagnostic: it measures the fragility of the evaluation protocol itself.